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Curtis G. Callan, Jr.,∗Steven B. Giddings,†‡

arXiv:hep-th/9111056v1 28 Nov 1991UCSB-TH-91-54EFI-91-67PUPT-1294hepth@xxx–9111056Evanescent Black HolesCurtis G. Callan, Jr.,∗Steven B. Giddings,†‡Jeffrey A. Harvey,# and Andrew Strominger†♮AbstractA renormalizable theory of quantum gravity coupled to a dilaton and conformal matterin two space-time dimensions is analyzed.The theory is shown to be exactly solvableclassically. Included among the exact classical solutions are configurations describing theformation of a black hole by collapsing matter.

The problem of Hawking radiation andbackreaction of the metric is analyzed to leading order in a 1/N expansion, where N is thenumber of matter fields. The results suggest that the collapsing matter radiates away all ofits energy before an event horizon has a chance to form, and black holes thereby disappearfrom the quantum mechanical spectrum.It is argued that the matter asymptoticallyapproaches a zero-energy “bound state” which can carry global quantum numbers andthat a unitary S-matrix including such states should exist.∗Department of Physics, Princeton University, Princeton, NJ 08544Internet: cgc@pupphy.princeton.edu†Department of Physics, University of California, Santa Barbara, CA 93106‡Internet: giddings@denali.physics.ucsb.edu♮Bitnet: andy@voodoo#Enrico Fermi Institute, University of Chicago, 5640 Ellis Avenue,Chicago, IL 60637 Internet: harvey@curie.uchicago.edu11/91

Following his ground-breaking work [1] on black hole evaporation, Hawking [2] arguedthat the process of formation and subsequent evaporation of a black hole is not governedby the usual laws of quantum mechanics: rather, pure states evolve into mixed states [3].This conjecture is hard to check in detail because of the many degrees of freedom andinherent complexity of the process in four spacetime dimensions. It would be useful tohave a toy model in which greater analytic control is possible.In this paper we investigate such a model.

It is a consistent, renormalizable theoryof quantum gravity in two spacetime dimensions coupled to conformal matter. It containsblack hole solutions as well as Hawking radiation, and is exactly soluble at the classicallevel.

As we shall see, the theory is just complicated enough to enable one to ask theinteresting questions concerning black hole evaporation, yet simple enough to obtain someanswers.We begin with the action in two spacetime dimensionsS = 12πZd2x√−ge−2φ(R + 4(∇φ)2 + 4λ2) −12(∇f)2(1)where g, φ and f are the metric, dilaton, and matter fields, respectively, and λ2 is acosmological constant. This action arises as the effective action describing the radial modesof extremal dilatonic black holes in four or higher dimensions[4,5,6]1; it is also closelyrelated to the spacetime action for c = 1 noncritical strings.

However, these connectionsneed not concern us here; the theory defined by the action (1) is of interest in its own rightas a renormalizable theory of two dimensional “dilaton gravity” coupled to matter.The quantization of related theories of 2D gravity has been considered in [7]. Gravi-tational collapse in related theories has been studied in [8].

The black hole solution of (1)in the absence of matter has appeared previously [9] as a low-energy approximation to anexact solution of string theory.The classical theory described by (1) is most easily analyzed in conformal gaugeg+−= −12e2ρg−−= g++ = 0(2)where x± = (x0 ± x1). The metric equations of motion then reduce toT++ = e−2φ 4∂+ρ∂+φ −2∂2+φ+ 12∂+f∂+f = 0T−−= e−2φ 4∂−ρ∂−φ −2∂2−φ+ 12∂−f∂−f = 0T+−= e−2φ 2∂+∂−φ −4∂+φ∂−φ −λ2e2ρ= 0 .

(3)The dilaton and matter equations are−4∂+∂−φ + 4∂+φ∂−φ + 2∂+∂−ρ + λ2e2ρ = 0(4)1 In the context of superstrings f does not have the usual dilaton coupling because it arisesfrom a Ramond-Ramond field.1

∂+∂−f = 0 . (5)The general solution of the dilaton, matter, and T+−equations (which do not involvef) may be expressed in terms of two free fieldsw = w+(x+) + w−(x−)u = u+(x+) + u−(x−)(6)ase−2φ = u −h+h−e−2ρ = e−w(u −h+h−)(7)whereh±(x±) = λx±Zew± .

(8)The matter equation of course impliesf = f+(x+) + f−(x−) . (9)The remaining constraint equations T++ = T−−= 0 may then be solved for u in terms off± and w±.

The general solution isu± = M2λ −12Zew±Ze−w±∂±f∂±f. (10)where M is an integration constant.We now consider solutions with f = 0, which implies that one can set u = M/λ.Conformal gauge leaves unfixed the conformal subgroup of diffeomorphisms.

This gaugefreedom can be fixed (on-shell) by setting w = 0. The general f = 0 solution is thene−2φ = Mλ −λ2x+x−= e−2ρ(11)up to constant translations of x.

It is readily seen [6] that for M ̸= 0, this correspondsto the r −t plane of the of the higher dimensional black holes of [4,5] near the extremallimit, or to the two dimensional black hole solution of [9], with M the black hole mass. Itis not immediately apparent that the parameter M corresponds to the black hole mass.This can be verified by a calculation of the ADM mass for this configuration as describedin [9] or by a calculation of the Bondi mass as is done later in this paper.

For M = 0one can introduce coordinates in which the metric is flat and the dilaton field φ is linearin the spatial coordinate. This “linear dilaton” vacuum has appeared in previous studiesof lower-dimensional string theories and also corresponds to extremal higher-dimensionalblack holes.2

HorizonDilaton Vacuum+f−wave+−−RLx +0oooo−0−Singularity0xxLinearLII+IRIa2−Figure 1. An incoming f−wave in classical dilaton gravityproduces a black hole metric (shaded region) with a horizon and singularity.From the above we may expect that any matter perturbation of the linear dilatonvacuum will result in the formation of a black hole.

To see that this is indeed the caseconsider the example of an f shock-wave traveling in the x−direction with magnitude adescribed by the stress tensor12∂+f∂+f = aδ(x+ −x+0 ) . (12)One then finds in the gauge w = 0 thate−2ρ = e−2φ = −a(x+ −x+0 )Θ(x+ −x+0 ) −λ2x+x−.

(13)For x+ < x+0 , this is simply the linear dilaton vacuum while for x+ > x+0 it is identicalto a black hole of mass ax+0 λ after shifting x−by a/λ2. The two solutions are are joinedalong the f-wave.

The Penrose diagram for this spacetime is depicted in Figure 1.The fact that any f-wave, no matter how weak, produces a black hole of course impliesthat weak field perturbation theory breaks down. The reason for this is simple.

From (1)it is evident that the weak field expansion parameter is proportional to eφ. Equation (13)3

shows that this parameter becomes arbitrarily large close to I±L or to the singularity andthat the weak field expansion diverges in this region.This has a higher dimensional interpretation as follows [5]. When (1) is taken as aneffective field theory for higher-dimensional dilatonic black holes, the 2D linear dilatonvacuum corresponds to the infinite throat in the extremal black hole solutions.

The centerof the black hole is at x+x−= 0. An arbitrarily small infalling matter wave then producesa non-extremal black hole with an event horizon and a singularity.So far the discussion has been purely classical.As a first step towards includingquantum effects, we now compute the Hawking radiation in the fixed background geometry(11).

This can be computed exactly for the collapsing f-wave because of the elegant relation[10] between Hawking radiation and the trace anomaly for 2D conformal matter coupledto gravity.The calculation and its physical interpretation is clearest in coordinates where themetric is asymptotically constant on I±R . We thus seteλσ+ = λx+e−λσ−= −λx−−aλ.

(14)This preserves the conformal gauge (2) and gives for the new metric−2g+−= e2ρ =([1 + aλeλσ−]−1,if σ+ < σ+0 ;[1 + aλeλ(σ−−σ++σ+0 )]−1if σ+ > σ+0(15)with λx+0 = eλσ+0 . By the standard one-loop anomaly argument, the trace T f+−of thestress tensor is proportional to the curvature scalar which is, in these coordinates, just thelaplacian of ρ.

The result is2⟨T f+−⟩= −112∂+∂−ρ . (16)One can then integrate the equations of conservation of T f to infer the following one-loopcontributions to T f++ and T f−−:⟨T f++⟩= −112∂+ρ∂+ρ −∂2+ρ + t+(σ+),⟨T f−−⟩= −112∂−ρ∂−ρ −∂2−ρ + t−(σ−).

(17)The functions of integration t± must be fixed by boundary conditions. For the collapsingf-wave, T f should vanish identically in the linear dilaton region, and there should be no2 It is assumed that the functional measure for the matter fields is defined with the metric g.One could imagine using instead the (flat) metric e−2φg, in which case there would be no Hawkingradiation.4

incoming radiation along I−R except for the classical f-wave at σ+0 . Using the formula forρ, this impliest+ = 0,t−= −λ24 [1 −(1 + aeλσ−/λ)−2].

(18)The stress tensor is now completely determined, and one can read offits values on I+R bytaking the limit σ+ →∞:⟨T f++⟩→0⟨T f+−⟩→0⟨T f−−⟩→λ248"1 −11 + aeλσ−/λ2#. (19)The limiting value of T f−−is the flux of f-particle energy across I+R.

In the far past ofI+R (σ−→−∞) this flux vanishes exponentially while, as the horizon is approached,it approaches the constant value λ2/48.This is nothing but Hawking radiation.Thesurprising result that the Hawking radiation rate is asymptotically independent of masshas been found in other studies of two-dimensional gravity.The total energy lost by the collapsing f-wave at some value of retarded time σ−canbe estimated by integrating the outgoing flux along I+R up to σ−. If the total radiatedflux is computed by integrating along all of I+R, an infinite answer is obtained, because theoutgoing flux approaches a steady state at late retarded times.

This is obviously nonsense— the black hole cannot lose more mass than it possesses. This nonsensical answer is, ofcourse, a result of the fact that we have neglected the backreaction of the radiation onthe collapsing f-wave.

As a first step toward analyzing the backreaction, it is useful toestimate, to leading order in the mass, the retarded time at which the integrated energy ofthe Hawking radiation on I+R equals the initial mass ax+0 λ of the incoming f-wave. Thisis given bye−λσ−= e−λσ+024.

(20)By this time, Hawking radiation has backscattered all the energy of the incoming f-waveinto outgoing flux on I+R.Unfortunately this picture cannot yet be taken seriously because the turn-aroundpoint at which all the energy has backscattered has coordinates (σ+0 , σ+0 +(log 24)/λ). Thevalue of the dilaton at this point is from (13) for small masse−2φ = 124(21)independent of σ+0 or a.

As we have stated, eφ is the loop expansion parameter for dilatongravity. Since this parameter is not small at the turn-around point, our one-loop calculationof the Hawking flux breaks down before the f-wave fully backscatters.The situation can be remedied by proliferating the number of matter fields.Thisintroduces a new small expansion parameter into the theory: 1/N, where N is the (large)number of matter fields [11].

For N matter fields the Hawking flux is N times as great5

and one finds that the f-wave has completely backscattered by (σ+0 , σ+0 + (log 24/N)/λ).For large N, the value of the dilaton at this point ise−2φ = N24(22)which indeed corresponds to weak coupling. This suggests that, for large N, the essentialphysics of Hawking radiation backreaction takes place in a weak coupling regime and shouldbe amenable to a semiclassical treatment.

In what follows, we will present some proposalsfor the development of such a fully consistent treatment of the scattering problem, alongwith some informed conjectures about the form of the solution.In a systematic expansion in1N , one must include the one-loop matter-induced con-tribution to the gravitational effective action at the same order as the classical action(1).This incorporates both Hawking radiation and backreaction.Because of the waythe dilaton varies with position, there is a region in spacetime where the O(N) one-loopmatter-induced gravitational action is of the same order as the strictly classical part andthe loop coupling constant is nonetheless small. As described above, it is precisely in thisregion where the essential backreaction physics will occur and a semiclassical treatment ofthe proper action should give meaningful answers.

To leading order in 1N , and in conformalgauge, the quantum effective action to be solved isSN = 1πZd2σhe−2φ(−2∂+∂−ρ + 4∂+φ∂−φ −λ2e2ρ)−12NXi=1∂+fi∂−fi + N12∂+ρ∂−ρi. (23)The last term is the Liouville term induced by the N matter fields and the conformalgauge constraints (the T±± equations of (3)) are modified by its presence in a way whichwill shortly be made explicit3.

We have also tuned the coefficient of the possible Liouvillecosmological constant (to be distinguished from the classical “dilatonic” cosmological con-stant λ) to zero. In a slight abuse of terminology, we nevertheless refer to the dynamicsgoverned by the last term in (23) as Liouville gravity.

Solving the quantum theory toleading order in1N is equivalent to solving the classical theory described by SN.Unlike S0, it does not appear possible to solve SN exactly, though it may be possibleto solve the equations numerically. At present the best we can do is make the followingeducated guess about the evolution of an incoming f-wave.

Consider a quantization ofthe system defined on null surfaces Σ(σ−) of constant σ−. The light-cone Hamiltonian P−evolves the system in the direction of increasing σ−.

The charges P± are not separatelyconserved because translation invariance is spontaneously broken. The combination H =P+ + P−generates an unbroken symmetry and is conserved for spacelike surfaces.

In fact3 In a systematic quantum treatment of this action one will find, at subleading order in 1/N,that the N in (23) is shifted by the ghost and gravity measures in order to maintain a net centralcharge c = 26.6

there are in general two conserved quantities, given by boundary terms at the two spatialinfinities. For the null surfaces Σ, the eigenvalue M(σ−) of H is given by a boundary termon I+R (assuming the boundary term on I−L vanishes) and is called the Bondi mass.

TheBondi mass is not conserved because radiation energy can leak out onto I+R.Now consider an initial state at σ−= −∞describing an incoming f1-wave as in (12),with the other N −1 f’s set to zero. In addition it is useful to let this wave be characterizedby the non-anomalous, left-moving global conserved charge Q1L =Rdσ+∂+f1.

Near σ−=−∞, e−2φ is very large and the extra Liouville term may therefore be neglected in thedescription of the incoming state on Σ(−∞), which is essentially described by (13). Asσ−increases away from I−R , M(σ−) will decrease.

From the point of view of the quantumeffective action SN, this is not due to Hawking radiation, but is simply a consequence ofthe extra Liouville term. As σ−→+∞, it is plausible that M(σ−) decreases to zero.However, the state on Σ(σ−) can not revert to the linear dilaton vacuum on I+L becauseit carries the conserved charge Q1L.The picture can thus be summarized as follows.

A state with non-zero charge Q1Land Bondi energy is incoming from I−R . As it evolves it loses its energy, but retains itscharge.

Asymptotically it approaches a zero-energy state with charge Q1L on I+L . This isillustrated in Figure 2.This picture can be corroborated by direct analysis of the Bondi energy associatedwith data on a null surface Σ corresponding to a charged f-wave.

Such data must satisfythe null constraint equations:0 = T++ = e−2φ(4∂+φ∂+ρ −2∂2+φ) + 12∂+f∂+f−N12∂+ρ∂+ρ −∂2+ρ + t+(σ+)0 = T+−= e−2φ(2∂+∂−φ −4∂+φ∂−φ −λ2e2ρ)−N12∂+∂−ρ . (24)The extra function t+ appearing in T++ is in agreement with (17) and is a consequenceof the anomalous transformation law for T++.

t+ is coordinate dependent and must befixed by boundary conditions, as in (18). The linear dilaton configuration remains as thevacuum solution of the full leading N theory:ρ = 0fi = 0φ = −λ2 (σ+ −σ−).

(25)The Bondi energy may then be defined for configurations which approach (25) on I+R (i.e.the configuration must not only be asymptotically flat, but presented in an asymptoticallyMinkowskian coordinate system). It is given by the surface term which must be addedto the integral of T++ + T+−over Σ to obtain the generator of time translations.

This7

+f−wave+−−RLx +0oooo−0−0xxLII+IRIDilatonLiouville RegionRegionFigure 2. An incoming f−wave in quantum dilaton gravityeventually propagates into the region dominated byLiouville gravity, for which the curvature is constrained to vanish and all excitations have zero energy.canonical procedure yieldsM(σ−) = 2eλ(σ+−σ−)(λδρ + ∂+δφ −∂−δφ)+ N12(∂−δρ −∂+δρ)(26)where δρ and δφ, are the asymptotically vanishing deviations of ρ and φ from (25), andthe right hand side is to be evaluated on I+R.

The first “dilaton” term was obtained inreference [9]. The term proportional to N, arising from matter quantum effects, actuallyvanishes due to the boundary conditions (25).

A modified formula is required in coordinatesystems (such as in (11)) for which the fields do not asymptotically approach (25).Let us first consider the energy, evaluated on a surface Σ, of a small amplitude f1-wavepacket localized in the ‘dilaton region’ where e−2φ is very large, i.e. at very large σ+ −σ−.8

Then the Liouville terms proportional to N may be neglected in solving the constraints.M will be given as before by the integrated value of 12∂+f1∂+f1 times the x+ coordinateof the center of the wave packet.Now, however, consider the case where the f1 wave-packet is localized on Σ in the‘Liouville region’ where e−2φ is very small. The dilaton gravity term is then very small, andthe action governing ρ and f 4 reduces to Liouville gravity coupled to conformal matter:SN(large φ) = 1πZd2σ N12∂+ρ∂−ρ −12NXi=1∂+fi∂−fi!

(27)with constraints0 = T++ = 12NXi=1∂+fi∂+fi−N12∂+ρ∂+ρ −∂2+ρ + t+(σ+)0 = T+−= −N12∂+∂−ρ . (28)The T+−constraint implies that the spacetime is in fact flat.

The Bondi energy of (26)reduces to its Liouville pieceM(σ−) = N12(∂−δρ −∂+δρ) . (29)Since there is no invariant one can associate to a flat metric one would expect this expres-sion to vanish.

That it does can be seen from direct evaluation of (29): if ρ approaches zeroon I+R as required by the boundary conditions (25), the derivatives of ρ and consequentlyM must also vanish. Thus all asymptotically flat states of Liouville gravity plus matterhave zero energy.We now have a plausible global picture of the scattering process.

The linear dilatonvacuum is divided into two regions characterized by e−2φ large or small compared to N12.This dividing line is timelike. For e−2φ ≫N12, the dynamics are essentially that of classicaldilaton gravity coupled to matter.

For e−2φ ≪N12, one has Liouville gravity coupled tomatter. An incoming f1 wave-packet on I−R begins in the dilaton gravity region where ithas non-zero Bondi energy.

However, it eventually crosses into the Liouville region, whereall excitations have zero energy. By energy conservation, all of the initial energy of thewavepacket must have radiated away to I+R.

There is no indication of an event horizonor singularity5: in the region where the singularity occurs in the classical solution, the4 The dynamics of φ are roughly governed by the free field ψ = e−φ. However it is not clearwhat range should be taken for ψ.5 In two dimensions, unlike in higher dimensions, we know of no local notion of an appar-ent horizon.

Global event horizons exist as usual when the spacetime is singular or otherwiseincomplete.9

quantum dynamics are governed by Liouville gravity (with no cosmological constant) inwhich the curvature is required to vanish.One expects, therefore, a unitary S-matrixevolving from I−to I+. One would hope to extract information about this S-matrix froma semiclassical treatment of the large-N action (23).While we find this picture compelling, we emphasize that it must at present be re-garded as speculative.We have not shown that an incoming f-wave does not in factproduce a singularity, or even that the large N equations of motion give a well-definedevolution.

One might try to substantiate our speculations by doing weak field perturba-tion theory in the amplitude of the f-wave. However preliminary calculations indicate thatweak field perturbation theory breaks down near the boundary of the dilaton and Liouvilleregions: the second-order perturbation is divergent.

Thus in order to settle the question anon-perturbative analysis of the large N theory (23) is probably needed6.In conclusion we have analyzed the process of black hole formation and evaporation,including backreaction, in the 1/N expansion of a two-dimensional model. A set of equa-tions describing the process were found, but have so far not been solved.

A qualitativeanalysis suggests that in this model would-be black holes in fact evaporate before an eventhorizon or singularity has a chance to form. Thus there is no indication that pure statesevolve into mixed states.

The implications of our results for four-dimensional black holesremain to be explored.AcknowledgementsWe are grateful to S. Hawking, G. Horowitz, J. Preskill and R. Wald for useful dis-cussions. After completion of this work we learned that some aspects of this theory havebeen considered by E. and H. Verlinde.This work was supported in part by DOE grant DE-AC02-84-1553, DOE grant DE-AT03-76ER70023, and NSF grant PHY90-00386.

S.B.G. also acknowledges the supportof NSF PYI grant PHY-9157463 and J.A.H.

acknowledges the support of NSF PYI grantPHY-9196117.6 Of course a singularity at large N does not imply a singularity of the full quantum theorysince the 1/N expansion breaks down as soon as fields grow to order N.10

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