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July 1992; Revised October 1992

arXiv:hep-th/9207095v2 14 Oct 1992COLO-HEP-288hepth@xxx/9207095July 1992; Revised October 1992Quantum Black Holes in Two DimensionsS.P. de Alwis⋆Dept.

of Physics, Box 390,University of Colorado,Boulder, CO 80309ABSTRACTWe show that a whole class of quantum actions for dilaton-gravity, which reduceto the CGHS theory in the classical limit, can be written as a Liouville-like theory.In a sub-class of this, the field space singularity observed by several authors isabsent, regardless of the number of matter fields, and in addition it is such thatthe dilaton-gravity functional integration range (the real line) transforms into itselffor the Liouville theory fields.We also discuss some problems associated withthe usual calculation of Hawking radiation, which stem from the neglect of backreaction. We give an alternative argument incorporating back reaction but findthat the rate is still asymptotically constant.

The latter is due to the fact thatthe quantum theory does not seem to have a lower bound in energy and Hawkingradiation takes positive Bondi (or ADM) mass solutions to arbitrarily large negativemass.⋆dealwis@gopika.colorado.edu

1. IntroductionThe theory of dilaton gravity coupled to scalar fields proposed by Callan et al,[1] (CGHS) has generated a flurry of activity on black hole physics.

What one hasis a simple toy model, within which the puzzling questions associated with Hawkingradiation [2] can be addressed in a systematic way. In the original work of CGHS aswell as in several subsequent papers, it was assumed that quantum effects to leadingorder could be included by just adding a piece to the action which reproduced theconformal anomaly.

However it was later realized that the consistent quantizationof the theory in conformal gauge, required that the cosmological constant termand/or the kinetic terms should get renormalized in a dilaton dependent manner,so that the theory becomes a conformal field theory (cft)[3, 4]. This requirementthat the theory be an exact cft (though not necessarily a soluble one) is not amatter of choice.

It is a necessary consequence of general covariance. In otherwords dilaton gravity coupled to matter fields must be a cft in exactly the sameway that string theory (i.e ordinary 2d gravity coupled to matter fields) is a cft.In this paper we will first review this argument and then consider the general-ization of previous solutions to the conformal invariance conditions.

We show thatthere is a subset of models which are free of the quantum black hole singularitypointed out in [5, 6], and which are such that the original range of integration forthe conformal factor and the dilaton, is transformed into itself for the Liouville the-ory fields. We will also argue that the calculations of Hawking radiation that havebeen given in the literature, are inconsistent with the constraints and equations ofmotion of the theory since they neglect back reaction.

There is no sensible approx-imation scheme in which the latter can be ignored. We then show that when theexact solution of the system of equations coming from quantum corrected actionis considered, the results differ from previous calculations.

However it turns outthat one cannot see the radiation turning offin this theory, the (Bondi) mass ofthe solutions of the theory can be arbitrarily negative, and the Hawking processcauses a positive mass solution to decay indefinitely to infinitely negative mass.2

Altough Liouvile theory has a positive definite spectrum the same is not true ofthe Liouville-like theory that is obtained from the CGHS theory. It is possible thatthe origin of the problem lies in the the CGHS theory itself, but a more rigorousquantum treatment of the Bondi mass may resolve this question.In the next section we review the quantization of the CGHS theory.

In thethird section we discuss a class of solutions to the integrability conditions for theconstraints and present arguments for taking the resulting exact conformal fieldtheory as a quantum theory of dilaton gravity. In the fourth section we demonstrateexplicitly how the classical singularities are tamed by quantum effects.

In the fifthsection we review the CGHS calculation of Hawking radiation in this model. Inthe sixth section we give an alternative calculation which is consistent with theconstraints (this is basically a detailed version of a calculation contained in thesecond paper of [3]) and in the final section we make some concluding remarks.2.

QuantizationThe CGHS theory is defined by the classical actionS = 14πZd2σ√−g[e−2φ(R + 4(∇φ)2 + 4λ2) −12NXi=1(∇fi)2]. (2.1)In the above G is the 2d metric, R is its curvature scalar, φ is the dilaton andthe fi are N scalar matter fields.

This action may be obtained as a low energyeffective action from string theory, in which case the f fields will arise from theRamond-Ramond sector. Note that the (zero mass) tachyon of 2d string theory isexcluded from this action.

If this field had been coupled then the theory wouldnot be solvable even at the classical level.⋆The quantum field theory of this classical action may be defined as⋆For a discussion of how in this case, 2d black hole solutions are affected far away from theblack hole by the presence of the tachyon, see [7].3

Z =Z [dg]g[dφ]g[df]g[V ol. Diff.] eiS[g,φ,f].

(2.2)The metrics which define these measures are usually given by,||δg||2g =Zd2σ√−ggαγgβδ(δgαβδgγδ + δgαγδgβδ)||δφ||2g =Zd2σ√−gδφ2,||δf||2g =Zd2σ√−gδijδfiδfj. (2.3)However we can be more general in these definitions as long as 2d diffeomorphisminvariance is preserved.

Now let us gauge fix to the conformal gauge g = e2ρˆg andrewrite the measures with respect to the fiducial metric ˆg. Following the work ofDavid and of Distler and Kawai [8], we may expect the action to get renormalized,except that unlike in their case the renormalization will be dilaton dependent (sincethe coupling is e2φ).

Thus in general we may expect the gauge fixed path integralto be written as [9,7,3]†,Z =Z[dXµ]ˆg[df]ˆg([db][dc])ˆgeiI(X,ˆg)+iS(f,ˆg)+iS(b,c,ˆg),(2.4)whereI[X, ˆg] = −14πZ p−ˆg[12ˆgabGµν∂aXµ∂bXν + ˆRΦ(X) + T(X)]. (2.5)S(b, c, ˆg) is the Fadeev-Popov ghost action, and we have written (φ, ρ) = Xµ.

Notethat all the measures in (2.4) are defined with respect to the 2d metric ˆg and thatin particular the measure [dXµ] is derived from the natural metric on the space||δXµ||2 =Rd2σ√−ˆgGµνδXµδXν. In the limit of weak coupling (e2φ << 1) wehave,† For alternative approaches to the quantization see [10].4

I →14πZd2σp−ˆg[e−2φ(4( ˆ∇φ)2 −4 ˆ∇φ. ˆ∇ρ) −κ ˆ∇ρ.

ˆ∇ρ+ ˆR(e−2φ −κρ) −4λ2e2(ρ−φ)](2.6)This is obtained from (2.1) by putting g = e2ρˆg, and including a very specifichigher order term; namely the usual conformal anomaly term. κ in the above isequal to 26−(N+2)6= 24−N6, if one includes the contribution of the transformationof the measure for φ and ρ.‡I is a generalized sigma model action and we have kept only renormalizableterms.The sigma model action introduces three (dilaton dependent) couplingfunctions G, Φ, and T, respectively the field space metric, dilaton, and tachyon.The only a priori restriction arises from the fact that the functional integral for Z in(2.4), must be independent of the fiducial metric ˆg, as is obvious from the expression(2.2) for it.

This implies that the following constraints should be satisfied:< T±± + t±± >= 0,(2.7)and< T+−+ t+−>= 0,(2.8)where Tµν is the stress tensor for the dilaton-gravity and matter sectors, and tµνis the stress tensor for the ghost sector. ((2.8) is equivalent to the equation ofmotion for ρ, and so is not an additional constraint).Furthermore one has tosatisfy the integrability conditions for these constraints, namely that they generatea Virasoro algebra with zero central charge.§ As is well known (see for instance [11]and references therein) this is equivalent to the requirement that the β-functions[12] corresponding to the coupling functions, G, Φ, and T, vanish.‡ We will justify this in more detail later on.§ In effect this means that the field space must be exactly like the target space of string theory,though here we do not give this space a space-time interpretation.

The only space-time inthe theory is the original one parametrized by the coordinates σ.5

βµν = Rµν + 2∇Gµ ∂νΦ −∂µT∂νT + . .

. ,βΦ = −R + 4Gµν∂µΦ∂νΦ −4∇2GΦ + (N + 2) −263+ Gµν∂µT∂νT −2T 2 + .

. .

,βT = −2∇2GT + 4Gµν∂µΦ∂νT −4T + . .

. ,(2.9)where R is the curvature of the metric G. These equations have to be solvedunder the boundary conditions that in the weak coupling limit (e2φ << 1) we get,comparing (2.5) with (2.6),Gφφ = −8e−2φ,Gφρ = 4e−2φ,Gρρ = 2κ,(2.10)Φ = −e−2φ + κρ,T = −4λ2e2(ρ−φ).(2.11)3.

From CGHS to LiouvilleLet us first discuss the renormalization of the field space metric and dilaton(G and Φ) and postpone the discussion of the tachyon T. The (renormalized) fieldspace metric may be parametrized as,ds2 = −8e−2φ(1 + h(φ))dφ2 + 8e−2φ(1 + h(φ))dρdφ + 2κ(1 + h)dρ2,(3.1)where h, h, and h are O(e2φ). If we are going to consider only O(e2φ) effects thenwe should certainly set h to zero.

But even if we consider the renormalizationfunctions h and h to all orders, it is consistent to limit ourselves to the class ofquantum versions of the CGHS theory which have h = 0, provided that we satisfythe beta function equations. This corresponds to confining ourselves to theories in6

which the field space curvature R = 0. In this case we can transform this metricto Minkowski form.

First put¶y = ρ −κ−1e−2φ + 2κZdφe−2φh(φ). (3.2)Then the metric becomesds2 = −8κP 2(φ)dφ2 + 2κdy2,whereP(φ) = e−2φ[(1 + h)2 + κe2φ(1 + h)]12.

(3.3)Puttingx =ZdφP(φ),(3.4)we haveds2 = −8κdx2 + 2κdy2(3.5)With this form of the metric, ignoring O(T 2) terms, we find from the first(graviton) β-function equation in (2.9), that ∂µ∂νΦ = 0.In other words Φ islinear in x, y. Demanding that we recover the CGHS Φ given in (2.11) in the weakcoupling limit we find the unique solution,¶ We will only consider theories with κ ̸= 0 i.e.

N ̸= 24.7

Φ = κy. (3.6)Substituting in the second (dilaton) equation in (2.9), we then getκ = 24 −N6.

(3.7)To determine T we consider the third equation of (2.9), to linear order and get,κ4∂2xT −1k∂2yT + 2∂yT −4T = 0. (3.8)This has solutions of the form T = eβx+αy, where κ4β2 −1kα2 + 2α −4 = 0.Now we need to impose the boundary condition that we recover the CGHS tachyongiven in (2.11) in the weak coupling limit.

To do so we expand the expression forx ((3.4), (3.3)) to getx ≃−12e−2φ +Zdφe−2φh + κ2φ + O(e2φ). (3.9)Then we find that −4κx + 2y = 2ρ −2φ + O(e2φ) so that the unique solution (con-fining ourselves to multiplicative renormalizations) obeying the required boundarycondition isT = −4λ2e−4κx+2y.

(3.10)For κ > 0 there is another (additive) term⋆satisfying the boundary condition.NamelyTnp = µe4√κx ≃µ exp(−2√κe−2φ).This is in fact a non-perturbative ambiguity. We will set µ = 0 in the rest of⋆I wish to thank Andy Strominger for pointing this out to me [13].8

the paper. In any case it is absent for κ < 0, since in that case we will have anoscillatory solution which will not vannish in the classical limit.It is convenient now to introduce rescaled fields,X = 2s2|κ|x,Y =p2|κ|y,(3.11)in terms of which the metric and the tachyon become,ds2 = ∓dX2 ± dY 2T = −4λ2e∓q2|κ|(X∓Y )In the above and in the equations below, upper/lower signs correspond to havingκ > 0/κ < 0 respectively.

In terms of the new field variables the functional integralbecomes,Z =Z[dX][dY ][df][db][dc]eiS[X,Y,f]+iSghost,(3.12)where,S = 14πZd2σ[∓∂+X∂−X ±∂+Y ∂−Y +Xi∂+fi∂−fi + 2λ2e∓q2|κ|(X∓Y )]. (3.13)Several comments need to be made about this functional integral.

First andmost obviously there is the question of the range of the integration. As we see from(3.4) and (3.3), in general the range of integration in X will not extend over thewhole real line.

What we then have is an approximate solution to the β-functionequations (2.9) valid only to leading order in the sigma model (α′) expansion and toleading order in the weak field expansion in T. On the other hand if we define the9

quantum theory by (3.12) with the range of integration for X being the whole realline, we have† a solution to the exact β-function equations. Thus this definition ofquantum dilaton-gravity theory, even if somewhat unorthodox, is a very compellingone.

It is on the same footing as for instance the definition of 2 + 1 dimensionalquantum gravity given by Witten[14] in which the functional integral is taken overall values of the vielbein field. Also let us point out that if we restrict the rangeof integration to be consistent with the original definition of the quantum theorythen, since we only have a solution to the leading order β-function equations, itseems as if we will need an infinite number of terms to satisfy the exact conformalinvariance conditions.

It is plausible to suppose that this theory is equivalent tothe one above with the unrestricted range of integration. This argument is alsoreinforced by the fact that, as in the usual Liouville theory, the integration rangeis effectively cut off(albeit softly) by the Liouville potential term.

Finally (andperhaps this is the most compelling reason for the quantum Liouville-like conformalfield theory) there exist choices of h and ¯h, for which when the integration rangesfor φ and ρ are as usual taken over the whole real line, the same is true for theranges for X and Y (see case d at the end of this section).The second comment is with regard to the approximation in which the dilatonand graviton loops can be ignored. By rescaling and translating the fields X, Y it iseasily seen that ¯h = κ so that the semiclassical approximation is valid only for largeκ.

Thus one might be inclined to believe that any (even qualitative) conclusionsderived for the N < 24 theory[3] are drastically effected by dilaton graviton loopcorrections. On the other hand for N = 1 we get κ = 3.8 which is of the sameorder as the relevant parameter in QCD where the approximation works quite well.Finally we comment on the different possibilities for the functions h and h.Three special cases have so far been discussed in the literature.a) h = h = 0. i.e.

the field-space metric of the classical CGHS Lagrangian† The theory is very much like Liouville theory which is an exact cft [15]. In fact it is lesssingular than Liouville.

So one expects it to be an exact cft as well.10

is not renormalized.However in this case the cosmological constant term T isrenormalized.b)h = −e2φ, h = −2e2φ. This is the case proposed by Strominger [16].

In thiscase both the metric G and the tachyon T are renormalized.c) h = 0, h = −κ4e4φ. This is the case considered in [17] where P 2 is a perfectsquare (see (3.3) from which we find P = e−2φ(1 + κ4e2φ)).

In this case the metricis (obviously) renormalized but the tachyon is not (as is easily seen from (3.10)and the expressions for x and y with the above value of P).d) In all of the above cases the transformation (3.4) has a singularity whenκ < 0. For instance in case a) it is at e2φ = −κ−1.

It is however quite easy to finda class of models which have no such singularity. Put h = ae2φ and h = be2φ.

Thenputting e2φ = z the condition for the absence of a singularity is that the quadraticequation z2P 2 = (a2 + κb)z2 + (2a + κ)z + 1 = 0 has no real roots. i.e.

we mustchoose κ2 + 4(a −b)κ < 0. Obviously there are many solutions to these conditionsbut one particular class is of particular importance since members of it naturallyallow the range of integration in the X, Y variables to go over the whole real line.The simplest member of this class has h = 0 and ¯h = −κ2e2φ.

In this case we havefrom (3.3),(3.4),(3.2),x =Zdφe−2φ(1 + κ24 e4φ)12= −12(κ24 + e−4φ)12 + |κ|4 sinh−1|κ|2 e2φ,(3.14)andy = ρ −κ−1e−2φ −φ. (3.15)Clearly as φ, ρ, range from −∞to +∞so do x and y.11

4. Exact solutionsThe equations of motion coming from (3.13) are as follows.⋆∂+∂−f = 0,(4.1)∂+∂−X =λ2s2|κ|eq2|κ|(X+Y ),∂+∂−Y = −λ2s2|κ|eq2|κ|(X+Y ).

(4.2)We have taken the case with the lower signs in (3.13) so that the discussion isfor N > 24. There is no qualitative difference in the other case so it is unnecessaryto write it out explicitly.† These equations are easily solved.

From (4.2) we have∂+∂−(X+Y ) = 0, so that X+Y =q|κ|2 (g+(σ+)+g−(σ−)), where g± are arbitrarychiral functions. Substituting into the X equation of motion and integrating wehaveX = −s2|κ|(u+(σ+) + u−(σ−)) + λ2s2|κ|σ+Zdσ+eg+(σ+)σ−Zdσ−eg−(σ−)= −Y +r|κ|2 (g+ + g−),(4.3)where u± are arbitrary chiral functions to be determined by the boundary condi-tions.By a coordinate choice we can set g± = 0.

In these coordinates (the analog ofKruskal-Szekeres coordinates for the black hole) we get⋆In this section and in section 6, wherever it is appropriate, all equations are to be understoodas being valid inside the functional integral, i.e. as expectation values of quantum operators.Since following the arguments of reference [15] the theory can be mapped into a free theoryit is plausible that the only quantum effects come from normal ordering.† It is also contained in [4] and the second paper of [3].12

X = −Y = −s2|κ|(u −λ2σ+σ−). (4.4)where u = u+ + u−.These solutions are of course the same as those of CGHS, except that they arefor X and Y and all the effects of the quantum anomalies are now incorporated inthe expressions for them in terms of ρ and φ.

To be explicit consider the case d)discussed at the end of the last section (h = 0, ¯h = −κ2e2φ);X =2s2|κ|Zdφe−2φ[1 + κ24 e4φ]12=p2|κ|Zdφ[1 + 4κ2e−4φ]12,andY =p2|κ|ρ +s2|κ|e−2φ −p2|κ|φ.In the weak coupling limit (e2φ << 1) we have from (4.4) the classical solutione−2φ = e−2ρ = u −λ2σ+σ−,(4.5)which exhibits the classical (black hole type) singularity on the curve wherethe right hand side vanishes. But the singularity is in the strong coupling regionwhere we have to use the strong coupling expansion (from the second line of theabove equation for X)X ≃p2|κ|[φ −e−4φκ2+ .

. .

].Then we have from (4.4),13

φ ≃κ−1(u −λ2σ+σ−),andρ ≃1κe−2κ−1(u−λ2σ+σ−).The metric (e2ρ) is clearly non-singular at the classical singularity.Differentiating the solution for X with respect to σ± we get2e−2φ∂±φ = −(∂±u± −λ2σ∓)¯P(φ),(4.6)where ¯P = e2φP, P being defined by (3.3). This equation gives the trajectoryof the apparent horizon (∂+φ = 0) introduced in [6] (once the unknown function uis determined) asσ−= 1λ2∂+u+(σ+).

(4.7)By differentiating the solution for Y and using (4.6) and the expression for Yin terms of ρ, φ, we haveκ∂−∂+ρ = (1 + ¯h)∂−¯h(∂+u+ −λ2σ−) ¯P −1 −(1 + ¯h)(∂+u+ −λ2σ−)∂−¯P ¯P −2+ λ21 −1 + ¯h¯P. (4.8)From this expression the curvature R = 8e−2ρ∂+∂+ρ is easily seen to be non-singular at the classical singularity in all the cases a) to d) discussed at the endof section 3 (as is obvious from the fact that the metric is non-singular there) andfurthermore in case d), it is seen that there are no curvature singularities anywherefor either sign of κ.14

5. Problems in Calculating Hawking RadiationBefore we calculate Hawking radiation we would like to comment on previouscalculations of this phenomenon in 2d dilaton gravity.

These comments may havea bearing on the original calculation [2] in 4d as well.In the CGHS calculation [1], the stress tensor anomaly is added to the classicalstress tensor trace to giveT−+ = e−2φ(2∂+∂−φ −4∂+φ∂−φ) −λ2e2ρ−2φ −N6 ∂+∂−ρ.From the conservation equation for the stress tensor the remaining componentsof the stress tensor are then determined to beT±± = e−2φ(4∂±ρ∂±φ −2∂2±φ) + T f±±,with the quantum (one loop) part of the stress tensor being given byT f±± = −N6 (∂±ρ∂±ρ −∂2±ρ + t±(σ±)),where t± are arbitrary chiral functions to be determined by the boundaryconditions. Of course in a consistent quantization ghosts have to be included andN →N −24 [16,3,4] and t± must be related to the ghost stress tensor [3, 4] butwe will ignore this for the moment.

The usual argument then goes as follows. Toleading order, Hawking radiation may be computed by substituting the classicalsolution (corresponding to the formation of a blackhole due to an incoming mattershock wave along σ+ = σ+0 ) into the quantum piece of the stress tensor T f, andthen imposing boundary conditions.

In terms of the asymptotically Minkowski15

coordinates ¯σ+ = 1λ log(λσ+), ¯σ−= −1λ log(−λσ−−aλ), the classical solution is2ρ = −log(1 + aλeλσ−),σ+ < σ+0 ,2ρ = −log(1 + aλeλ(σ−−σ++σ+0 )),σ+ > σ+0 .Substituting this in T f−−and demanding that the latter vanishes for σ+ < σ+0one determines t−(σ−) = −λ24 (1 −1(1+ aλeλσ−)2). Then observing that ∂ρ, ∂2ρ →0when σ+ →∞( I+R in the Penrose diagram) we haveT f−−→N24λ2(1 −1(1 + aλeλσ−)2.This determines the Hawking radiation rate at time like future infinity to beN24λ2 in agreement with earlier calculations (see for instance [18]).This calculation however neglects back reaction.

This is of course true for allprevious calculations of Hawking radiation. In the original calculations [2,18] onequantized in a fixed background metric which means that back reaction is ignored.But there is no sensible approximation in which back reaction can be ignored.

Backreaction is of the same order as the radiation! Within the context of this toy modeland our explicit solution of it, this problem can be resolved.

But before we do itlet us elaborate on this question further.The point is that the one loop (matter) corrected theory has an action (equation(23) of [1] ) and associated equations of motion and constraints. Aside from thedilaton equation, these correspond to (2.8) and (2.7) and read in this notation,T−+ = T cl−+ + T f−+ = 0,T∓∓= T cl∓∓+ T f∓∓= 0.

(5.1)One has to now find a consistent solution to this set of equations (and the φequation of motion). Such a solution will have a classical piece plus a one loop16

quantum correction. Now in calculating T f to order ¯h it is sufficient to substitutethe classical part of the solution into it.

But to the same order one should keepthe result of substituting the O(¯h) correction to the classical solution into T cl.One should not just keep the former as Hawking radiation and ignore the latter.In fact the classical solution, by definition, satisfies the classical equations T cl−+ =0, T cl±± = 0, so that in order to satisfy (5.1), the leading quantum correction tothe classical solution when substituted into T cl must give a value which exactlycancels the value obtained by substituting the classical solution into T f.TheCGHS calculation of course agrees with the calculations involving quantization ina fixed background, since keeping the background fixed is tantamount to ignoringthe quantum correction to the classical solution, and is of course inconsistent withthe quantum corrected equations of motion and constraint.A related point is that the energy-momentum conservation equation and theequation of motion for ρ make T±± chiral fields as in conformal field theory.⋆Thisis because in any conformal gauge the stress tensor conservation law (which isa consequence of general covariance and the matter-dilaton equations of motion)takes the form∂±T∓∓+ ∂∓T+−−2∂∓ρT+−= 0and the equation of motion for ρ is equivalent to the first equation of (5.1), sothat ∂±T∓∓= 0. Since this is automatically true for t it is also true separately forthe non-ghost part of the stress tensor.

Now how can we identify the ”radiation”part of the stress tensor.As we argued earlier it does not make sense to justsubtract offthe ”classical” part of the stress tensor. One can subtract the classicalvalue of the classical stress tensor (i.e.

the value when the classical solution issubstituted into it). But by definition this is zero, so we are left with the wholestress tensor.

Also as we’ve seen, T−−is independent of σ+, and hence cannot bezero in the region σ+ < σ+0 , and non-zero for σ+ > σ+0 . Indeed since T in this⋆Indeed the theory is, as we argued earlier, a conformal field theory.17

section is defined to include the ghost contribution (the translation is −N6 t± →t±±) it is zero everywhere, for that is the equation of constraint (second equationof (5.1)).6. A Proposal for Calculating Hawking RadiationHow then can we identify Hawking radiation?

In general relativity there is adefinition of the energy left in a system which is asymptotically flat, after radiationhas been emitted for a certain time. This is the so-called Bondi mass.

This is de-fined relative to some reference static solution and must be given in asymptoticallyMinkowski coordinates. So if δTµν is the first variation of the stress tensor aroundthe reference solution, then for a solution (static or non-static) which asymptoti-cally approaches the static solution at future null infinity, the Bondi mass is givenas (¯σ± are the asymptotically Minkowski coordinates)M(¯σ−) =I+RZd¯σ+δT 0+ = −I+RZd¯σ+(δT++ + δT+−).

(6.1)In the above the integral is to be evaluated at the future null infinity line I+R, i.e.at ¯σ+ →∞. Now the linearized stress tensor satisfies the linearized conservationequation∂∓δT±± + ∂±δT+−= 0.

(6.2)Using this we find from (6.1),∂−M(¯σ−) = −I+RZd¯σ+(∂−δT++ + ∂−δT+−)= +I+RZd¯σ+(∂+δT+−+ ∂+δT−−)= (δT+−+ δT−−)I+R(6.3)18

This equation gives the rate of decay of the Bondi mass. We may thereforeidentify the negative of the right hand side as the radiation flowing out to futurenull infinity.To proceed we need the exact solutions of our quantum corrected equations ofmotion (4.3) or (4.4).

Once a coordinate system is chosen, these solutions are givenin terms of two unknown chiral functions u±(σ±) which need to be determined fromthe constraint equations and the boundary conditions. As we argued in the lastsection the boundary conditions that have been used in the past, do not make sensebecause of the chirality of the stress tensor, so we have to proceed in an alternativemanner.

Let us first impose the constraint equations (2.7).The stress tensor calculated from (3.13) isT±± =12(∂±X∂±X −∂±Y ∂±Y ) +r|κ|2 ∂2±Y + 12Xi∂±fi∂±fi=e−2φ(4∂±φ∂±ρ −2∂2±φ + O(κe2φ)) + 12Xi∂±fi∂±fi + κ(∂±ρ∂±ρ −∂2±ρ),(6.4)andT+−= −rκ2∂+∂−Y −λ2eq2|κ|(X+Y ). (6.5)In the coordinate system in which g± are zero we have from (4.3),T±± =12Xi∂±fi∂±fi +rκ2∂2±Y=12Xi∂±fi∂±fi + ∂2±u±.Hence the constraint equations (2.7) become,19

∂2±u± + 12Xi∂±fi∂±fi + t±± = 0. (6.6)Now we have the problem of determining the ghost stress tensor t. This, aswell as the non-ghost stress tensors T X,Y , T f, transform like connections undercoordinate transformation because of the conformal anomaly.

It is only the sumwhich transforms as a tensor (since the conformal anomalies cancel between thetwo). Thus under a conformal coordinate transformation σ± →σ′± = f±(σ±),T ′f±±(σ′) =∂f±∂σ±−2[T f±±(σ) + N12Df±],T ′X,Y±±=∂f±∂σ±−2[T X,Y±± (σ) + 26 −N12Df±],t′±±(σ′) =∂f±∂σ±−2[t±±(σ) + −2612 Df±],(6.7)where Df is the Schwartz derivative defined by,Df = f′′′f′ −32f′′f′2.At this point we do not know how to proceed without making an assumptionabout the boundary conditions.

We assume that in a preferred coordinate system,namely one which covers the whole space (i.e. including the region behind theclassical horizon) and is asymptotically Minkowski, the expectation value of thematter stress tensor T f−−vanishes.

These coordinates are related to the Kruskal-Szekeres coordinates byˆσ+ = 1λ log(λσ+), ˆσ−= −1λ log(−λσ−). (6.8)This condition seems to correspond to Hawking’s boundary condition, and the20

reasoning is that there should be no f-particle energy coming in from I−L . Nowfrom the point of view of the exact theory the total (including ghosts) stress tensoris zero, so that it is difficult to see what objective meaning this condition has.Nevertheless in order to be as close as possible to the original calculation, let usimpose,ˆT f±± = 0,ˆT X,Y±± + ˆt±± = 0.The latter follows from the first equation and the constraint.

However it stillleaves us the freedom of choosing the separate values of the ghost and X, Y stresstensors. Let us put ˆt−−= αλ224 = −ˆT X,Y−−i.e.

we have in the ˆσ frame, an arbitraryconstant influx of ghost stress energy balanced by a constant outflow of X, Ystress energy. On I−R there is incoming f stress energy, which following CGHS[1] we take to be ˆT f++ = aλσ+0 δ(ˆσ −ˆσ0).Then we may take ˆt++ = αλ224 andˆT X,Y++ = −aλσ+0 δ(ˆs −ˆs0) −αλ224 to be consistent with the constraints.Then by putting σ′ = ˆσ in (6.7), we get in the σ frame,t±± = −26 −α241σ+2, T f−−= −26241σ−2,T f++ = −aλσ+0 δ(ˆs −ˆs0) −N24Using these values in (6.6) we find,u+ =a+ + b+σ+ −a(σ+ −σ+0 )θ(σ+ −σ+0 ) −¯N24 log |σ+|,u−=a−+ b−σ−−¯N24 log |σ−|,(6.9)where¯N = N + α −26(6.10)21

.We now need a reference static solution. This is obtained from (4.4) and (6.9)by putting a = a± = b± = 0 in the latter;X0 = −Y0 =s2|κ|(λ2σ+σ−+¯N24 log(−σ+σ−)), f = 0.This solution is in Kruskal-Szekeres coordinates, and we need to transform thisinto the asymptotically Minkowski coordinates⋆¯σ± defined by σ+ = 1λeλ¯σ+, σ−=−1λe−λ¯σ−.

Under a coordinate transformation X transforms as a scalar, and (sinceρ(σ) →ρ(¯σ) + λ2(¯σ+ −¯σ−)) Y transforms as Y (σ) →Y (¯σ) +q|κ|2 λ(¯σ+ −¯σ−).Hence we have in the new coordinate system,X0 = −s2|κ|eλ(¯σ+−¯σ−) −¯N24λ(¯σ+ −¯σ−) +¯N24 log λ2= −Y0 +r|κ|2 λ(¯σ+ −¯σ−). (6.11)This solution corresponds to the linear dilaton solution of the classical equa-tions.

To obtain the Bondi mass of a general solution which asymptotically tendsto the above static solution, we need to linearize the stress tensors around thelatter. From (6.4) and (6.5) we have using (6.11),δT++ + δT+−= −s2|κ|∂+hλeλ(¯σ+−¯σ−)(δX + δY )i+s2|κ|¯N24λ∂+(δX + δY )−r|κ|2 λ∂+δY +r|κ|2 (∂+(∂+δY −∂−δY ).Substituting into (6.1) we get⋆For the static solution these are the same as ˆσ defined above.22

M(¯σ−) = −Zd¯σ+(δT++ + δT+−)="r2κλe(¯σ+−¯σ−)(δX + δY ) −s2|κ|¯N24λ(δX + δY )+r|κ|2 λδY −r|κ|2 (∂+δY −∂−δY )#I+R. (6.12)Using (3.4) and(3.2) we find that when e2φ << 1 this expression tends (notsurprisingly) to the expression given by CGHS (equation (26) of [1]) except for theghost terms.Static solutions corresponding to black holes (in the classical limit) are obtainedby putting a = b± = 0 and a± ̸= 0.

Then for ¯σ+ >> 1, −δY =q2κ(a++a−) = δX;and we have from (6.12), a constant Bondi (ADM) massM(¯σ−) = λ(a+ + a−).The parameters a± can be of either sign and hence we may have negativemass solutions of the theory.Of course the classical theory has such solutionstoo, but there these correspond to naked singularities, whereas here these are non-singular solutions (as we argued in section 4)†. However one might ask whetherit is the case that we cannot generate these unphysical solutions dynamically, bystarting with positive mass solutions, in which case we might choose to ignorethem.

Unfortunately this is not the case. To see this, let us compute the Bondimass of the analog of the dynamic CGHS solution corresponding to the formationof a black hole by an incoming matter shock wave, and its decay by Hawkingradiation.

This solution is obtained in the σ frame by putting a± = b± = 0, a ̸= 0,† This point has been emphasized by Giddings and Strominger [19].23

in (6.6) and substituting in (4.4). Then in the region outside the classical horizonwe transform to the asymptotically Minkowski coordinates σ defined by,σ+ = 1λeλ¯σ+,σ−= −1λe−λ¯σ+ −aλ2,to get,X = −s2|κ|"M0λ θ(¯σ+ −¯σ+0 ) + eλ(¯σ+−¯σ−) −¯N24 log eλ¯σ+λ e−λ¯σ−λ+ aλ2)!

!#= −Y +rκ2λ(¯σ+ −¯σ−),where we have put M0 = λaσ+0 the mass of the classical black hole. Comparingwith the static solution we find,δX = δY = −s2|κ|M0λ θ(¯σ+ −¯σ+0 ) −¯N24 log1 + aλe+λ¯σ−.Substituting into (6.12) we get,M(¯σ−) = M0 −¯N24λ log(1 + aλeλ¯σ−) −¯N24λ1 + λae−λ¯σ−.In the infinite (light cone time) past ¯σ−→−∞the Bondi mass tends to theclassical black hole mass M0, but at future infinity ¯σ−→+∞one gets an infinitelynegative value.This unphysical conclusion is equivalent to the statement that the Hawkingradiation rate does not go to zero asymptotically.

This rate may be calculatedeither from the left hand side, or as the negative of the right hand side, of (6.3)−dM(¯σ−)d¯σ−=¯N24λ2(1 + λae−λ¯σ−)2 →¯N24λ2. (6.13)Now so far α has been kept arbitrary, but perhaps the most natural choice isα = 26, so that (see (6.10)) ¯N = N. This choice corresponds to the decoupling of24

the ghosts from the Hawking radiation which is positive regardless of the number ofmatter fields. This agrees with the two dimensional analog of the original Hawkingresult [2], as well as that of [1] asymptotically but back reaction still modifies theσ−dependence.

Unfortunately although the formalism allows this value of ¯N itis certainly not the only possibilty. Perhaps this choice has to made on physicalgrounds.

It is also possible that our analysis of the Bondi mass is not the completequantum mechanical story, and a proper treatment would resolve both this issueas well as the question of positivity.7. ConclusionsWhat progress have we made in understanding quantum black holes, and inparticular the phenomenon of Hawking radiation, from this work?Firstly let us stress that even if we leave aside our argument for regardingthe Liouville-like theory as the complete quantum theory, it still gives us the onlyconsistent treatment of the semi-classical theory (i.e.

to first order in κe2φ). Aswe pointed out in section three, if we just include the leading order correctionsthen the field space curvature is zero, and one immediately has a soluble semi-classical theory.

All of the above calculations are then still valid except that wecannot draw some of the conclusions that we’ve drawn from them. Thus we can nolonger explicitly demonstrate the taming of the classical singularity, and of coursethere is no need to conclude that the Hawking radiation does not stop, and thatpositive mass black holes radiate into negative mass solutions.

Nevertheless onehas a consistent semi-classical picture of black hole radiation and back reaction. Inparticular it should be emphasized again that our remarks about the inconsistenciesassociated with the usual calculation of Hawking radiation which ignores backreaction, are valid already at the semi-classical level.

To belabor the point, thecalculations with the Liouville-like theory, when interpreted in terms of the ρ, φvariables, and considered as being valid to O(κe2φ), are the correct semi-classicalresults coming from the classical CGHS theory.In particular, they show that25

the same semi-classical physics is obtained whatever options are chosen for thefunctions h, ¯h (as discussed in section three) simply because ¯¯h is zero at the semi-classical level. In other words we may use the exactly soluble conformal field theoryto make the calculations, provided we interpret the result as being valid only atthe semi-classical level.Secondly we have shown that there is a class of quantum dilaton gravity the-ories, namely those for which the field-space curvature is zero to all orders in e2φ(¯¯h = 0, see discussion after equation (3.1)) whose exact quantum treatment ispossible since they can be transformed into a Liouville-like theory.⋆These theoriesallow for the first time a complete quantum mechanical treatment (including theeffects of dilaton-graviton loops) of a theory of gravity with classical black holesolutions.

Unfortunately, as we have shown, these theories may not be physical. Itis an open question whether it is possible to find a soluble theory with ¯¯h ̸= 0 whichdoes not have this problem, but we believe that this is unlikely.

After all as wehave explicitly demonstrated, quantum mechanics does what one expects it to do,namely it tames the classical singularities, including the naked ones! However itthereby eliminates the usual argument (in the classical theory) for eliminating neg-ative mass solutions on the grounds that such spaces are not globally hyperbolic.It is possible that the problem is not so much with the soluble class of models thatwe have treated, as with the original classical dilaton-gravity theory itself, whichdoes not have a positive definite field space metric.

On the other hand it is alsopossible that the fault lies with our rather heuristic treatment of the Bondi massin the quantum theory, and that a rigorous quantum treatment may resolve thisissue.⋆The objection raised by some authors on the range of integration has been answered insection 3. In particular for the sub-class d) there can be no objection on these grounds.26

8. AcknowledgementsI wish to thank GeoffHarvey, Greg Moore, and Andy Strominger, for discus-sions.

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