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Two Dimensional Quantum Dilaton Gravity and the Positivity of Energy

arXiv:hep-th/9302144v2 11 Aug 1993.COLO-HEP-309hepth/9302144February 1993,Revised August 1993Two Dimensional Quantum Dilaton Gravity and the Positivity of EnergyS.P. de Alwis⋆Dept.

of Physics, Box 390,University of Colorado,Boulder, CO 80309ABSTRACTUsing an argument due to Regge and Teitelboim, an expression for the ADMmass of 2d quantum dilaton gravity is obtained. By evaluating this expression weestablish that the quantum theories which can be written as a Liouville-like theory,have a lower bound to energy, provided there is no critical boundary.

This fact isthen reconciled with the observation made earlier that the Hawking radiation doesnot appear to stop. The physical picture that emerges is that of a black hole in abath of quantum radiation.

We also evaluate the ADM mass for the models withRST boundary conditions and find that negative values are allowed. The Bondimass of these models goes to zero for large retarded times, but becomes negativeat intermediate times in a manner that is consistent with the thunderpop of RST.⋆dealwis@gopika.colorado.edu

In reference [1](CGHS), a theory of dilaton gravity coupled to matter was pro-posed. Subsequently it was argued in [2, 3] that the quantum version of the theorymust be a conformal field theory (CFT), and furthermore that it can be trans-formed into a solvable Liouville-like theory.

In this note we address the question ofthe positivity of energy in this theory. It has been claimed [4] that these theoriesare sick since they do not have a lower bound to energy.

Since all quantum dilatongravity theories that have been studied so far can be transformed in to the Liouville-like theory (because they all have zero field space curvature) it is important tocheck whether this is indeed the case. Related to this statement about the absenceof a lower bound to energy is the observation [3, 2, 5] that the black hole solutionof the quantum theory seems to radiate forever.Recent work by Park and Strominger [6] seems to indicate that there might be aresolution of this problem.

They established a positivity theorem using argumentsderived from supersymmetry considerations, for fields satisfying certain asymptoticconditions. However the expressions for the ADM mass given there do not givewell defined answers for the solutions of the classical theory since the asymptoticconditions of the theorem are violated.

Furthermore the situation for the Liouville-like theory was left unresolved and it was not clear why there appeared to be aconflict with previous work [4].In this work we resolve these issues. Indeed we are able to do so directly for thequantum theory since the complete space of solutions to the Liouville-like theoryis known.

We derive an expression for the ADM and Bondi mass using argumentsgiven by Regge and Teitelboim [7]. We show that the positivity theorems applyto the mass as defined by this procedure provided there is no critical boundaryin the space time such as the one in reference [10].Our argument shows thatthe expression used in [8,1,4] actually needs to be modified.

Furthermore unlessone imposes boundary conditions on some critical line[10], corresponding to r = 0in four dimensions, there will be a contribution from the negative infinite end of2

the space.† Classically when the matter stress tensor has zero expectation value(giving static solutions) the ADM mass is zero, while for the dynamic solutions (i.e.in the presence of collapsing matter) the mass is positive if the incoming matterhas positive energy. In the quantum case, in the no-boundary theory, the staticsolutions again have zero mass.In the dynamical case with collapsing matter,there is an infinite contribution from the negative end of the space.

i.e. this theorydescribes black hole collapse in an infinite bath of radiation.

The Bondi mass canalso be defined. Again in the theory without a boundary the Bondi mas is infinite(and positive) at any finite retarded time, but at positive infinite retarded time,it becomes equal to the energy of the collapsed matter.

This is consistent withthe fact that the ADM mass in this model is infinite. The Hawking radiation inthis picture is just this infinite bath which comes to an end at infinite retardedtime leaving behind the original mass which collapsed.

This is the way that theformalism resolves the positivity of energy with Hawking radiation which lasts foran infinitely long time. It does not reduce the energy from M0 to negative infinitybut from positive infinity to M0.

This is perhaps not a situation which can giveany insight into the four dimensional physics of black hole evaporation, but withinthe well defined formalism of this theory of two dimensional quantum gravity, withno phenomenological boundary, it seems to be the only interpretation possible.With the boundary conditions of Russo Susskind and Thorlacius (RST)[10] onthe other hand, we find in the quantum theory, that static solutions with negativeor zero masses are allowed. In the dynamic case the ADM mass is just that of thecollapsing matter and thus is positive if the latter is positive.

We also calculatethe Bondi mass with RST boundary conditions. Here we find that at negativeinfinite retarded time, it goes to the mass of the collapsing matter, in agreementwith the ADM mass of the model, decreases with increasing time, and for positiveinfinite retarded time, goes to zero.

However at an intermediate time the energybecomes negative, and is discontinuous across a certain null line; i.e. we encounter† This expression is given already in the second and third papers of [2] but it was evaluatedonly at one end of the space.3

the thunderpop of RST[10]. The existence of negative Bondi energy is of courseconsistent with our previous result that there are negative energy static solutions.The Liouville-like action for quantum dilaton gravity [2, 3, 5] is⋆S = 14πZd2σ[∓∂+X∂−X ± ∂+Y ∂−Y + 2λ2e∓√2N (X∓Y )] + Sf + Sb,c,(1)where Sf is the conformal matter action, and Sb,c is the reparametrization ghostaction.

The quantum theory is then given by functionally integrating with thenaive, translationally invariant, measures. The field variables X, Y are related tothe original variables φ (the dilaton) and ρ (half the logarithm of the conformalfactor) that occur in the CGHS action gauge fixed to the conformal gauge (gαβ =e2ρηαβ), by the following relations;Y =√2N[ρ + N−1e−2φ −12NZdφe−2φh(φ)],(2)X = 2r12NZdφP(φ),(3)whereP(φ) = e−2φ[(1 + h)2 −Ne2φ(1 + h)]12,(4)N being the number of matter fields.In the above, the functions h(φ), h(φ)parametrize quantum (measure) corrections that may come in when transformingto the translationally invariant measure (see [2, 3, 5] for details).

The statementthat the quantum theory has to be independent of the fiducial metric (set equalto η in the above) implies that this gauge fixed theory is a conformal field theory.The above solution to this condition was obtained by considering only the leading⋆We will work with N, the number of matter fields, very large, so that the difference betweenN and κ = 24−N6will be ignored.4

terms of the beta function equations, but it was conjectured in [2, 3] (because of itsresemblance to the Liouville theory,) that it is an exact solution to the conformalinvariance conditions.† It should be noted here that this latter statement is strictlyvalid only when P has no zeroes. When N is greater than 24 this implies somerestrictions on the possible quantum corrections, but as shown in [5] there is alarge class which satisfies these conditions.Let us calculate the time translation generator by following the argument ofRegge and Teitelboim [7].This goes as follows.Suppose the Hamiltonian ofthe theory is given as the integral of the stress-tensor over a spatial slice, H =RΣ dΣµT0µ, where as usual time derivatives of the fields have been eliminated infavor of canonical momenta π.

Then if one is to get Hamilton’s equations of motionone should be able to write δH =RdΣµ(Aµδπ + Bµδφ). In general however whenthe space-time is not spatially closed there will be a boundary term on the righthand side of this equation, so in order to get Hamilton’s equations one shouldredefine H by adding a boundary term whose variations will cancel this extraterm.

The resulting expression is the generator of time translations. In additionin a generally covariant theory, there are constraints which imply that the totalstress tensor (for matter plus gravity) is (weakly) zero, so that the total energy ofa solution is given by evaluating just this boundary term.

By Hamilton’s equationsit is indeed conserved. In 3+1 dimensions this boundary is the 2 sphere at infinityand for asymptotically flat solutions the corresponding energy is just the ADMenergy.

In our two dimensional case the boundary of the 1-space is the set of pointsσ = ±∞. To obtain the required expression let us write down the Hamiltonian asthe space integral of T00.H0 =12Zdσ"2(Π2X −Π2Y ) + 12(X′2 −Y ′2) + 2rN12Y ′′ −4λ2e√12N (X+Y )#+ Hf + Hb,c,† This statement has now been proved in [11].5

where ΠX, ΠY , are momenta canonically conjugate to X, Y, and a prime denotesdifferentiation with respect to the space-like coordinate σ. Then we have for thevariation,δH0 =12Zdσ[4(ΠXδΠX −ΠY δΠY )− X′′ + 4λ2r12N e√12N (X+Y )!δX + Y ′′ −4λ2r12N e√12N (X+Y )!δY ]+ δHf + δHb,c + 12[X′δX −Y ′δY + 2rN12δY ′]∞−∞,(5)assuming that the matter and ghost configurations are bounded.Thus we need to add a boundary term and redefine the Hamiltonian asH = H0 + H∂,such that δH∂cancels the boundary term in (5).

H∂cannot be defined forarbitrary configurations. But for the space of configurations which either vanishasymptotically or go to an arbitrary solution of the equations of motion, this maybe defined.

This is because the general solution [3,2] to the equations of motion isgiven byX = −r12N (u+(σ+) + u−(σ−)) + λ2r12Nσ+Zdσ+eg+(σ+)σ−Zdσ−eg−(σ−)= −Y +rN12(g+ + g−),(6)where u±(σ±) are arbitrary chiral functions to be determined by the constraints6

and g±(σ±) reflect the arbitrariness in the choice of conformal frame.⋆For thisspace of configurations and variations within this space, we have,H∂= −rN12[12g′(σ)X −X′]∂,(7)where g = g++g−. Now since H0 = 0 (weakly) is a constraint of the theory theenergy is entirely given by the boundary term.

We should however measure energyrelative to the linear dilaton vacuum (LDV) which corresponds to the solution withu± = 0 in (6). Defining ∆X = X −X0 where X0 is the LDV solution, we haveour final expression for the ADM energy,EADM = −rN12[12g′(σ)∆X −∆X′]∞−∞(8)It is instructive to express this in terms of the original variables of CGHS using(2) and dropping all quantum corrections i.e.

O(Ne2φ) terms. Then we get,EADM = ∆[e−2φ(λ + 2φ′]∞−∞(9)At this point it is incumbent upon us to discuss in what coordinate systemthis expression is expected to be valid.

As we observed after (5) the above resultsare valid provided that the matter and ghost configurations are bounded. In thequantum theory we cannot make this assumption in every conformal frame.

Thereason is that the stress tensors of matter, dilaton gravity and ghosts are not tensorsseparately due to quantum anomalies (Schwartz derivative terms).† It is only the⋆In Kruskal-like coordinates (x± of reference [1]) the latter are zero while in the asymp-totically Minkowski coordinates (σ = ± 1λ ln(±λx±)) g± = ±λσ±. These coordinates werecalled ˆσ in reference [2, 5] while the Kruskal coordinates were called σ.

In this paper westick to the notation of [1].† The point is that the normal ordering that is necessary to define the stress tensor is framedependent.7

total stress tensor that is a tensor and it is zero in every frame.Thus we take thepoint of view that the expressions for the quantum ADM and Bondi masses arecorrect only in asymptotically Minkowski coordinates.Let us evaluate (8) for the static quantum solution [3, 2]X = −r12NM0λ + X0(10)where M0 is a constant, and X0 is the quantum solution corresponding to thelinear dilaton vacuum of the classical theory which is given byX0 = −r12Neλ(σ+−σ−) −N24λ(σ+ −σ−). (11)Thus in the formula for the ADM energy (8), we must put ∆X = −q12NM0λ .the result isEADM = 0It should be noted that this result is true even in the classical limit and is dueto the contribution from the negative end of the space.

This is of course consistentwith positive energy theorems since this configuration is obtained in the situationwhere the expectation value of the matter stress tensor is zero.In the earliercomputation of the ADM energy [2, 4] (following [8, 1] only the contribution fromthe σ →∞was kept. However the time translation generator that one gets fromthe Regge Teitelboim argument requires the evaluation of the integral at both endsσ = ±∞.

In addition the expression (9) differs from the expressions given in [8]and [1]. In fact this can be calculated directly from the classical action using theRegge-Teitelboim argument to get exactly the same result as (9).

The originalexpression actually does not give a well defined answer for the ADM mass of thestatic black hole since the solution does not go to the LDV as σ →−∞.8

The apparent existence of negative mass static solutions was pointed out asa problem for the Liouville-like theory [4] since these solutions are non-singular(unlike the corresponding static solutions which had time-like singularities, andhence could be ruled out on physical grounds). Our result shows however that thecorrect mass for such configurations is always zero.To get non-zero mass one has to put in matter.For simplicity of presen-tation we will explicitly consider only the case of an incoming shock wave off-matter T f++ = aeλσ+0λδ(σ+ −σ+0 ), T f−−= 0 [1], but the generalization to ar-bitrary bounded configurations is trivial.The conformal frame in which thissolution is asymptotically Minkowski is related to is related to the σ frame byσ+ = σ+, σ−= −1λ ln(e−λσ−−aλ).Then the solution is (see for example [5])−rN12X =Mλ + eλ(¯σ+−¯σ−) −N24 log eλ¯σ+λ e−λ¯σ−λ+ aλ2)!

!for σ+ > σ+0=eλσ+(e−λσ−+ aλ2) −N24 log eλσ+λ e−λσ−λ+ aλ2)! !,for σ+ < σ+0(12)In the above we have put M = aeλσ+0 the mass of the classical black hole.

Inevaluating the ADM mass this needs to be compared with the LDV solution in thesame conformal frame i.e. (11) with σ replaced by σ.

Then we getEADM = M + Nλ24 ln1 + aλeλ(τ−σ)|σ→−∞+ N24λ. (13)Thus the quantum anomaly in the dynamical solution gives an infinite contri-bution to the ADM mass from the negative infinite end of the space-like line.

Itshould be noted that the classical solution has a well defined mass equal to M inthese coordinates. The source of the infinite radiation bath is quantum mechanical9

(N →N¯h in the above). It comes from the fact that due to the quantum anomalythe solution in the region σ+ < σ+0 does not go to the LDV as σ →−∞once weinsist that the solution for σ+ > σ+0 is asymptotically Minkowski.

It should benoted that the energy though infinite is positive.If we want to represent the situation corresponding to the absence of a radiationbath we have to put a boundary as done in [10]. We should stress here that todo this we need not necessarily use the RST choice of functions h, h in (4).

Thephysics may be discussed entirely in terms of the X and Y fields of the Liouville-like model and the difference between the two types of models lies in whether onechooses to put a boundary or not.The RST boundary condition puts ∂±X = f = 0, on a critical curve X = Xc,which is regarded as the left boundary of space-time, wherever it is time-like.Let us first consider the static solutions (10), (11).The RST boundary is atX = Xc = −q12N ( N24 −N24 ln N24). For M0 = 0, i.e.

for the LDV, this is the lineσ =12λ ln N24. For M0 > 0 on the other hand there is no solution to the boundarycurve equation and it is not clear how to evaluate EADM.

On the other hand forM0 < 0 there is a boundary, and since from the boundary conditions there will beno contribution to the ADM mass from this end of the space one getsEADM = M0. (14)In other words there is no positivity in the theory with boundary.

This is ofcourse consistent with the fact that there is a negative energy thunderpop [10] inthe dynamical solutions of this theory and as we shall see later, the Bondi massalso reflects this phenomenon.Let us now consider the collapse scenario. From the explicit solution it maybe seen that the boundary curve X = Xc is time-like for σ+ < σ+0 .

Hence theboundary condition may be imposed there, and in particular it follows that thereis no contribution to the ADM mass at the left end of the space. It should be10

stressed that this is the case since the boundary is given by a fixed value of X (atwhich ∂±X = 0) and thus its contribution cancels between the collapse solutionand the LDV. Thus with the RST boundary conditions,EADM = aeλσ+0 ≡M(15)If incoming energy is positive (a > 0) then the ADM mass is positive.

Clearlythis generalizes easily to arbitrary bounded configurations of matter so that forthese dynamical solutions we have explicitly demonstrated the positive energy the-orem for quantum dilaton gravity. It should be pointed out that in deriving theADM mass for the theory with RST boundary conditions, we need only the con-ditions at the time-like boundary for σ+ < σ+0 .

In order to obtain the completephysical picture one needs to impose boundary conditions on the critical curvewhen it becomes time-like again somewhere above the line σ+ = σ+0 . As explainedby Russo et al [10] this leads to a thunderpop - a burst of negative energy to I+L .We will rediscover this effect when we calculate the Bondi mass.Let us now discuss the issue of Hawking radiation.

As discussed at lengthin section 5 of [5] the usual calculation of Hawking radiation [1] cannot reallybe justified in a situation in which one takes back reaction and the constraintsof general covariance into account. Since this point does not seem to be widelyappreciated it is perhaps necessary to reiterate it here.

The constraints tell us that(the expectation value of) the total stress tensor is zero; i.e.T X,Y±± + T f±± + T b.c±± = 0(16)where the last term is the contribution of the reparametrization ghosts, togetherwith the equation of motion for the conformal factor T+−= 0. The usual argumentsare equivalent to the following.

Since only f-matter is propagating only the fstress tensor should contribute to radiation. Now we have the following anomalous11

transformation law for the stress tensor, from the σ coordinate system which coversthe whole space, to the σ system which covers only the region outside the horizon.Tf−−(σ−) = T f−−(σ) + N24λ21 −1(1 + aλeλσ)2where the second term on the RHS is the Schwartz derivative for the transformationfrom σ to σ.Then it is argued that σ−is the appropriate coordinate on I−Rand since there should be no incoming radiation on this null line one must setT f−−(σ) = 0 so the Hawking radiation observed on I+L is just given by the Schwartzderivative term. However this argument ignores the constraint equation (16).

Infact the counting of propagating degrees of freedom merely tells us that there areN of them, but this does not imply that the energy propagated is given just by thef stress tensor. In fact the physical propagating state must be dressed by X, Yfields and possibly ghost fields as well.

For the above result to be compatible withthe constraints there has to be an inflow of X, Y, b.c field energy to compensate forthe outflow of f energy.Thus we believe [5] that the evaluation of the Hawking radiation must be doneby calculating the Bondi mass of the system. This quantity, like the ADM mass,could be non-zero for open systems even though the expectation value of the totalstress tensor is zero.

Since it must give us the total energy minus the energy thathas been radiated away up to a given retarded time, it must be evaluated on a linewhich is asymptotic to σ−= const. at σ+ →∞(i.e.

on I+R) and to σ+ = σ+1 < σ+0on σ−→∞(i.e. on I+L ).⋆Then, in analogy with the expression (8) for the theADM mass, we have for the Bondi massEBondi(σ) =rN12 [−λ∆X + (∂+∆X −∂−∆X)]σ+=+∞σ−=∞,σ+=σ+1(17)We have to substitute in the above the difference between the dynamical solu-⋆One may also take the line to be asymptotic to a space like line at the σ →−∞end.12

tion evaluated in the σ frame and the LDV solution. Then we getEBondi(σ) =M0 −N24 ln(1 + aλeλσ−) −N24λ(1 + λae−λσ−)−"−N24 ln(1 + aλeλσ−) −N24λ(1 + λae−λσ−)#σ−→∞(18)This gives an infinite Bondi mass for any finite value of σ−, but this is areflection of the fact that the ADM mass is infinite.

On the other hand in the limitσ−→∞the Bondi mass becomes equal to M0 i.e. the initial incoming matterenergy.

In [5] by contrast the Bondi mass was identified (just as with the ADM)mass as the value evaluated at one end i.e. I+R.

However that led to the resultthat the Hawking radiation did not stop and the Bondi mass decayed indefinitely.In view of our discussion of the ADM mass, we believe now that the problem wasthe fact that both ends of the line (space-like in the case of ADM, light-like in theBondi case) need to be considered. This is the correct time translation generatorin the ADM case and the object that satisfies positivity.We have shown that quantum CGHS theory is soluble and satisfies positivity ofenergy.

Unfortunately it is not possible to get much insight into four dimensionalblack hole physics from it, since the conformal invariance forces us to a situationwhere one has to start with an infinite bath of radiation. It is clear that to makecontact with four dimensional physics one should have a time like boundary (cor-responding to the origin of polar coordinates in 3+1 dimensions) as is done in themodels with the so-called quantum singularity [12, 13].

However, in that case it isnot clear whether conformal invariance, which is a consequence of general covari-ance, can be preserved. Within the strict interpretation of the formalism of twodimensional quantum dilaton gravity, the theory without boundary seems to bethe only consistent one.However for phenomenological purposes and to check that (17) gives the phys-ical picture that one expects in the Hawking evaporation of black holes let us13

evaluate the Bondi mass with RST boundary conditions. In this case the lowerlimit in (17) is replaced by any point on the critical curve X = Xc for σ+ < σ+0 .Then there is no contribution from this end to the energy.

At the upper end how-ever there are now two regions to consider. Calling the point where the apparenthorizon (∂+X = 0) and the critical curve intersect (σ+s , σ−s ), we have the regionσ−< σ−s (region I of RST; see [10] for figure) and the region between the time likeboundary and σ−= σ−s (region II of RST).

In region II the black hole has decayedand the solution is taken to be the LDV. In region I the solution is the collapsesolution for σ+ > σ+0 .

Thus we have−rN12∆X =0 in II,=Mλ −N24 ln(1 + aλeλσ−) in I.Computing the Bondi mass from (17) we have,EBondi(σ−) =M0 −N4 λ ln(1 + aλeλσ−) −N4λ1 + λae−λσ−in I,=0 in II. (19)Thus we seem to have a physical picture of Hawking radiation when the RSTboundary conditions are imposed.

For σ−→−∞, EBondi →M; while at lateretarded times EBondi →0. However there is an unphysical feature in the modelin that some time before σ−= σ−s = −ln[ aλ(1 −e−4MλN )] the Bondi energy goesnegative.

Indeed the energy flow is discontinuous at σ−= σ−s .EBondi(σ−s + 0) =0EBondi(σ−s −0) = −N4 (1 −e−4MλN )This is just the effect of the thunderpop of RST [10] and is caused by the fact14

that when the collapse solution is matched to the LDV along the null line σ−= σ−s ,the result is continuous but not smooth.In conclusion, what we have shown is that in the absence of a boundary inthe two dimensional space one is forced to an interpretation where the black holeis immersed in an infinite bath of quantum radiation.However although boththe ADM and Bondi masses are infinite (the latter actually goes to the collapsingmass at infinite time) they are positive.On the other hand if RST boundaryconditions are imposed one has a physical picture of black hole evaporation at theprice however of a negative energy thunderpop. The latter is consistent with thefact that there are negative energy static solutions in the theory.Acknowledgements: While this paper was being prepared for publication theauthor had a discussion on the present work with A. Bilal and I. Kogan, duringwhich they showed me a paper [14] by themselves in which formula (9) is derivedin a somewhat different manner.

This work is partially supported by Departmentof Energy contract No. DE-FG02-91-ER-40672.REFERENCES1.

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