---

EQUILIBRIUM TWO-DIMENSIONAL

arXiv:hep-th/9302009v1 3 Feb 1993EQUILIBRIUM TWO-DIMENSIONALDILATONIC SPACETIMES*Michael CrescimannoCenter for Theoretical PhysicsLaboratory for Nuclear Scienceand Department of PhysicsMassachusetts Institute of TechnologyCambridge, Massachusetts 02139U.S.A.Submitted to: Physical Review DTypeset in TEX by Roger L. GilsonCTP#2174January 1993* This work is supported in part by funds provided by the U. S. Department of Energy(D.O.E.) under contract #DE-AC02-76ER03069, and by the Division of Applied Mathematicsof the U.S. Department of Energy under contract #DE-FG02-88ER240660

ABSTRACTWe study two-dimensional dilaton gravity coupled to massless scalar fields and look fortime-independent solutions. In addition to the well-known black hole, we find another classof solutions that may be understood as the black hole in equilibrium with a radiation bath.We claim that there is a solution that is qualitatively unchanged after including Hawkingradiation and back-reaction and is geodesically complete.

We compute the thermodynamicsof these spacetimes and their mass. We end with a brief discussion of the linear responseabout these solutions, its significance to stability and noise, and a speculation regarding theendpoint of Hawking evaporation in four dimensions.1

I.INTRODUCTIONTwo-dimensional gravity has emerged as a useful toy model for addressing some basicphilosophic questions about semiclassical and quantum gravity. Dilation gravity is the low-energy effective theory of string theory and so solutions of this theory arise naturally as theσ-models backgrounds associated to conformal field theories.

An example of these are thecoset models1, 2, 3 among which was found two-dimensional σ-models, including that of theblack hole,4, 5, 6 which is a solution of two-dimensional dilaton gravity with a Minkowski sig-nature. Besides just the graviton and dilaton, a two-dimensional gravity also arises naturallyfrom dimensional reductions of ordinary Einstein gravity in four and five dimensions (seeRefs.

[7,8,9]). Realistic string theory should also have low-energy “matter” fields.

Callan, etal.10 showed that including massless scalar fields to dilaton gravity in two dimensions yields atoy model rich with phenomena such as gravitational collapse and Hawking radiations. Thismodel has subsequently been explored by many authors, who addressed a host of questionsranging from the structure of the singularity,11 of the Hawking radiation and its back reactionon the metric,12, 13 information loss in black hole evaporation,14, 15, 16 and approaches to thefull quantum theory.17−22 There are several very readable reviews on this subject; see forexample Refs.

[8,23].Although two-dimensional dilaton gravity is somewhat different than ordinary Einstein–Hilbert gravity, there are many features of the model that are generic to other dimensionsand other metric theories of gravitation. Certainly one difficulty in trying to answer some ofthe above questions is that they concern (inherently) non-equilibrium phenomena.

Since theGreen’s functions, that arise naturally when studying quantum fields in these backgrounds2

are typically equilibrium Green’s functions they are often not suitable for addressing thesenon-equilibrium questions beyond low orders in perturbation theory.There are, however, many interesting questions to be answered about the equilibriumproperties of semiclassical gravity, such as the nature of the equilibrium state and the powerspectrum of noise in the spacetime itself. These issues have been somewhat neglected, as theydo not allow one to directly answer the interesting non-equilibrium questions posed above.Alternatively, these notions are computable and rigorously defined.Here we study (static) solutions to two-dimensional dilatonic gravity coupled to matterand try to answer some of the equilibrium questions above.

We will find that although theordinary dilaton black hole is not an equilibrium solution (that is, semiclasically with realisticboundary conditions or once back-reaction is included) there does exist a solution which isnon-singular, geodesically complete and static. This solution seems to persist even with theback-reaction included.

It is interpreted as the black hole in equilibrium with a heat bath ofradiation. We then compute its thermodynamic properties and describe the fluctuations aboutthis solution.

The idea of studying the two-dimensional dilatonic black hole in equilibriumwith a radiation bath is not new (see for example the discussion in Refs. [24–26,16]).

Insome earlier works these solutions were discarded on the basis that they are not finite energysolutions. This is trivially true for black holes in a radiation bath in any open spacetime andso here we accept this as a simple fact and instead focus on what we feel are more pressingphysics issues.At the end we remark on whether something like these solutions may berealized approximately.

Principally, then, in this note we aim to clarify and extend thoseearlier works.Let us consider a naive picture of a black hole of mass M in thermal equilibrium in1 + 1 dimensions, in a box of length L. Let the radiation in the box be of massless bosons3

of temperature T. Here we follow closely the line of reasoning presented in the initial pagesof Ref. [27].

As suggested by earlier investigations of the two-dimensional black hole,6 thetemperature of the black hole, TBH is fixed and independent of the mass. The total energyand entropy of the system isU = M + aLT 2S = MTBH+ 2aLT,(1)where a depends on the number of species of bosons in the thermal bath.

Extremizing theentropy at fixed energy yields T = TBH and the second derivative of the entropy with respectto temperature at this extremum is negative,d2SdT 2 =1TBH∂2M2T 2U= −2aLTBH< 0(2)suggesting that the system is stable. As we show later, when we write down a solution tothe field equations that correspond to a black hole in a heat bath, this picture is too naive;in 1 + 1 dimensional dilaton gravity the presence of the bath very dramatically affects thegeometry of the spacetime.To begin with, consider the low-energy effective action of string propagation in a two-dimensional manifold (M, g)I = IG + IMIG = 12πZMd2x √−g e−2φ R + 4(∇φ)2 + 4λ2+ 1πZ∂Me−2φK dǫIM = 12ZMd2x √−gNXΛ=1(∇fi)2(3)where K is the trace of the second fundamental form on the boundary, (this term is necessaryfor the IG to depend on the fields g, φ and their first derivatives only) and where fi aremassless scalars minimally coupled to gravity.4

We look for equilibrium (i.e. globally static) solutions by requiring the metric possess aglobal time-like Killing vector.

Any two-dimensional metric that has a global time-like Killingvector may be put in the form,ds2 = −Ω2(r)dt2 + dr2 . (4)The non-vanishing Christoffel symbols with this metric are Γ100 = ΩΩ′ and Γ010 = Ω′/Ωwherethe prime denotes ∂/∂r.

The scalar curvature is R = −2Ω′′/Ωand the most general staticand covariantly conserved tensor has the formTµν =" A −ΩΩ′ A′bΩbΩAΩ2#(5)where A is an arbitrary function of r and b is a constant. The equations of motion that followfrom Eqs.

(1) are, in general,1π e−2φ ∇µ∇νφ + gµν(∇φ)2 −∇2φ −λ2+ 12T (f)µν = 0(6a)1π e−2φ R + 4∇2φ −4 (∇φ)2 + 4λ2= 0(6b)1√−g ∇µgµν√−g ∇νfi = 0(6c)whereT (f)11 = 12NXi=1(∂rfi)2 + 1Ω2 (∂tfi)2and T (f)µν is the stress tensor of the fields fi. Since DµT (f)µν = 0, Eq.

(6a) implies thatR + 4∇2φ −4(∇φ)2is a constant, so Eq. (6b), while an independent equation, is consistent with Eq.

(6a). Equation(6b) is thus essentially a statement of the gravitational Bianchi identity.5

To look for static solutions to Eq. (6) with the metric of Eq.

(4) we learn immediatelythat T (f)µνmay not be of the most general form of Eq. (5).

Indeed, (6a) for µ = 1, ν = 0implies that b = 0.Since the matter fields fi that we are minimally coupling to gravityare massless we expect that (at least classically) T (f)µνis traceless. This forces A = 2A0, aconstant.

A0 may be thought of as the local energy density. It is easy to check that constantA is consistent with the equations of motion for the fi, Eq.

(6c), but we are, for now, notinterested in explicit classical solutions for the fi. The remaining equations of motion in thismetric ansatz read;φ′′ −(φ′)2 + λ2 + πA0Ω2 e2φ = 0(7a)φ′′ + Ω′Ωφ′ −(φ′)2 + λ2 −Ω′′2Ω= 0(7b)−Ω′Ωφ′ + (φ′)2 −λ2 + πA0Ω2 e2φ = 0(7c)Eliminating the terms involving A0 permits one to integrate once, findingφ′ = Ω′Ω+ cΩ(8)where c is a constant of integration.

These equations of motion are invariant under togetherrescaling Ωand by shifting φ by a constant, and so there are only really three possibilities forc; either 0 or ±λ. It is simple to show that the solutions with c = ±λ are: a) Ω= tanh λr,φ = −ln(cosh λr)+φ0; the LDV Ω= 1, φ = −λr+φ0; and b) Ω= cosh λr, φ = ln(sinh λr)+φ0,respectively.

Regions a) and b) correspond to those of the maximally extended black holesolutions10,28−31 and a), the LDV, and b) are solutions with A0 = 0.For c = 0, Eqs. (7) reduce toΩ′′2Ω−Ω′Ω2+ λ2 = 0(9)6

which has the general solution1Ω= B+e√2 λr + B−e−√2 λr ,φ = lnΩ+ φ0(10)with B±, φ0 being real numbers.These solutions all have A0 =λ2π e−2φ0 ̸= 0.They are static spacetimes filled withradiation density, A0.Before including back-reaction or studying the thermodynamics of these spacetimes webriefly describe their geometry. These solutions, Eq.

(10), all approach constant curvatureR = −4λ2 asymptotically, r →±∞.There are three different geometries possible fromEq. (10), depending on the relative sign of B+ and B−.

Note that as described above, wemay scale Ωsuch that, without loss of generality, B+ = 1. Three different geometries resultfrom whether B−is positive, zero, or negative.

Essentially they areΩ=12 cosh√2 λrB−> 0(I)e−√2 λrB−= 0(II)12 sinh√2 λrB−< 0(III)Solutions I and II are geodesically complete and III has a naked singularity at r = 0.Solution II is another linear dilaton solution (LDV) φ = −√2 λr+φ0. Solution II is a constantcurvature metric and therefore has, in addition to the trivial time translation Killing vector,two other Killing vectors, one associated with translations in r and the other akin to a boost.This is in strong analogy to the LDV solution (flat space) with A0 = 0.

It is instructive tocompare these solutions with the black hole solution. In null coordinates x+ and x−the blackhole metric (with A0 = 0) is6, 10ds2BH = −dx+dx−Mλ −λ2x+x−.

(11)7

In the coordinatesx± =rM4λ"√2λ cosh√2 λr ± t#the metric of spacetime III isds2III = −dx+dx−Mλ −λ22 (x+ + x−)2. (12)The classical solution in conformal gauge has the general solutionds2 = −e2ρdx+dx−(13)with∂+∂−e−2ρ = −λ2 ,∂+∂+e−2ρ = −t+(x+) ,∂−∂−e−2ρ = −t−(x−)(14)where t±(x±) are the right and left moving parts of the T (f)µν .

Thus, solutions I and III maybe interpreted as a two-dimensional dilatonic black hole in a radiations bath where the naivetemperature of the black holeTBH ∼λ2πis the same as that of the radiation bath. This isquite different than known solutions to general relativity in four dimensions where classicallythere are static cosmological solutions with TBH > Tcosmo (see Refs.

[32,33]), where Tcosmois a cosmological temperature associated with the cosmological horizon. In four dimensionalEinstein gravity it seems technically formidable to find analytic solutions corresponding to ablack hole in equilibrium with a heath bath.

We will remark on the relation of the solutionsof the two-dimensional model presented here and ordinary gravity in four dimensions later.The interpretation of spacetimes I and III as the two-dimensional dilaton black hole ina radiation bath is, although correct, not quite the naive model of the black hole ”in-a-box”discussed after the introduction. The presence of uniform, static energy density A0 drastically8

modifies the geometry of the spacetime, and there is no smooth limit as A0 →0; the equationswith and without the bath (A0 = 0 and A0 ̸= 0, respectively) are very different. It is stilluseful for taxonomic purposes to regard the solutions I, II and III as related to a), the LDVand b) respectively, of the black hole spacetime but including the effects of a radiation bath.

*For example, the curvature scalar of I is positive near r ≈0 while that for III is strictlymore negative than −4λ2.Also note that spacetimes I and III only have a single globalKilling vector. That is, the r-translation isometry of spacetime II is broken spontaneously inspacetimes I and III, as expected of black hole backgrounds.

Later, in computing asymptoticmasses we will again see how natural it will be to think of the solutions I, II and III ascorresponding to the patches a), the LDV and b), respectively, of the black hole spacetime.It is easy to compute the Hawking radiation12 of, and back-reaction on these spacetimes.Following Refs. [8,10] this may be done in two steps.

First, Hawking radiation arises from theconformal anomaly of the matter fields,34,35⟨Tµνgµν⟩= −N12R(15)Now, following Ref. [10] and using metrics I, II and III, one may easily compute semiclassicallywhat the Hawking flux at r = ∞.

It is simple to show that this flux is negligibly smallcompared to the ambient matter (A0 ̸= 0) at infinity.Thus spacetimes I,II and III donot semiclassically decay, unlike the black hole without a radiation bath (A0 = 0 and withtrivial boundary conditions at r = ∞.) Furthermore, due to the fact that these metrics arefunctions of (x+ + x−), we find that at infinity x+ →∞or x−→∞the components of* Note although regions a) and b) (see discussion after Eq.

(8)) of the black hole metric arerelated by duality,29−31,38 when matter is included, duality is no longer a symmetry; there isno duality symmetry relating solutions I and III.9

the full renormalized stress tensor T++ (x+, x−) and T−−(x+, x−) are equal and in the limitproportional to A0.Of course it may be argued that there is nothing really “asymptotic” about r ∼∞. It issimple to show that for the spacetimes I, II, III a time-like trajectory reaches r = ∞in a finiteproper time proportional to 1/λ.

Indeed it is easy to find coordinates in which one may studythe maximal extensions of spacetimes I, II and III. Figure 1 contains the Penrose diagramsfor these spacetimes.

It is natural to regard spacetime II as the “vacuum” to which we shouldcompare the solutions I and III, as was the LDV for the black hole without a radiation bath(A0 = 0). In this sense, “asymptotic” will mean r ∼∞since there the solutions I and IIIapproach solutions II.It is somewhat more interesting to study the back-reaction of the Hawking radiation inthese solutions.

For the black hole it was hoped that studying the back-reaction would furtherclarify gravitation collapse, subsequent evaporations, and the problem of information loss.In the metric ansatz, Eq. (4), the most general, static, covariantly conserved stress tensor(see Eq.

(5)) satisfying Eq. (15) is:Tµν =2A0 −α(Ω′)22−ΩΩ′′bΩbΩ2A0Ω2 −α2Ω′Ω2(16)where A0 and b are again constants and α = N/12.To understand how the Hawking radiation back-reacts on the metrics I, II and III, we canbegin by putting Eq.

(16) into Eq. (6a).

This gives the equations of motion with back-reaction,φ′′ −(φ′)2 + λ2 + πe2φΩ2 A0 −α2 (Ω′)22−ΩΩ′′! != 0(17a)φ′′ + Ω′Ωφ′ −(φ′)2 + λ2 −Ω′′2Ω= 0(17b)−Ω′Ωφ′ + (φ′)2 −λ2 + πe2φΩ2 A0 −α(Ω′)24!= 0(17c)10

Note that for metric II, the LDV solution, the terms in Eq. (17) due to the back-reactionare indeed negligibly small in the r →∞region.

This is also true for the other solutions(I,III). Thus our notion of “asymptotic,” as described above for these solutions is unchangedby including the back-reaction.

Although for A0 = 0 the LDV solution remains a solutioneven including back-reaction, for A0 ̸= 0 it is simple to show that there can be no exactlyLDV solution (no solution with φ = −βr + φ0 for some β).The asymptotic solution to Eq. (17) are, (r →∞)φ(r) →√2 λr + φ0 + e−2√2 λr ω0 + 2αA0 √23 λr −16!

!Ω(r) →e−√2 λr 1 + e−2√2 λr ω0 + α2√23 λr!! (18)where ω0 < 0, ω0 = 0, ω0 > 0 correspond to asymptotic views of the solutions I, II and III,respectively.Although no exact solution to Eq.

(17) with A0 ̸= 0 (or A0 = 0) is known, we can ascertainmany properties of any solution. For example, letting y = e−2φ + πα, it is straightforward toshow that Eq.

(17) imply"4λ2 + Ω′Ω ΩΩ′′(Ω′)2!′ #y = π 4αλ2 −αΩ′′Ω+ 4A0Ω′′Ω(Ω′)2! (Ω′y)′ = −2πΩ A0 −α (Ω′)24!.

(19)We now study these equations near a power law singularity, such as that found in spacetimeIII. For the metric to behave asΩ∼rγ ,as r →0(γ ̸= 0,real)(20)(which could indicate either a horizon γ > 0 or a singularity γ < 0) using Eq.

(19) we find thatincluding back-reaction only allows γ = 1. That is, roughly speaking, including back-reaction11

is not consisent with simple power-law singularities in the metric. Of course, Eq.

(15) is aone-loop result and near curvature singularities we might expect that there may be othercontributions to ⟨Tµνgµν⟩that would dominate. By “singularity” we really mean curvaturesapproaching “1” in units of λ2/α.

As a consequence of this we see that including back reactionwill strongly modify the singularity at r = 0. This is also true for the singularity of the blackhole in the absence of a bath (A0 = 0).Similarly one may show that back-reaction also strongly modifies solutions II in thestrong coupling region r →−∞.

However, for metric I the corrections due to including theback-reaction appear to be mild almost everywhere. It can also be shown that Eqs.

(19)are consistent with Ω′ vanishing linearly. Since this is the distinctive feature of the “throat”region (r ≈0) of the metric in I, this indicates that there may be an exact semiclassicalsolution (i.e.

with back-reaction included) that is qualitatively metric I.We now associate “asymptotic” masses to the spacetimes discussed above.This willsupport the notion that spacetimes I, II, and III should be thought of as a), the LDV and b),respectively but with spacetime filled with a radiation bath.Before continuing we note that, as described earlier, we have made a special choice indefining what we mean by “asymptotic.” With this definition we are not a priori guaran-teed that all “definitions” of mass will lead to the same expressions.Below we discuss athermodynamic definition of “asymptotic’ mass. However, first, we would like to point outthat one may compute an “asymptotic” mass by simply comparing the geodesics near r = ∞of, say, spacetime I to those of the “vacuum” spacetime II.

Recall the “vacuum” spacetimeII is anti-deSitter space and, as usual, we consider geodesics in the covering space shown inFig. 1.

One may compute the “transit time” to cross a wedge: that is, the proper time for an12

observer to enter and leave the region marked II. As expected, it is the same for all time-liketrajectories and is ∼1/λ.

Now, when one solves the geodesic equation for geodesics nearr ∼∞in spacetime I, one finds that their transit time is a little bit slower than that ofspacetime II. This difference is attributable to an extra overall attractive mass M ∼A0/λin spacetime I as compared with spacetime II.

The natural position to associate to this masswould be near r = 0 in spacetime I, since that is where the metric deviates appreciable fromthe ”vacuum” metric of spacetime II. Again, one may even more simply compare the instan-taneous accelerations of a body near r →∞but initially at rest in both spacetimes I and II.One again finds that the extra acceleration towards r = 0 in space I may be attributable to amass M ∼A0/λ.

These masses are to be thought of as contravariant quantites with respectto the metric II. It should also be possible to derive an ADM-like6,37 mass formula directlyfrom the equations of motion, but we do not pursue that here.We now compute the equilibrium thermodynamic functions for the spacetimes I, II andIII.

This will give a deeper understanding of these solutions and will be convenient for dis-cussing issues of stability and fluctuation below. In what follows we proceed essentially fromRefs.

[28,36].To understand thermodynamic properties of the spacetime semiclassically, it is mostconvenient to compute the free energy. This is done by simply evaluating the action Eq.

(3)on a Euclidean continuation of the solutions in question. In our metric ansatz Eq.

(4), thisyieldsI = 1πZ∂Me−2φΩ′Ω−2φ′dΣ + Bmat(21)where we have computed the action in some box, the boundary ∂M of which is located atsome fixed parameter distance rw. Bmat is a bulk term due entirely to the IM of the radiationbath.13

Call Tw the temperature that is seen by an inertial observer at the wall. Tolman’s relation(which is a consequence of thermal equilibrium) indicates,Tw =˜TΩ(rw)(22)where ˜T is some constant temperature.

Later it will be identified with the temperature ofthe ambient radiation.This relation implies that in the Euclidean continuation of thesespacetimes, the t-direction of the manifold at r = rw is periodic, with extent Ω/ ˜T. Thuswe find that, for fixed A0, Bmat is the same linear function in rw for any solution.

We arefundamentally interested in distinguishing the solutions I, II and III from one another and soone can show Bmat, the bulk term, may be ignored; differences between the solutions will bemanifest in differences in their IG only.Following Ref. [28] it is useful to consider the free energy F as a function of Tw andD, the total dilaton charge within the box.

D may be conveniently defined as the chargeassociated with the currentjµ = ǫµν∂ν e−2φ(23)where ǫµν is the antisymmetric covariant tensor (ǫ01 = √−g) so ∇µjµ = 0. The charge withinthe box is thusD =Z rwjadΣa = e−2φw(24)where φw = φ(rw) is the value of the dilation field at the wall.It is now easy to compute the free energy F = −TwI from Eq.

(21). For the spacetimeII,F II = −√2 λπD(25)14

and so, the entropy in the gravitational field isSII = −∂F II∂TwD= 0 .As in the case without a radiation bath (A0 = 0), the pure LDV solution has zero entropybut finite energy U II = F II +TwSII = F II ̸= 0. This may seem curious for a vacuum solution,but is expected since this is energy tied up in the dilaton field.We now compute the free energy of spacetime I. WithD = 4e−2φ0 cosh2 √2 λrw,Tw = T cosh√2 λrw(26)we find the free energy of spacetime I to beF I = −√2 λDTπTwsTwT2−1(27)where T is a constant that we will relate to the temperature of the radiation bath (seeEq.

(22)). Again using S = −∂F∂TwDwe find the entropy to beSI =√2 λDTπT 2wsTwT2−1.

(28)Note that the entropy is positive and that its “asymptotic” value is zero, which is consistentwith the fact that the metric of I has no horizon. The energy U I = F I + TwSI isU I = −√2 λD"TwT2−2#πTwT sTwT2−1.

(29)Thus, an “asymptotic” observer would ascribe a mass to this spacetime,M I =limrw→∞U I −U II= 6√2 λe−2φ0π= 6√2 A0λ(30)15

where we have made use of the fact that e−2φ0 =πA0λ2 .Thus, as in the black hole case(A0 = 0) we again find that the mass of the spacetime is related to the constant term of thedilaton. This reinforces the interpretation of metric I as that of the two-dimensional dilatonblack hole in equilibrium with a radiation bath.

It also concurs with the comparison of thetransit times described in the beginning of this section.The fact that this analysis has yielded an asymptotic mass for a spacetime that asymp-totically has zero entropy seems to be at odds with the expectations that S = M/T.28, 26This is simply due to the fact that the quantities F, M, Tw are really to be thought of ascontravariant (i.e. possessing an upper index) quantities while S and T are really globallydefined (scalar quantities).

Roughtly speaking, if the “asymptotic” region is not simply flatspace, general covariance suggests one should expect a relation rather like S = M/Tw. ThusS may vanish although M is non-zero.A final aside: Were we to repeat this computation of the mass for spacetime III wewould find MIII = −2√2 A0/2λ.

Negative mass solutions are not necessarily excluded sincethe dilaton contributes to the equations of motion in such a way that the dominant energycondition is violated and so the mass of spacetime is not bounded below. For spacetime III,it is related to the fact that this spacetime has a naked singularity.

This identification of themass is also consistent with the taxonomy described earlier, that we really should think of I,II and III as region a), LDV, b) of the extended black hole solution but in a bath of radiation.It is straightforward to go beyond the static ansatz Eq. (4) and study the full linearresponse of the equations of motion.Linearizing Eq.

(6) reveals that the static solutionsare stable only to time-dependent perturbations that vanish spatially at least as Ω4 in theasymptotic region. However, such perturbations seem not to contribute to the asymptotic16

mass, and so do not correspond to throwing mass into the “hole” (neck at r = 0). Time-dependent perturbations larger than this represent spacetimes that are asymptotically notconstant curvature.

Indeed there are two possible scenarios with regard to “throwing mass”into the “hole” in these spacetimes. Either the perturbation causes a shift in the dilaton,thereby increasing the mass of the spacetime or it causes collapse to ensue in which some ofthe radiation energy of the bath falls ,with the perturbation, into the hole, perhaps forminga singularity.

A cursory analysis indicates that the latter possibility occurs, but more workremains to be one. For a related discussion in four dimensions, see Refs.

[27,39].More fundamentally, the study of equilibrium spacetimes including effects like Hawkingradiation should include a discussion of the equilibrium power spectrum of noise in the space-time. The most natural way to do this is by treating the thermal bath quantum mechanicallyand understanding Hawking radiation as dissipation of (gravitational) energy from the geom-etry of the space to the bath.

For a stable, equilibrium spacetime it should be possible tounderstand the coupling between the hole and the bath that is represented by the Hawkingradiation in terms of linear transport coefficients. At this time it seems that to compute lineartransport coefficients one would need a clearer picture (for example, a realization of Hawkingradiation as arising from a Hamiltonian in the bath’s phase-space co-ordinates) of Hawking’seffect than the cursory one presented here.

Nonetheless, the power spectrum of noise couldbe expressed in terms of these transport coefficients. Work is underway to ascertain howcomplete this view of equilibrium is for the spacetimes presented here and their cousins infour dimensions.17

CONCLUSION AND SPECULATIONSWe have found static solutions of two-dimensional dilaton gravity coupled to matter thatmay be interpreted as a two-dimensional black hole at equilibrium in a bath of radiation.We have computed the thermodynamic potentials and identified the mass attributable to theblack hole. We have also included the Hawking radiation’s back-reaction on the metric anddilaton in these solutions.Of course, this toy model of gravitations is somewhat different than higher-dimensionalEinstein–Hilbert gravity.

However, we feel some ideas discussed here may be fruitfully ex-plored in Einstein–Hilbert gravity in four dimensions.We conclude with one intriguing speculation. There are many issues surrounding thefinal stages of black hole evaporation.Information loss and related questions aside, it isvalid to ask simply how would one describe the final stage of the evaporation.

Note that ourspacetime I has two interesting properties: it has no horizon and the “mean density” (∼Mλ)of the “hole” is equal to the energy density of the bath A0. This is essentially due to thefact that in two dimensions Newton’s constant is dimensionless.

Our intuition would suggestthat, for four-dimensional black holes, immersing the black hole in a radiation bath at thesame temperature as that of the hole would make neither the horizon evaporate nor tame thesingularity. This results from the fact that for macroscopic black holes in four dimensions,the radiant energy density at the hole’s temperature is much smaller than the “mean density”of the hole.

As evaporation proceeds, and somewhat before one gets to temperatures of thePlanck scale, the equilibrium radiation energy density outside the hole becomes comparableto the “mean density” of the hole itself. Furthermore, near the endpoint of the evaporation,since the area of the horizon is so minute and the radiation density is so high, any single18

(interacting) quanta would likely see many radiation lengths of other quanta between it andspatial infinity. Thus locally, near the evaporating hole, something akin to a quasi-equilibriumstate (with respect to time scales of O (1/Mp)) may be achieved.

In such a state in analogy tothe solution I presented here, we conjecture the horizon may “evaporate” and the singularitymelt away. It would be interesting to investigate the validity of this “picture” of the finalstage of black hole evaporation in a more realistic model.ACKNOWLEDGEMENTSThe author wishes to thank D. Freed, D. Z. Freedman, M.Ortiz and M. D. Perryfor conversations and assistance and D. Cangemi, M. Leblanc, N. Rius and I. M. Singer forencouragement and support.NOTE ADDEDAfter completion of this project, the author received two papers, Refs.

[40,41] whichdescribe cosmological solutions that are somewhat related to the static spacetimes discussedhere.19

REFERENCES1. K. Bardakci and M. B. Halperin, Phys.

Rev. D3, 2493 (1971).2.

P. Goddard, A. Kent and D. Olive, Phys. Lett.

B152, 88 (1985).3. K. Bardakci, M. Crescimanno and E. Rabinovici, Nucl.

Phys. B344, 344 (1990).4.

G. Mandal, A. Sengupta and S. R. Wadia, Mod. Phys.

Lett. A6, 1685 (1991).5.

S. Elitzur, A. Forge ad E. Rabinovici, Nucl.

Phys. B359, 581 (1991).6.

E. Witten, Phys. Rev.

D44, 314 (1991).7. G. T. Horowitz, USCBTH–92–32, HEPTH/9210119.8.

J. A. Harvey and A. Strominger, EFI-92-41, HEPTH/9209055.9.

S. B. Giddings and A. Strominger, UCSB-TH-92-01, HEPTH/9202004.10. C. Callan, S. B. Giddings, J.

A. Harvey and A. Strominger, Phys. Rev.

D45, R1005(1992).11. B. Birner, S. B. Giddings, J.

A. Harvey and S. Strominger, UCSB-TH-92-08; EFI-92-16;HEPTH/9203042.12. S. W. Hawking, Phys.

Rev. D14, 2460 (1976).13.

S. B. Giddings and W. M. Nelson, UCSBTH-92-15, HEPTH/9204072.14. A. Peet, L. Susskind and L. Thorlacius, Phys.

Rev. D46, 3435 (1992).15.

J. G. Russo, L. Susskind and L. Thorlacius, Phys. Lett.

B292, 13 (1992).16. L. Susskind and L. Thorlacius, Nucl.

Phys. B382, 123 (1992).20

17. A. Bilal and C. Callan, PUPT-1320, HEPTH/9205089.18.

S. P. DeAlwis and J. Lykken, Fermilab-Pub-91/198-7.19. S. P. DeAlwiss, COLO-HEP-284, HEPTH/9206020; COLO-HEP-288, HEPTH/9207095;COLO-HEP-280, HEPTH/9205069.20.

A. Mikovic, QMW/PH/92/16, HEPTH/9211082.21. U. H. Danielsson, CERN-TH.6711/92, HEPTH/9211013.22.

K. Hamada, UT-Komaba 92-9, HEPTH/9210101.23. S. B. Giddings, UCSBTH-92-36, HEPTH/9209113.24.

S. B. Giddings and A. Strominger, UCSBTH-92-28, HEPTH/9207034.25. A Strominger, UCSBTH-92-18, HEPTH/9205028.26.

V. P. Frolov, in Quantum Mechanics in Curved Space-Time, J. Audretsch and V. de Sab-bata, eds. (Plenum Press, NY, 1990), p. 141.27.

G. W. Gibbons and M. J. Perry, Proc. R. Soc.

London A358, 467 (1978).28. G. W. Gibbons and M. J. Perry, HEPTH/9204090.29.

A. Giveon, Mod. Phys.

Lett. A6, 2843 (1991).30.

E. Kiritsis, Mod. Phys.

Lett. A6, 2871 (1991).31.

A. A. Tseytlin, Mod.

Phys. Lett.

A6, 1721 (1991).32. L. F. Abbott and S. Deser, Nucl.

Phys. B195, 76 (1982).33.

G. W. Gibbons and S. W. Hawking, Phys. Rev.

D15, 2738 (1977).21

34. P. C.W.

Davies, S. A. Fulling and W. G. Unruh, Phys. Rev.

D13, 2720 (1976).35. S. M. Christensen and S. A. Fulling, Phys.

Rev. D15, 2088 (1977).36.

G. W. Gibbons and S. W. Hawking, Phys. Rev.

D15, 2752 (1977).37. R. Arnowitt, S. Deser and C. W. Misner, “The Dynamics of General Relativity,” inGravitation: An Introduction to Current Research, L. Witten, ed.

(J. Wiley & Sons,Inc., New York, 1962).38. E. Kiritsis, LPENS-92-30, HEPTH/9211081.39.

D. J. Gross, M. J. Perry and L. G. Yaffe, Phys.

Rev. D25, 330 (1982).40.

F. D. Mazzietelli and J. G. Russo, UTTG-28-92, HEPTH/9211095.41. M. Yoshimura, TU/92/416.22

Figure 1The physical maximally extended spacetime diagrams of I, II and III(There isno physical reason to extend spacetime beyond a strong coupling region. )23

---

출처: arXiv:9302.009 • 원문 보기