PHYSICAL REVIEW D, VOLUME 61, 064002

Two-dimensional dilaton gravity and spacetimes with finite curvature at the horizonaway from the Hawking temperature

O. B. Zaslavskii*Department of Physics, Kharkov State University, Svobody Square 4, Kharkov 310077, Ukraine

~Received 17 May 1999; published 16 February 2000!

It is shown that static solutions with a finite curvature at the horizon may exist in dilaton gravity attemperaturesTÞTH ~includingT50) whereTH is the Hawking temperature. Hawking radiation is absent andthe state of a system represents thermal excitation over the Boulware vacuum. The horizon remains unattain-able for an observer because of the thermal divergences in the stress energy of the quantum fields there.However, the curvature at the horizon is finite, when measured from outside, since these divergences arecompensated by those in the gradients of the dilaton field. The spacetimes under consideration are geodesicallyincomplete and the coupling between the dilaton and gravity diverges at the horizon, so we have a ‘‘singularitywithout a singularity.’’

PACS number~s!: 04.70.Dy, 04.60.Kz

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In recent years great interest has been focused ondimensional~2D! theories of dilaton gravity. One of the reasons for this consists in the possibility of tracing in detail tprocess of black hole evaporation since on the semiclaslevel spacetime evolution is described by differential eqtions following directly from the Lagrangian even if quatum effects are taken into account@1#. It turned out that sucha kind of theory possesses a rather rich set of exactly sable models. These circumstances give us hope to get ininto subtle features of black hole evolution. Especially iportant is the question of whether or not the final geomecan be regular. In Ref.@2# Bose, Parker, and Peleg~BPP!proposed an exactly solvable model in which, under a cerchoice of parameters, after evaporation a black hole leaan everywhere regular geodesically complete geomwhich infinitely extends to a region of strong couplin~‘‘semi-infinite’’ throat!. In so doing, quantum fields are inradiationless state in which the effects of vacuum polarition tend to zero at Minkowski infinity. In Ref.@3# Cruz andNavarro-Salas~CN! suggested a more general model intpolating between the BPP and Russo-Susskind-Thorla~RST! @4# ones. The qualitatively new feature of the Cmodel consists in that, although a static radiationless geetry with curvature finite everywhere is still possible, it bcomes geodesically incomplete.

Meanwhile, the CN model possesses some other intring features which were not noticed in@3#. Namely, we willshow below that at the boundary of spacetimeg00→0, so itrepresents a Killing horizon. At this point stresses of qutum fields due to back reaction diverge since the solutunder discussion is radiationless~temperatureT50) whereasthe usual condition of their finiteness demandsT5TH whereTH is the Hawking temperature~see, for instance,@5#!. Thus,the intimate connection between regularity of a geometrya horizon and equality of temperaturesT5TH ceases to exisand we have a static geometry with a finite curvature athorizon atTÞTH and the infinite quantum stresses. In th

*Email address: aptm@kharkov.ua

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respect the significance of the CN model is beyond the ctext of concrete problems of 2D dilaton gravity: relative simplicity of the 2D case enables us in the best possible wareview unbiasedly some fundamentals of black hole physand elucidate opportunities which probably may occur in4D world but remain hidden because of complexity of tsituation in the latter case.

In the present paper we argue also that there exiswhole range of temperatures for which properties of spatime sketched above hold true. We exploit exactly solvamodels of 2D gravity@6,7# ~with respect to which the CNone is the particular case! and show that the results remavalid for a whole class of such theories provided the copling between gravitation and dilaton obeys some genrestrictions.

Let us consider the system described by the action

I 5I 01I PL ~1!

where

I 051

2pEMd2xA2g@F~f!R1V~f!~¹f!21U~f!#,

~2!

I PL is the Polyakov-Liouville action@8# incorporating effectsof Hawking radiation and its back reaction on the bacground metric for a multiplet of N scalar fields, and thboundary terms omitted. In what follows we will use thquantitiesTmn defined according to

dI 51

2E d2xAugudgmnTmn , ~3!

so the field equations which are obtained by varying a mehave the formTmn50.

In the conformal gauge,

ds252e2rdx1dx2 , ~4!

the Polyakov-Liouville action reads I PL52(2k/p)*d2x]1r]2r5(k/p)*d2xAg(¹r)2. After integration by

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O. B. ZASLAVSKII PHYSICAL REVIEW D 61 064002

parts the action~1! is reduced~up to boundary terms! to I5(1/2p)*d2xAg@V(¹f)212¹r¹F12k(¹r)214l2eh#,where by definitionh5*dfv, v5U8/U. In what followswe restrict ourselves by exactly solvable models of dilagravity @6,7# that implies the constraint on the form of thpotential V5v(u2kv/2)1g(u2kv)2 where u5F8. Inwhat follows we consider only the simplest choiceg50, so

V5vS u2kv

2 D . ~5!

Introducing new fieldsV andx instead off andr accordingto F̃[F2kh52kV, h52(x2V2r), after simple rear-rangement we obtainI 5(1/p)*d2xAg$k@(¹x)22(¹V)2#12l2e2(x2V2r)%. Corresponding equations of motion hathe form 2k]1]2V52l2e2(x2V), 2k]1]2x52l2e2(x2V). One can choose the gaugex5V whence]1]2F̃52l2. This equation should be supplemented byconstraint equationsT115T2250 @Tmn52(dI /dgmn)#.The expressions for the classical parts ofT11 andT22 fol-low directly from the general expression for covariant coponents

Tmn(0)5

1

2p$2~gmnhF2¹m¹nF !2Ugmn12V¹mf¹nf

2gmnV~¹f!2%, ~6!

which can be obtained by varying the actionI 0. The formulafor the quantum contribution can be obtained from the cservation law and conformal anomaly and has the form

T66(PL)5

2k

p@~]6r!22]6

2 r1t6# ~7!

where the functiont6(x6) are determined by the boundaconditions. From Eqs.~6! and ~7! we have the equation]6

2 F̃52kt6 . To find the explicit form oft6 , let us imposethe condition that at the right infinity quantum fields shoube at a finite temperatureT. Then in asymptotically flat co-ordinatess65t6s connected withx6 according tolx6

56e6lx6 stresses take the formT66(s) 52pT2/12 at s

→` whereds252ds1ds2 . On the other hand, asymptotcally 2r'2 ln(2l2x1x2) to match the region of a lineadilaton vacuumf52sl. Substituting this into Eq.~7! andperforming transformation of tensor components betwtwo coordinate systems, we havet65 1

4 x622@12(T2/T0

2)#.

Then, integrating the equations of motion forF̃ we obtain

ds25g~2dt21ds2!, g5exp~2r12y!,

y5ls, 2r52E vdf, ~8!

F̃~f!5 f ~y![e2y2By1C, B5k~12T2/T02!, T05l/2p.

The expressions~8! describe generic static configurationgravitational and dilaton fields in exactly solvable modelsdilaton gravity at finite temperatures. Hereafter we conc

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trate ourselves onF̃(f) such thatF̃(f) has one simple mini-mum at some realf0 where F̃8(f0)50. The well knownexample is the RST model@4#. Let the coefficientB.0. Thestructure of spacetime depends crucially on the relationsbetweenf min5f(y0) @the minimum value off (y) achieved inthe point y0] and F̃(f0). It follows from Eq. ~8! thatdf/dy5 f 8(y)/F̃8(f). If f min, F̃(f0), the dilaton valuechanges monotonically fromf52` at right infinity y5`to f5f0 where the spacetime is singular and cannotcontinued further. For the RST model the structure of spatime at the equilibrium temperatureT5T0 is analyzed indetail in @9# ~see also generalization in@7#!. If f min.F̃(f0),the dilaton field takes its values in the limits (2`, f1)wheref1,f0 , F̃8(f) changes its sign nowhere.

In what follows we dwell upon the special caseF̃(f0)5 f min , soC5C* where

C* 5F̃~f0!1C0 , C052B

2 S 11 ln2

BD . ~9!

Then f 8(y) and F̃8(f) turn into zero simultaneously andf/dy does not changes its sign, so the dependencef(y) ismonotonic. As a result,f→` when y→2`, i.e. at theevent horizon. In the vicinity of the pointy0 one can use thepower expansion, so the equalityF̃(f)5 f (y) turnsinto @ F̃9(f0)/2#(f2f0)21@ F̃-(f0)3/6#(f2f0)31•••

5@ f 9(y0)/2#(y2y0)21@ f-(y0)/6#(y2y0)31••• whence itis clear that the Riemann curvatureR52l2g21(d/dy)@g21(dg/dy)#522l2(d2f/dy2)e22f22y is finite inthe pointy0.

Let us consider the concrete example. We assume thafunctions f (y) and F̃(f) obey the conditions describeabove which guarantee the absence of the singularity outthe horizon. We do not specify the exact form ofF̃(f) in thewhole region and only assume that in the vicinity of thorizon it reads

F̃'e22f1bf, f→`. ~10!

The condition~5! of exact solvability implies thatV'(b22e22f)v1kv2/2 near the horizon. We find that, ay→2`,

f→`, g;e2uyu(B/b21), R;e22uyu(2B/b21). ~11!

Thus, if B/b,1 the surfacey52` represents a horizonwhere g50. In the Schwarzschild gaugex5*dsg, ds2

52gdt21g21dx2 we have a usual behaviorg;x2xh nearthe horizon located atx5xh . If 2B/b>1, the Riemann cur-vature on this surface is finite. It is remarkable that boconditions are consistent with each other. Thus, if1

2 <B/b,1 or, equivalently,

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TWO-DIMENSIONAL DILATON GRAVITY AND . . . PHYSICAL REVIEW D 61 064002

we have a black hole with the Riemann curvature finiteerywhere including the horizon. This inequality is not satfied by T5T0 when, as is well known, a horizon is regul~the usual Hartle-Hawking state!. However, there is no contradiction here since atT5T0 the coefficientB50 andasymptotic behavior of such a solution@7,9# has nothing todo with Eq. ~11!. Thus, according to Eq.~12!, the Hartle-Hawking state is not included in the set of states undercussion and there are two possibilities to have the horiwith a finite curvature which cannot match continuously:T5T0 ~the isolated point! or T obeys the inequality~12! ~thewhole range of temperatures!.

In the particular case of the RST modelb5k, so Eq.~12!is reduced, according to Eq.~8!, to the condition

0,T<T0

A2. ~13!

The caseT50 is now trivial since for solutions under discussionC50 according to Eq.~9!, Eq. ~8! gives use22f

1kf5e2y2ky and, with the condition of asymptotical flanessf(`)52`, we have a linear dilaton vacuum solutiof52y with a flat spacetime. Therefore, for the RST moda horizon with a finite curvature may exist only at nonzetemperature. However, for a generick,b<2k it is possiblefor such a horizon to exist even atT50.

The caseT50 is tractable for more detailed investigatioLet us consider the example

F̃5e2g(f)2kg~f!. ~14!

Then, taking into account that for this model the quantityC*depends on temperature according toC* 5C0(0)2C0(T),so C* 50 at T50, we find from Eq.~8! the solutiony5g(f). Let us choose, for instance,

g52f21

2ln~11e2f!. ~15!

Theng5(11e2f)21. After simple calculations we find theRiemann curvatureR54(12g)(g22)23. At the horizonf5`, g50 the curvature remains finite in spite of quantustresses diverge there. For instance,Ty

y(PL)52(l2/24)(11 1

2 e22f)22g21.Thus, we arrive at a rather surprising conclusion: spa

times with a finite curvature at the horizon may exist atemperature different from the Hawking one and, moreovtheir temperature may be equal to zero. This conclusiothe main result of the present paper.

What is the physical reason for such an unusual behavIt is instructive to look at fields equationsTm

(0)n1Tm(PL)n50.

The classical part of the effective stress-energy tensorTm(0)n

contains terms with gradients of the dilaton fieldf. For in-stance,Tx

(0)x5(2p)21@V(¹f)22U12¹0¹0F#. A simpleestimate based on Eqs.~10!, ~11! shows that near the horizowhere y→2` and f'2(B/b)y, Tx

(0)x;g21→`. On theother hand, in the Schwarzschild coordinates we hTx

(PL)x52(p/6)@T22(1/4p)(dg/dx)2#g21;g21→` if T

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ÞTH @5#. Thus, each part ofTmn diverges separately and

regular metric near the horizon arises as a result of mucompensation of these divergences. Meanwhile, the uproof of the fact thatT5TH relies strongly upon the regularity of the stress-energy tensor of quantum field at therizon. The corresponding criteria@5# appeal to the behavioof Tm

(PL)n but do not take into account properties ofTm(0)n ,

i.e., they were considered in isolation from the dynamic cotents of the theory whose solution a given metric represeUsually, such an approach is justified since a classical pafield equation is regular, as was tacitly assumed in@5#, butthe present case is exceptional in that either quantum or csical contributions to field equations diverge near the hozon.

Thus, the curvature may be finite everywhere even whthe stress-energy tensor describing the back reaction of qtum fields diverges on the horizon. The situation is sharcontrasted with that in general relativity where the streenergy tensor singular at a horizon is inconsistent withregularity of a metric and of the Einstein tensor. The reasconsists in that now we have, apart from a metric, one mclassical field—the dilaton one coupled to a metric. Graents of this field compensate the divergencies due to breaction of quantum fields.

It can be readily seen from the above formulas thatproper distance between the horizon and any other poil5l*2`

y dyAg is finite. Therefore, the spacetime region ouside the horizon is geodesically incomplete. The dilaton fif cannot be continued further across the horizon sincefdiverges there. The situation with continuation across sucsurface leads, in general, to complex dilaton values@10#. Toavoid complex dilaton field, one may try to redefine thelaton field choosing, say,x(f) instead off where x is aSchwarzschild-like coordinate. However, any redefiniti

cannot abolish the fact that the quantityF̃ describing thecoupling between the dilaton and curvature diverges athorizon. There is more deep reason to believe that anytempts of continuation across the horizon for our solutioshould be rejected. The point is that any observer approaing the horizon from outside moves along a nongeodepath and perceives fluxes of quantum field which becoinfinite at the horizon. Therefore, the horizon remains in funaccessible. The surfacef5` acts as a boundary of spactime and the usual criteria of geodesic completeness is phcally irrelevant for the solution under discussion. Of cournothing prevents making a formal analytic continuationthe metric itself but, if the dynamic contents of the theo~interaction between gravitational, dilaton and quantum slar fields! is taken into account, the region inside a horizseems not to have a physical meaning for the modequestion—at least, if one starts from the outside regiThus, actually the loss of information inside the horizonmuch more severe than in a usual case where at leasobserver brave enough to dive into a black hole can getformation about the region inside the horizon. In fact tnotion of an event horizon changes its ordinary meansince there are no events inside a horizon at all.

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O. B. ZASLAVSKII PHYSICAL REVIEW D 61 064002

It is seen from Eq.~10! that F̃ diverges at the horizonwheref5` and so does the total entropy including the cotribution of quantum fields which for exactly solvable moels is proportional~up to a constant! to the horizon value ofthis function@7#. These divergences have the same naturthose in the Schwarzschild background where entropyquantum fields is infinite ifTÞTH . The difference betweenthese two situations manifests itself, however, in that foblack-hole metric in general relativity disparity between twtemperatures makes a horizon singular and in fact destrocompletely whereas in our case the horizon remains regin the sense that the Riemann curvature remains finite th

The features of spacetime discussed above are sharewhole classes of exactly solvable models of dilaton gravfor which ~i! F̃8(f) has a simple zero at somef0, ~ii ! F̃;f at f→`, ~iii ! the constant in the solution is chosenC5C* in accordance with Eq.~9!. These features, howevehave nothing to do, for example, with the BPP modelwhich F̃5e22f and the choiceC5C* leads to a regularsemi-infinite ‘‘throat’’ @2# ~generalization to a finite temperature is considered in@11#!. The fact that unusual properties othe solution under discussion are intimately connected wquantum terms in the action reveals itself in the Eq.~8! andits classical limit directly. Indeed, it follows from Eqs.~8!,~10! that in the limitk50 the pointf0 whereF̃850 movesto f05`, the coefficientsB505C* , b;k50 and Eq.~8!

with C5C* turns into F̃5e22f5e2y whose solution is alinear dilaton vacuumf52y.

Usually, the temperature of Hawking radiation in a blahole background is determined by characteristics of a sp

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time ~say, an event horizon radius of a Schwarzschild ho!.The fact that temperature may be arbitrary actually methat Hawking radiation as such is absent, so we deal wthermal excitation over the Boulware state which cantreated at any temperature. In particular, the choiceT50corresponds to the Boulware vacuum. Therefore, the typsolution considered in the present paper might shed lighthe fate of a black hole after evaporation, being a candidon the role of a ‘‘remnant.’’ This, however, needs furthtreatment based on dynamic scenarios and is beyondscope of the present paper.

While usually the notion of singularity implies that it igeometry which exhibits singular behavior, in our casevergencies manifest themselves in the dynamic characttics ~components of the stress-energy tensor! and in the cou-pling between the dilaton field and curvature. Thedivergences along with the geodesic incompleteness ofmetric in fact mean that the solution in question is singularspite of the metric itself being regular—at least, in thestricted sense: as one approaches the horizon from outthe Riemann curvature remains finite. So, we have ‘‘sinlarity without singularity.’’

In fact, we are faced with a qualitatively new typeobjects in gravitation which occupies an intermediate plabetween regular black holes and naked singularities. It isinterest to elucidate whether coupling between gravitatand dilaton or other classical fields can produce the satype of solutions in the four-dimensional case.

I am grateful to Sergey Solodukhin for fruitful discussioand to Alessandro Fabbri for helpful correspondence.

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3444 ~1992!; 47, 533 ~1992!.@5# D. J. Lorantz, W. A. Hiscock, and P. R. Anderson, Phys. R

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4265 ~1995!.@7# O. B. Zaslavskii, Phys. Rev. D59, 084013~1999!.@8# A. M. Polyakov, Phys. Lett.103B, 207 ~1981!.@9# S. N. Solodukhin, Phys. Rev. D53, 824 ~1996!.

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