24 February 2000

Ž .Physics Letters B 475 2000 33–38

Quantum extreme black holes at finite temperature and exactlysolvable models of 2d dilaton gravity

O.B. Zaslavskii 1

Department of Physics, KharkoÕ Karazin’s National UniÕersity, SÕobody Sq. 4, KharkoÕ, 61077, Ukraine

Received 12 November 1999; accepted 20 January 2000Editor: P.V. Landshoff

Abstract

It is argued that in certain 2d dilaton gravity theories there exist self-consistent solutions of field equations with quantumterms which describe extreme black holes at nonzero temperature. The curvature remains finite on the horizon due tocancellation of thermal divergences in the stress-energy tensor against divergences in the classical part of field equations.The extreme black hole solutions under discussion are due to quantum effects only and do not have classical counterparts.q 2000 Elsevier Science B.V. All rights reserved.

PACS: 04.60.Kz; 04.70.Dy

1. Introduction

w xIn 1 it was argued that thermodynamic reason-ings allow one to ascribe nonzero temperature T toextreme black holes since the Euclidean geometryremains regular irrespectively of the value of T. As aresult, it was conjectured that an extreme black holemay be in thermal equilibrium with ambient radia-tion at any temperature. This conclusion was criti-

w xcized in 2 where it was pointed out that the pre-w xscription of 1 is suitable only in the zero-loop

approximation and does not survive at quantum levelsince the allowance for quantum backreaction leadsto thermal divergences in the stress-energy tensorthat seems to destroy a regular horizon completely,so the demand of regularity of this tensor on the

1 E-mail: aptm@kharkov.ua

horizon seems to enforce the choice Ts0 for ex-treme black holes unambiguously.

The aim of the present paper is to show that indilaton gravity there exists possibility to combine afinite curvature at the horizon with divergences ofthe stress-energy tensor T n. As a result, extremem

black holes with T/0 and finite curvature on thehorizon may exist. Namely, the above mentioneddivergences may under certain conditions be com-pensated by the corresponding divergences in theclassical part of field equations due to derivatives ofthe dilaton field. We stress that the geometries dis-cussed below are self-consistent solutions of fieldequations with backreaction of quantum fields takeninto account. In spite of the geometry itself turns outto be regular in the sense that the curvature measuredfrom outside is finite at the horizon, the solution asthe whole which includes, apart from the metric, the

Ždilaton field as well, is singular. The similar result

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00060-5

( )O.B. ZaslaÕskiirPhysics Letters B 475 2000 33–3834

was obtained quite recently for nonextreme blackw x.holes 3 .

It was observed earlier that ‘‘standard’’ extremeblack holes with Ts0 exhibit some weak diver-

w xgences at the horizon 4 . The phenomenon we arediscussing is qualitatively different. First, these di-vergences are inherent in our case to the components

n w x nT themselves, whereas in 4 T were assumed tom m

be finite and divergences revealed themselves in theenergy measured by a free falling observer which is

Ž 1 0. Ž 0proportional to the ratio T yT rg x and1 0 00

x1are a temporal and spatial coordinates in the.Schwarzschild gauge . Second, divergences consid-

w xered in 4 arise for a generic extreme black holewith Ts0, whereas in our case they are due toT/0 entirely. Third, backreaction of quantum field

w xproduces a singularity at the horizon in 4 , whereasin our case the curvature remains finite there.

It is worth noting that in quantum domain theissue of existence of extreme black holes is non-triv-

w xial by itself. In Ref. 5 it was argued that theexistence of quantum extreme black holes is incon-sistent with field equations at all. However, thisconclusion was derived for the particular class of

w xCGHS-like models 6 , so its region of validity isvery limited, as, in fact, the authors themselves noteat the end of their paper. In the present paper weshow that for a certain class of dilaton gravity theo-

Ž .ries i extreme black holes with quantum correctionsŽ .taken into account do exist; ii the temperature of

quantum fields in their background may be nonzero.We exploit the approach which was elaborated ear-

w xlier in Refs. 7,8 , where it was applied to nonex-treme black holes and semi-infinite throats.

2. General structure of solutions

Consider the action of the dilaton gravity

Is I q I , 1Ž .0 PL

where

12 'I s d x yg F f RŽ .H0 2p M

2qV f =f qU f 2Ž . Ž . Ž . Ž .

w xand the Polyakov-Liouville action 9 incorporatingeffects of Hawking radiation and its backreaction onthe black hole metric can be written as

2k =cŽ .

2 'I sy d x yg qc R . 3Ž .HPL 2p 2M

The function c obeys the equation

IcsR , 4Ž .where Is= = m, ksNr24 is the quantum cou-m

pling parameter, N is number of scalar masslessfields, R is a Riemann curvature. We omit theboundary terms in the action as we are interestedonly in field equations and their solutions.

A generic quantum dilaton-gravity system is notintegrable. However, if the action coefficients obeythe relationship

kvVsv uy , 5Ž .ž /2

where vsU XrU, the system becomes exactly solv-w xable 10,7 . Then static solution are found explicitly

and even for a finite arbitrary temperature T ofw xquantum fields measured at infinity 8 :

ds2 sg ydt 2 qds 2 , gsexp yc q2 y ,Ž . Ž .0

ysls , c s v dfs lnU f qconst.Ž .H0

F Ž0. f s f y 'e2 y yByqC ,Ž . Ž .Bsk 1yT 2rT 2 , T slr2p , 6Ž .Ž .0 0

and the auxiliary function c s c q gs , g s0Ž . Ž0.2l TrT y1 . Here F sFykc , ysls , l0 0's U r2. At right infinity, where spacetime is sup-

posed to be flat, the stress-energy tensor has the form

p Nn ŽPL . 2T s T 1,y1 . 7Ž . Ž .m 6

The coordinate s is related to the Schwarzschild onex, where

ds2 sydt 2 gqgy1dx 2 , 8Ž .by the formula xsHgds . We assume that the func-

Ž .tion v f changes its sign nowhere, so in whatfollows we may safely use the quantity c instead of0

Žf in fact, this can be considered as reparametriza-.tion of the dilaton .

( )O.B. ZaslaÕskiirPhysics Letters B 475 2000 33–38 35

Ž .The solutions 6 include different types of object– nonextreme black holes, ‘‘semi-infinite throats’’and soliton-like solutions, depending on the bound-ary conditions imposed on the function c on the0

w xhorizon 8 . Now we want to elucidate whether theŽ .models 5 include also extreme black holes with a

finite curvature at the horizon and under what condi-tions. Near a horizon the metric of a typical extreme

Žblack hole must have the standard form subscript‘‘h’’ indicates that a quantity is calculated at the

.horizong02gfg xyx f 9Ž . Ž .0 h 2y

or, equivalently,

c f2 yq ln y2 10Ž .0

At the right infinity the function c must obey the0

relation

c f2 y 11Ž .0

that a spacetime have the Minkowski form and, thus,c ™q` in accordance with general properties in-0

w xdicated in 11 .Ž . Ž .Provided Eqs. 9 and 10 are satisfied, the

Hawking temperature

1 dg ldgT s s limH ž /4p dx 4p gdyy™y`h

l 1 dc0s lim 1y s0 12Ž .ž /2p 2 dyy™y`

as it should be for an extreme black hole, the Rie-d22 y1 y1 2mann curvature Rsyl g ln gfy2 g l is2 0dy

Ž .finite. As near the horizon y™y`, f y fyBy,we obtain that the function

B c 20Ž0.F fy c y ln 13Ž .0ž /2 4

at c™y`.Ž . Ž .If Eq. 13 is satisfied, the function c f which0

Ž .is the solution of 6 has the desirable asymptoticŽ .10 and, therefore, the metric function behaves ac-

Ž . Ž .cording to 9 . To achieve the behavior 11 atinfinity, it is sufficient to enforce the condition F Ž0.

fec0 at c™q`.As we are looking for solutions of field equations

which are regular in the region between the horizon

and infinity, we must exclude a possible singularityof curvature. Since R syl2 gy1 d2c rdy2 and0

XŽ . XŽ . Ždc rdys f y rF c prime denotes derivative0 0.with respect to a corresponding argument , the dan-

) Ž0.XŽ ) .gerous point is c where F c s 0 and0 0

dc rdy may diverges. Let B)0. Then dc rdy is0 0

finite in the corresponding point y) , providedXŽ ) .f y s0. It is achieved by the choice Cs

B BŽ0. ) )Ž . Ž .F c q ln y1 'C . If B-0, the function0 2 2Ž0.Ž .F c must be monotonic to ensure the absence of0

singularities. For Bs0, as we will see below, ex-treme black holes do not exist.

Let us consider an example for which all above-mentioned conditions are satisfied:

F Ž0.se2g yBgqC ) , 14Ž .Ž . Ž .where gsg c . Then Eq. 6 has the obvious0Ž .solution: ysg c . If we choose the function g in0

Ž .such a way that the solution of this equation c y0Ž .has the needed asymptotics 10 , we obtain an ex-

Ž .treme black hole in accordance with 9 . It is alsoŽ0.X XŽ .clear that F s0 in the same point where f y s0,

so the condition of absence of singularities is satis-fied.

Let us look now at the behavior of the compo-nents of the stress-energy tensor of quantum fieldnear the horizon. For definiteness, let us choose theT 1 component. It can be written in the Schwarzschild1

Ž w xgauge as see, for example, 2 ; it is assumed that thefield is conformally invariant, our definition of T n ŽPL .

m

differs by sign from the quantum stresses discussedw x.in 2

X 2p g1ŽPL . 2T s T y . 15Ž .1 ž /6 g 4p

Since in our case T/T , the expression in squareH

brackets remains nonzero at the horizon, T 1 ;gy1 ;1Ž .y2 2xyx ;y , so stresses diverge strongly.h

3. Properties of solutions

w xIt was observed in 7 that for black holes de-Ž .scribed exact solutions 6 the Hawking temperature

T slr2p is nonzero constant irrespectively of theH

particular kind of the model that generalizes thew x w xearlier observation 12 for the RST model 13 .

( )O.B. ZaslaÕskiirPhysics Letters B 475 2000 33–3836

Therefore, it would seem that black holes enteringthe class of solutions under discussion may benonextreme only. The reason why, along with nonex-treme holes, the class of solutions contains, neverthe-less, extreme ones too, can be explained in terms of

w x w xthe function c . It was assumed in 12 and 7 thatthis quantity is finite on the horizon to ensure thefiniteness of the Riemann curvature. Then the deriva-tive dcrdy™0 at the horizon, where y™y`, and

Ž .T slr2p in accordance with 12 . However, weH

saw that even notwithstanding c diverges on thehorizon, we can find the solutions with a finite R onthe horizon, provided the function in question has the

Ž .asymptotic 10 . Thus, we gain qualitatively newtypes of solutions due to relaxing the condition ofthe regularity of c on the horizon.

It is worth paying attention to two nontrivialŽ .features of solutions under discussion: i the charac-

Ž .ter of solutions in the classical ks0 and quantumŽ . Ž .k/0 cases is qualitatively different, ii extremeblack holes with a finite curvature at the horizon arepossible even if T/T s0. First, let us consider theH

Ž .point i . Even if Ts0, in the quantum case thecoefficient B/0 and f™` at the horizon, whereas

Ž .in the classical case Bs0 f™Csconst. There-fore, in the classical case the value of the couplingcoefficient F is finite on the horizon: according toŽ . Ž0.6 , F sF sC. As a result, in the vicinity of theh h

horizon c sc plus exponentially small correc-0 0 hŽ .tions, so c y is regular and a nonextreme black0

hole becomes nonextreme, as explained in the prece-dent paragraph. On the other hand, F Ž0. must divergeat the horizon in the quantum case. Thus, we arriveat a rather unexpected conclusion: for exactly solv-

Ž .able models of dilaton gravity 5 extreme blackholes are due to quantum effects only and disappearin the classical case ks0. This fact is insensitive tothe value of temperature, so classical extreme blackholes of the given type are absent even if Ts0. Forthe same reasons quantum extreme black hole areabsent in the exceptional case TsT , when Bs0.0

w xIt was observed in 5 that for a rather wide classof Lagrangians the typical situation is such thatclassical extreme black holes may exist, whereasquantum correction to the action destroy the charac-ter of the solution completely and do not allow theexistence of extreme black holes. In our case, how-ever, the situation is completely opposite: extreme

Žblack holes are possible in the quantum but not.classical case. It goes without saying that our result

does not contradict the existence of extreme blackholes in the classical dilaton gravity theories ingeneral but concerns only the special class of La-

Ž . Žgrangians 5 and their classical limit. It is worthŽ .recalling that the condition 5 was imposed to en-

sure the solvability of the quantum theory, whereasin the classical case every dilaton-gravity model isintegrable, so there is no need in supplementary

Ž .. w xconditions like 5 . The results of either 5 or thepresent paper show clearly that quantum effects notonly may lead to quantum corrections of the classicalmetric but radically change the character of thegeometry and topology.

The most interesting feature of solutions obtainedin the present paper is the possibility to have black

w xholes at T/T s0. In the previous paper 3 weH

showed that nonextreme black holes with T/TH

may exist. The reason in both cases is the same and Irepeat it shortly. The usual argument to reject the

.possibility T/T relies on two facts: 1 this in-H

equality makes the behavior of the stress-energytensor of quantum fields singular in the vicinity of

.the horizon, 2 in turn, such a behavior of thestress-energy tensor is implied to inevitably destroy aregular horizon by strong backreaction. Meanwhile,a new subtlety appears for dilaton gravity as com-pared to the usual case. As, in addition to a metricand quantum fields, there is one more object –dilaton, there exists the possibility that divergencesin the stress-energy tensor are compensated com-pletely by gradients of a dilaton field to give a metric

Ž .regular at the horizon at least, from outside . Andfor some models, provided the conditions describeabove are fulfilled, this situation is indeed realized.

.Thus, dilaton gravity shares the point 1 with general.relativity but the condition 2 may break down.

Ž .It is worthwhile to note that, as follows from 6 ,the temperature which determines the asymptoticvalue of the energy density at infinity, is the functionof the given coefficient B which enters the form of

Ž0.Ž . 'the action coefficient F c : TsT 1yBrk , so0

the solution under discussion has sense for BFk

only. In particular, for ‘‘standard’’ extreme holeswith Ts0 the coefficient Bsk/0. It follows

Ž . Ž . Žfrom 5 , 10 , that near the horizon i.e. in the limit. Ž .y™y` the coefficient V in the action 2 behaves

( )O.B. ZaslaÕskiirPhysics Letters B 475 2000 33–38 37

v 2 Ž .like Vf kyB , so the condition BFk ensures2

Ž .2the right sign of the term with =f .The fact that for a given B we obtain the fixed

value of the temperature is contrasted with that fornonextreme black holes with T/T where the tem-H

perature is not fixed by the form of the action butw xtakes its value within the whole range 3 . This can

be explained as follows. In the extreme case we mustachieve T s0 that entails fine tuning in the asymp-H

Ž .totic behavior of c y that forces us to choose the0

coefficient at the linear part of F Ž0. equal to Bexactly. Meanwhile, in the nonextreme case, wherethe value of T is not fixed beforehand, the corre-H

sponding coefficient is bounded only by two condi-tions which guarantee the existence of the horizonand the finiteness of the curvature. These conditionsresult in two inequalities, so the constraint is less

Ž w x .restrictive see 3 for details .It is worth recalling that Trivedi has found nonan-

alytical behavior of the metric of extremal blackŽ .2 Žholes near the horizon: gfa xyx qa xy1 h 2

.3qd Ž Ž . w x.x see Eq. 27 of 4 . In our case correctionshŽ .to the leading term of 9 also may be nonanalytical,

Ž0.Ž .depending on the properties of the function F c0Ž Ž .for the example 14 all is determined by the choice

Ž ..of the function g c . Moreover, it is readily seen0

that in our case the nonanaliticity may reveal itself inthe leading term of g. Indeed, let the function ghave the asymptotic

dgfg xyx 16Ž . Ž .0 h

with d)2 that is compatible with the extremalitycondition T s0. This case can be handled in theH

Ž . Ž .same manner as for ds2. From 6 and 8 itfollows that now we must have near the horizon theasymptotics

c f2 yqa ln yy , 17Ž . Ž .0

Ž . Ž .with asdr dy1 instead of 10 . One more casearises when gfg eyx , so y;e x, the horizon lies0

Ž .at xs`. Then we obtain the same formula 17with as1.

From the physical viewpoint, the model consid-w xered in 4 represents charged black holes in the

dilaton theory motivated by the reduction from fourdimensions. The corresponding model is not exactlysolvable either due to the presence of the electro-magnetic field or due to the form of the action

coefficients which do not fall into the class of ex-w xactly solvable models 7,10 . On the other hand, our

model is exactly solvable and deals with unchargedblack holes. Their extremality is due to quantumeffects entirely.

The relevant physical object in dilaton gravityincludes not only a metric but both the metric anddilaton field. In the situations analyzed above thecoupling F Ž0. between curvature and dilaton de-scribe inevitably diverges at the horizon as seen from

Ž .Eq. 6 in the limit y™ y` corresponding to thehorizon. Therefore, although the curvature is finiteon the horizon, the solution as the whole exhibitsingular behavior. Moreover, even the metric part ofthe solution can be called ‘‘regular’’ only in the veryrestricted sense: the region inside the horizon hardlyhas a physical meaning at all since an outer observercannot penetrate it because of strong divergences inthe stress-energy tensor on the horizon surface, sothe manifold is geodesically incomplete. All thesefeatures tell us that in fact we deal with the class ofobjects which occupies the intermediate positionsbetween ‘‘standard’’ regular black holes and nakedsingularities. In the previous paper we described thenonextreme type of such objects, in the present paperwe found its extreme version.

4. Summary

We have proved that extreme black holes do existin exactly solvable models of dilaton gravity withquantum corrections taken into account. It turned outthat they are contained in general solutions of ex-actly solvable models. The existence of extremeblack holes depends strongly on the behavior of the

Ž Ž ..action coefficients near the horizon see Eq. 13and is insensitive to their concrete form far from itwhere it is only assumed that their dependence onthe dilaton field ensures the absence of singularities.

We found solutions which shares features of bothblack holes and naked singularities and represent‘‘singularities without singularities’’. We showed thatquantum fields propagating in the background ofextreme black holes may nonzero temperature atinfinity. In fact, this means that such fields cannot becalled Hawking radiation since the geometry does

( )O.B. ZaslaÕskiirPhysics Letters B 475 2000 33–3838

not enforce the value of the temperature. Theseproperties of solutions with T/T are the same asH

w xin the nonextreme case 3 . The qualitatively newfeature inherent to extreme black holes consists inthat the type of solutions we deal with is verysensitive to quantum effects: taking the classicallimit destroys completely our solutions with T/TH

s0, so the solutions under discussion are due toŽquantum effects only. In contrast to it, solutions

describing nonextreme black holes with T/T con-Hw x .sidered in 3 have sense in the classical limit.

The separate issue which deserves separate con-sideration is what value of entropy should be as-cribed to the objects we found.

Acknowledgements

I am grateful to Sergey Solodukhin for valuablecomments.

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