PHYSICAL REVIEW D VOLUME 47, NUMBER 10 15 MAY 1993

Semiclassical extremal black holes

Sandip P. TrivediLauritsen Laboratory ofHigh Energy Physics, California Institute of Technology, Pasadena, California 9I125

(Received 13 November 1992)

Extremal black holes are studied in a two-dimensional model motivated by a dimensional reductionfrom four dimensions. Their quantum-corrected geometry is calculated semiclassically and a mild singu-larity is shown to appear at the horizon. Extensions of the geometry past the horizon are not unique butthere are continuations free from malevolent singularities. A few comments are made about therelevance of these results to four dimensions and to the study of black hole entropy and information loss.

PACS number(s): 97.60.Lf, 04.60.+n, 11.17.+y

I. INTRODUCTION

Hawking's discovery of black hole radiance [1,2] raisesseveral intriguing questions. It shows that black holeshave an entropy which can be elegantly expressed interms of their geometry but which remains mysterious interms of any underlying microstates. It also suggests thatbecause of the thermal nature of the outgoing radiation, aloss of quantum coherence might occur in processes in-volving black holes. Extremal black holes provide a con-venient setting in which to address both these issues.Their zero temperature suggests that their entropyshould be explained in terms of a degeneracy of groundstates. It also makes them convenient toy laboratories inwhich to study scattering and a potential loss of quantumcoherence.

In this paper we study a model consisting of dimen-sionally reduced gravity and electromagnetism coupled totwo-dimensional scalar fields. Classically, this model hasReissner-Nordstrom black hole solutions, obtained fromdimensionally reducing the usual four-dimensionalcharged black hole solutions. In this paper we concen-trate for the most part on the extremal black holes whichin Planck units have a mass equal to their charge andhave zero Hawking temperature. We show that, contraryto expectations, the quantum stress tensor of a scalar fieldin the background of such an extremal black hole blowsup at the horizon. This raises the possibility of theirgeometry being drastically modified in the vicinity of thehorizon and their entropy being very different from whatclassical considerations would suggest. A semiclassicalcalculation of their quantum-corrected geometry shows,however, that this is not true. For large black holes, thevalue of the dilaton at the horizon and, hence, their en-tropy, ' stays large. A singularity does appear at the hor-izon, but it is very mild. For example, tidal forces andthe curvature stay finite at the horizon. This suggeststhat there should be a continuation of the geometry pastthe horizon. In fact, there is more than one such con-

The entropy of these black holes depends on the dilaton andis large if the value of the dilaton at the horizon is large.

II. THE MODEL

We start in four dimensions and make a spherical sym-metric ansatz for the metric, which gives

c& =g pdx dx +e dQ

Here g & is the two-dimensional metric in the r-t planeand e ~, which we call the dilaton, is the square of theradius of the two-sphere. The Einstein-Hilbert actionthen takes the form

SG = Id x &g e ~[R +2(VQ) +2e~&] . (2)

We note that this differs from the action considered byCGHS [10] in the form of the dilaton potential, i.e., thelast term above. The term considered here prevents theaction SG from scaling simply under the transformationP—+/+ c.

tinuation, even when we restrict ourselves to static solu-tions. In one of these, the causal structure of spacetime ismuch like the classical extremal solution. But there isanother continuation possible in which the causal struc-ture is much different and in which there are nomalevolent singularities. We conclude with a brief dis-cussion of the relevance of our results to the study offour-dimensional extremal black holes and to the study ofblack hole entropy and information loss.

The study of quantum effects in two-dimensional blackholes was first undertaken in the 1970s in some very in-teresting papers which include Refs. [3—7]. More recent-ly, considerable interest was renewed by the discovery ofa two-dimensional black hole solution in the work ofMandal, Sengupta, and Wadia [8], and Witten [9]. Thissolution was then used to study questions related toHawking evaporation in the work of Callan, Giddings,Harvey, and Strominger (CGHS) [10]. Two recent paperswhich review the subsequent developments are Refs. [11,12]. Papers especially relevant to the work presentedhere include Refs. [13—18]. Papers which discuss dimen-sionally reduced models include Refs. [19—21]. To ourknowledge, the first published reference to the idea of us-ing extremal black holes for studying issues related toHawking radiation is in the paper of Preskill et al. [22].

4233 1993 The American Physical Society

4234 SANDIP P. TRIVEDI 47

Proceeding similarly, the Maxwell field can be dimen-sionally reduced to give an action

S = — d x&ge ~I'EM 4G(3)

(4)

and a dilaton—2P 2

where F now refers to the field strength of a two-dimensional gauge field.

Spherically symmetric charged black hole solutions ofthe original four-dimensional theory continue to be solu-tions of this theory. They are given by a metric

2

ds = — 1 — + dt+ drr 1 2M/—r+Q /r

III. QUANTUM STRESS TENSOR

For a black hole with a mass much bigger than thePlanck mass, the curvature, as measured, for example, bya nonvanishing invariant such as R & &R ~~, is small allthe way from infinity to the horizon. Thus we would ex-pect quantum effects associated with curved spacetime tobe small in this region. In fact, this expectation is notmet for an extremal black hole. As we show below, thequantum stress tensor of a massless scalar field divergesat the horizon of an extremal black hole no matter howlarge its mass.

We work in Schwarzschild gauge where the metric isgiven by

ds = fdt —+ dr—2

The conservation equations for the stress tensor then im-ply that [23]

The corresponding field strength isT] —c]

and that

(9)

Here M is the mass and Q the charge of the black hole.We will be mainly interested here in extremal black holesfor which M=Q.

In order to incorporate quantum effects in a manage-able way, we couple the above theory to N conformalycoupled scalar fields in two dimensions. In total, the ac-tion of the two-dimensional theory is then

d2 —2$R + —2/2 q 2+2 ~24G 4

T„"= I f 'T„"dr + (10)

where rh refers to the position of the horizon. An ambi-guity in state of the scalar field is rejected in the two ar-bitrary constants c& and c2. Consider now an observerfreely falling into the black hole with a four-velocity(po/f, —(po —f )' ). She sees an energy density

2 2

T U"U =T" —1 —T' —2T" ( f)'—Pv P f t f t fQ PO

The parameter N allows us to consider the theory in thelarge-N limit in which N~~ and A~O while keepingNfz fixed. This allows us to systematically incorporatethe quantum effects due to scalar loops, which go as Nfi,while keeping the other fields classical. The scalar fieldsare taken to be conformally coupled in the interest oftractability, but, as a result, the model we consider herewith electrically charged black holes does not retain anobvious four-dimensional interpretation. However, allour conclusions will go through, unchanged, for a closelyrelated model obtained by dimensionally reducing a sys-tem consisting of fermions coupled to a magneticallycharged black hole in four dimensions. The scalar fieldswill then correspond to the bosonized version of theCallan-Rubakov modes of the fermions [13,17]. Weshould add, though, that from a strictly four-dimensionalpoint of view, even in this model, the other modes of thefermion fields will contribute to the quantum stress tensorand their neglect cannot be justified.

2I would like to thank A. Strominger for pointing this out tome.

Substituting from Eqs. (9) and (10), we see that

T U"U =pv

2(c2 ci )po + —T"+— f 'T"drf2 IJ f p„P

j f'TI'dr, (12)

T"=—24~ '

and 1'Hopital's rule, we see that, close to the horizon,

(13)

T„U"U =constX (14)

And this diverges since for an extremal black hole f has adouble zero at the horizon. In other words, if 6~ is theproper time taken to reach the horizon, the energy densi-ty diverges like

T„U"U —1/6w .

We note that the analysis above was very general

where f ' refers to the derivative off with respect to r.This shows that the leading divergence goes as

[(c2—c, )/f ]po. So we restrict ourselves to states inwhich c2 =c, . Then, using the trace anomaly

SEMICLASSICAL EXTREMAL BLACK HOLES 4235

without restrictions to any particular state of the scalarfield. In fact, for these black holes, the Boulware, Unruh,and Hartle-Hawking states are the same and considera-tions similar to those above at the past horizon wouldsingle it out as having the minimal divergence.

This divergence can be better understood by regardingan extremal black hole as the limit of a nonextremal one.A nonextremal black hole has an outer and an inner hor-izon, and these come together in the extremal limit. Letus take rh in Eq. (10) to refer to the outer horizon. Then,as before, we see that c, must equal c2 for the leadingdivergence to vanish at the outer horizon. Furthermore,Eq. (14) shows that the stress tensor stays finite at theouter horizon, since f has a single zero. Now let us focuson the inner horizon. Since rh in Eq. (10) was taken tomean the outer horizon and c, =c2, we see from Eq. (12)that the leading divergence goes as

pO If' rinner f' router

f2(16)

And this diverges since the quantity within square brack-ets does not cancel and f is zero, albeit a simple zero, atthe inner horizon. If 5w is the proper time taken by afreely falling observer to reach the horizon, this impliesthat

IV. QUANTUM-CORRECTED GEOMETRY

We turn now to calculating the quantum-correctedgeometry of the extremal black hole. We expect the con-formal gauge to be most convenient for this purpose sincein this gauge the quantum stress tensor of the scalar fieldscan be expressed as a local function of the metric. Butthe extremal solution is static and the related Killing vec-tor is most easily expressed in the Schwarzschild gauge.It is in this gauge also that the classical metric is mosteasily described, whereas its description in the conformalgauge is at best implicit. Our strategy will therefore be toderive the equations in the conforrnal gauge, but then todo a coordinate transformation and express them in theSchwarzschild gauge. The conformal coordinates we usefor this purpose are defined as follows. We first start withthe Schwarzschild coordinates

ds = —f(r)dt + dr1

f(r)

constP» (g )2

In summary, we find that if we adjust the quantumstate of the scalar field so that the stress tensor is finite atthe outer horizon, it diverges at the inner horizon. It isperhaps not so surprising then that in the extremal casewhen the two horizons come together the divergence per-sists, although in a softened form.

ds = —f(r )dv du, (19)

where

U=t+r(20)

are the required conformal coordinates. To illustratewhat we mean by a change of coordinates, we considerthe T„constraint equation. This is given as

t), e t — t3,f—r3, e t +—e ~B,e ~t), e

[f1/2t)2f —1/2+ t ( v ) ] (2 1 )U

f (e &)"———1 2 2 „1(e )'

4 —2$

4 2ff" (f')' +—c—. (23)

Here the primes denote derivatives with respect to r, andsince the dilaton and metric are functions of r alone, wehave replaced t+(v) by a constant C. Proceeding similar-ly, the equation of motion of the Maxwell field gives

2 —4p(24)

The equation obtained by varying the trace of the metricthen becomes

[f(e &)']' —2+2 = gf" . —(25)

The equation obtained by varying the dilaton is

[(—2P)']2 f(e

—2P)~ QQ2

(e—2$)2 —2P (e

—2P)2(26)

Finally, the T„„constraints gives back Eq. (23).We expect extremal black holes to have zero tempera-

ture and seek solutions with C=0 and f'=0 at the hor-izon (where f=0). If x represents the coordinate dis-tance from the horizon, this suggests that, close to thehorizon,

Here t+ (v) is an arbitrary function of v, which representsthe ambiguity in the quantum state of the scalar fields,and

NA'

12. -

Now, since the solution is static, we convert, by using Eq.(20), the derivatives with respect to v to those withrespect to r. This gives

and then define a new coordinate r + as

(18)and

f=a,x'+a,x'+'

e '&=d„+d,x'+'.

(27)

(28)

The metric then becomes Equations (25) and (26) then show that

4236 SANDIP P. TRIVEDI 47

and that

d2

0+dh

1

4+de(30)

Further, Eq. (25) then shows that

d2=&z 6+2o;, 5

(31)

Finally, Eq. (26) shows that 5 is given by the equation

—3++9+24//(dp, —g)5=-2

These values of the parameters can also be shown to beconsistent with Eq. (23) (with C set equal to 0). Like theirclassical counterparts, these solutions have only one freeparameter, which we can take to be d&, the value of thedilaton at the horizon. ~a2~ can be set equal to 1 by re-scaling x. The other parameters are then uniquely deter-mined. Numerical computations show that with +2= —1

the solution evolves to an asymptotically Aat geometry asx~ ao. As a special case of Eq. (32), note that for largeblack holes, where dh ))g, 6=2//dh.

V. DISCUSSION

What do we learn from these solutions? Extremalblack holes are dangerously close to becoming nakedsingularities. And we might have thought that thediverging stress tensor would cause a singularity to ap-pear at the horizon and even drive the dilaton to a smallvalue, thereby changing the entropy of the black holedramatically. However, this does not happen. The valueof the dilaton field at the horizon remains a free parame-ter and for large black holes (dh ))g), the entropy stayslarge and remains as mysterious as ever.

For generic values of dI„5 is not an integer and thesolution is nonanalytic in x. This nonanalyticity impliesa very mild singularity at the horizon. The secondderivative of the curvature as seen by a freely falling ob-server diverges as she crosses the horizon, but the tidalforces and the curvature stay finite. Thus there should bean extension of the geometry past the horizon. In fact,there are two obvious static extensions. These corre-spond to replacing x + in Eq. (27) above by x + or byx ~x~ and correspondingly extending the dilaton field.We will call these the even and odd extensions, respec-tively. It can be shown that both of these satisfy thejunction conditions which arise from Eqs. (23), (25), and(26). The odd continuation results in a spacetime muchlike the classical extreme black hole spacetime in which

FIG. 1. Penrose diagram of the odd extension.

we can hit a timelike singularity within a finite propertime of crossing the horizon. The even continuation,though, results in an entirely different spacetime, inwhich we pass from one asymptotically Aat universe toanother without encountering any malevolent singulari-ties at all. The corresponding Penrose diagrams areshown in Figs. 1 and 2, respectively.

We have not investigated, as yet, whether any of theseextensions are relevant for a black hole formed from col-lapse; but it is not inconceivable that at least some part ofspacetime outside a collapsing "star" is described by ei-ther of these. In this case it should be possible to beginthe collapse from one asymptotically Aat universe andopen out into another. Information thrown in from oneasymptotically Bat universe could then end up in another.Indeed, such black holes would be the "ultimate" rem-nants [10,17,25]. Information would not just be hiding insome long tube, but would have found its way into anoth-er universe and be lost forever. This model might alsofurnish a simple context in which to explore the produc-tion of such remnants in external fields.

We cannot say much about four-dimensional blackholes, since we do not know how the stress tensor of afour-dimensional scalar field behaves. If we do interpret

)0

(0

FIG. 2. Penrose diagram of the even extension.

For special values of dz (within the semiclassical regime), 5becomes an integer and depending on its value either of thesecan become the analytic continuation, in which even the mildsingularity disappears.

4A spacetime with the same causal properties has been foundby Horne and Horowitz [24].

sThese calculations are currently in progress [26].

47 SEMICLASSICAL EXTREMAL BLACK HOLES 4237

the two-dimensional metric and the dilaton as com-ponents of a spherically symmetric four-dimensionalmetric, we find that the curvature and hence componentsof the tidal force felt by a freely falling observer blow upat the horizon. However, the divergence is quite mildand the tidal impulse stays finite, which suggests thatthere should be an extension of the geometry past thehorizon. If the four-dimensional stress tensor behavessimilarly to the two-dimensional case and blows up, forexample, no faster than 1/5r [Eq. (15)], we would expecta mild singularity to form at the horizon. In particular,the area of the horizon would stay large and so would theentropy.

Is there a loss of information in scattering quanta offthese extremal black holes? Unfortunately, we cannotanswer this question conclusively here. It is clear thatsome information regarding the kinds of scalar quantathrown in will be lost in scattering. But in the large-Nlimit considered here, the entropy of the black hole,which goes as 1/A, is large. If this entropy has an ex-planation in terms of underlying microstates, this wouldsuggest a large degeneracy of ground states. And we can-not exclude the possibility that the lost information ishiding in correlations between the ground states and theoutgoing radiation. To settle this issue would requirekeeping track of very subtle correlations (beyond leadingorder in N) in the outgoing radiation. We hope in subse-quent work to return to this problem.

Finally, a few comments regarding nonextremal blackholes in contact with a heat bath. Classically, these blackholes have an outer and inner horizon. Numerical calcu-lations show that the behavior of the geometry inside theouter horizon depends, for a given value of the dilatonfield, on the electric charge. If the electric charge issmall, the dilaton decreases until it reaches its criticalvalue

I would like to thank J. Preskill for pointing this out and forthe subsequent argument about the large-1V limit being inade-quate.

7At this value the kinetic-energy term for the metric becomesdegenerate in conformal gauge.

(33)

xf=a,x—2 ln(x)

(34)

and the dilaton behaves like

e ~= —const Xln(x),

where x is the coordinate distance from the horizon.This shows that the curvature (which is related to thesecond derivative of f) and hence the tidal forces as feltby a freely falling observer blow up at the inner horizon.But it is straightforward to show that the divergence ismild enough for the tidal impulse to stay finite, whichsuggests that there should be an extension of thegeometry past the inner horizon. If the metric and dila-ton above are regarded as components of a sphericallysymmetric metric in four dimensions, though, there arecomponents of the tidal impulse that blow up, which sug-gests that the four-dimensional black holes might behavedifferently.

ACKNO%'LEOGMENTS

It is a pleasure to acknowledge the help M. Bucher, G.Horowitz, J. March-Russell, E. Poisson, A. Ridgway, K.Thorne, E. Verlinde, E. Witten, and P. Yi have given.This work greatly benefited from conversations with J.Preskill, A. Strominger, and F. Wilczek, and it is a spe-cial pleasure to thank them for their generous insight andencouragement. This research was supported in part bythe U.S. Department of Energy under Contract No.DEAC-03-81ER4050.

A. Ori has found a similar divergence in his study of massinflation in four dimensions [27].

and a spacelike singularity forms. However, once thecharge is large enough, the dilaton does not go to its criti-cal value. Instead, an inner horizon forms at which themetric component f [Eq. (8)] behaves like

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