PHYSICAL REVIEW D VOLUME 46, NUMBER 4

Rotating dilaton black holes

15 AUGUST 1992

James H. Horne* and Gary T. HorowitzDepartment ofPhysics, Uniuersity of California, Santa Barbara, California 93I06 95-30

(Received 3 April 1992)

It is shown that an arbitrarily small amount of angular momentum can qualitatively change the prop-erties of extremal charged black holes coupled to a dilaton. In addition, the gyromagnetic ratio of theseblack holes is computed and an exact rotating black string solution is presented.

PACS number(s): 97.60.Lf, 04.20.Jb, 11.17.+y

I. INTRODUCTION

All stationary black holes in general relativity coupledto electromagnetism are described by the Kerr-Newmansolutions. These solutions contain three parameters, themass M, charge Q, and angular momentum J, whichsatisfy the inequality M )Q +J /M . (When this in-equality is violated, the event horizon disappears and thespacetime describes a naked singularity. ) It has beenknown for a long time that, if one minimally couples cer-tain additional matter fields such as scalars, neutrinos, orcharged fermions, then the black-hole solutions do notchange. It has recently become clear that, if one couplesgravity to more complicated forms of matter, then theblack-hole solutions do change. In particular, if one cou-ples to an SU(2) Yang-Mills theory, new black-hole solu-tions exist [1] (although they have been shown to be un-stable [2]). If one includes a Higgs field as well as thegauge field, there are additional solutions which can beinterpreted as black holes inside non-Abelian monopoles

In this paper we will focus on perhaps the simplest ex-tension of the standard matter, gravity coupled to aMaxwell field and a dilaton. The action is

S=f d x&—g [—R+2(V4) +e F ],

where 0. is a free parameter which governs the strength ofthe coupling of the dilaton to the Maxwell field. ' When+=0, the action reduces to the usual Einstein-Maxwellscalar theory. When a= 1, the action (1) is part of thelow-energy action of string theory. However, we will

consider this theory for all values of o, . Since changingthe sign of a is equivalent to changing the sign of 4, it issuf5cient to consider only o, ~ 0.

Nonrotating charged black-hole solutions of (1) havebeen found [4,5] and are reviewed below. Certain qualita-tive features of the solutions turn out to be independent

of a. For example, for a given mass, there is always amaximum charge that can be carried by the black hole.If the charge is less than this extremal value, there is aregular event horizon. However, other qualitativefeatures depend crucially on a. For example, for a%0there is no inner horizon. More importantly, in the ex-tremal limit, the area A of the event horizon and the sur-face gravity ~ are discontinuous functions of n. This hasled to the view [7] that properties of black holes are notuniversal but depend on the details of the matter La-grangian. In [7], it was shown that, for a ) 1, infinite po-tential barriers form around extremal black holes. Thisled to the interpretation that the extremal black holesbehave like elementary particles.

However, the black-hole solutions which have beenstudied so far represent only a special case of the generalblack-hole solution since rotation has been ignored. Tosee whether qualitative properties of generic black holesdepend on the matter content, one must consider rotatingcharged black holes. This is the subject of the present in-vestigation. We have not yet found explicit solutionsdescribing rotating charged black holes for arbitrary a.However, for a=&3, the solutions are known [8]. Byconsidering this one exact solution and some general ar-guments, we are led to the conclusion that, at least interms of properties such as the area of the event horizonand the surface gravity, rotating black holes do not de-

pend qualitatively on a. For example, for a=&3, thearea of the event horizon of an extremal black hole is

simply proportional to the angular momentum:

~ =8~(Z[ . (2)

Thus, the area is nonzero in general, and vanishes only inthe limit of no rotation. In addition, the surface gravityvanishes in the extremal limit whenever the angularmomentum is nonzero. These properties are more analo-gous to the usual Kerr-Newman solution (a =0) than thenonrotating a=&3 solution. Nevertheless, we will argue

Electronic address: jhh@cosmic. physics. ucsb. edu~Electronic address: gary@cosmic. physics. ucsb. edu

In recent papers, this parameter has been called a. We adopta different notation here to avoid confusion with the standardangular momentum parameter for rotating black holes.

In the thermodynamic description, the entropy is simplyS=—'A and the temperature is T=a./2m. Since it has been ar-

gued that the thermodynamic interpretation breaks down near

the extremal limit [6], we will focus on the geometrically well-

defined quantities A and ~.

46 1340 1992 The American Physical Society

46 ROTATING DILATON BLACK HOLES 1341

that, when the quantum effects of particle creation aretaken into account, the late time behavior of rotating di-laton black holes still resembles that of elementary parti-cles rather than traditional black holes.

Although we do not yet have explicit rotating black-hole solutions for all values of a, we will present the solu-tions in the limit of slow rotation. This is sufhcient tocompute the gyromagnetic ratio of the black hole, i.e.,the ratio of the magnetic dipole moment to the angularmomentum. This is of interest since in the Einstein-Maxwell theory, black holes have a gyromagnetic ratiocorresponding to g=2 (the value for electrons) ratherthan g= 1 (the value for classical matter). We will seethat this is modified when a%0. Additionally, extendedblack-hole or black string solutions to low-energy stringtheory have recently been found in various dimensions[9—11]. We will present an exact rotating charged blackstring solution in five dimensions, which is closely relatedto the rotating dilaton black hole with a= &3.

We begin by reviewing the unrotating charged black-hole solutions to (1). The field equations are

1K-2r+

(1—a )/(1+a )

(13)

Thus, for a ( 1 the surface gravity goes to zero in the ex-tremal limit, for a=1 it approaches a constant, and fora & 1 it diverges. Holzhey and Wilczek [7] have shownthat, for a) 1 as the black hole approaches its extremallimit, there are potential barriers outside the horizonwhich increase without bound. Thus, nearly extremalblack holes do not absorb low-energy incoming radiation.In addition, any radiation which is absorbed by the blackhole is likely to be radiated very quickly since the temper-ature is proportional to K. The nearly extremal blackholes thus act more like elementary particles than tradi-tional black holes.

case a=O when it is a nonsingular inner horizon. Thus,they describe black holes only when r &r+. In the ex-tremal limit, r =r+, it is clear from (6) and (10) that, foraAO, the area of the event horizon goes to zero. The sur-face gravity K is

P (&2ac FPv)

—0P 7 (3)

II. ROTATING BLACK HOLESq2@+ e

—2a@F2 p2

g —2q @q @+2e—2a@F F p ig —2a@F2

The spherically symmetric solutions take the form [4,5]

dgdrA,

2

' 2a/(1+a2)

(4)

(5)

(6)

ds 2 5—a sin8X

2a sin 8(r +a —b, ) d dX

Let us now consider rotating black holes. When a=O,the rotating charged solution is the familiar Kerr-Newman solution

e24

Ftr r

where

r+1—(1—a )/(1+a )r

(7)

(9)

where

(r +a ) b,a sin 8 . 28—d~2sin 8d

+ dr +Xd8—

Qr aQr sin 8

(14)

and

R=r 1—a /(1+a )r

(10)

X=r +a cos 0,h=r +a +Q 2Mr, —

(15)

(16)

The two free parameters r+, r are related to the physi-cal mass and charge by

M= +2

r+r1+a

1 —a r

1+a1/2

(12)

When a =0, this solution reduces to the standardReissner-Nordstrom solution of Einstein-Maxwell theory.However, for a&0 the solution is qualitatively different.For all a, the surface r =r+ is an event horizon. Thesurface r =r is a curvature singularity except for the

3 =87r[+D +Q l4++D J]—(17)

where D2= M(M —Q ). The —extraction of energy also

and a is the angular momentum J divided by the mass.This solution has both an event horizon and an inner hor-izon at the zeros of A. Perhaps the key difference be-tween a rotating and nonrotating black hole is the ex-istence of an ergosphere. (For a general discussion, see[12].) In the Kerr-Newman solution, g«&0 in a regionoutside the event horizon. This allows energy to be ex-tracted classically either by the Penrose process for pointparticles or by superradiant scattering for fields. Howev-er, the area of the event horizon cannot decrease in theseclassical processes. The area can be expressed as

1342 JAMES H. HORNE AND GARY T. HOROWITZ 46

reduces the angular momentum in such a way that 3 in-creases.

For any a, the rotating uncharged black hole is simplythe Q=O limit of (14) and is called the Kerr solution.The Kerr metric has an event horizon atr =M +V M —a and an inner horizon atr =M —+M —a . The extremal limit has a finite areaand zero surface gravity. We have seen that, for a& 1,the extremal limit of the unrotating charged black holehas zero area and infinite surface gravity. Which of thesetwo situations, zero rotation or zero charge, is likely tohave qualitative properties similar to the general rotatingcharged black hole? First, consider a large unchargedblack hole which is rotating near its extremal rate. Sup-pose we add a small amount of charge. Since the eventhorizon is large and stable, we do not expect the chargeto drastically change the geometry in its vicinity. Thissuggests that extremal rotating black holes with smallcharge will resemble the uncharged black holes. In par-ticular, the area of the event horizon should be large andthe surface gravity should be small. Furthermore, thepotential barriers outside the horizon should remainmodest. Now, suppose we start with a nearly extremalunrotating black hole and try to add a small amount ofangular momentum. In this case, the event horizon isvery small and the potential barriers are very large.Thus, one cannot add angular momentum without send-ing it in with a large amount of energy which cansignificantly alter the properties of the solution. Thus,there is little reason to expect that black holes with smallangular momentum will resemble the zero angularmomentum limit.

III. THE KALUZA-KLEIN SOLUTION

To determine the behavior of extremal black holes withsmall angular momentum, as well as to verify the abovearguments for large angular momentum, one needs to ex-amine exact solutions. There is one value of a for whichan exact solution is known: a=&3. This solution wasdiscussed briefiy in [13]and in more detail in [8]. Its gen-eral properties are reviewed below. However, the ex-tremal limit was not explicitly considered, and we will seethat it has some unusual properties. The a =v'3 solutionis well within the a) 1 regime of [7] where the extremalblack holes are supposed to behave like elementary parti-cles. For this value of a, the action (1) is simply theKaluza-Klein action which is obtained by dimensionallyreducing the five-dimensional vacuum Einstein action. Inother words, given a five-dimensional spacetime with atranslational symmetry in a spacelike direction, define afour-dimensional metric g„, vector potential A„, and di-

laton W by

ds =e (dx +23 dx")

solution of Einstein's equation, takes the product with Ito obtain a five-dimensional translationally invariantsolution, and finally boosts the solution in the extra direc-tion. When reinterpreted in four dimensions, the solutionhas nonzero charge and a nontrivial dilaton field [13,14].Starting with the Schwarzschild solution, this procedureyields the charged black hole (6) for a=&3. To obtainrotating black holes, one simply starts with the Kerrsolution and repeats the same procedure. The resultingmetric is [8]

ds = — dt — dr dp8+1—u

+ B(r +a )+a sin 8—sin Hdg2 ~ 2 ZB

+B dr 2+Br de2 (19)

where

u ZB=1+'U2

1/2

Z =, ho=r +a —2mr,2mr

(20)

and m and a are the mass and rotation parameters of theoriginal Kerr solution and U is the velocity of the boost.X is defined in (15). The Kaluza-Klein solution has a vec-tor potential

lnB .3(22)

The physical mass M, charge Q, and angular momentumJ are given by

U2

M=m 1+2(1—v )

(23)

1 —u

ma

&1—v'

(24)

(25)

It is easy to see that, if U=O, the solution reduces to theoriginal Kerr solution. Furthermore, if a=O, the solu-tion agrees with (6)—(10) if one sets a =&3, identifies

2mr+ 2 )

1 —u

2mU

1 U

(26)

A,=, A&= —a sin8, (21)2(1 —u )8 2 1 —u'&'

and a dilaton field

+ g dx)"dx (18)

Then the field equations (3)—(5) are equivalent to thefive-dimensional vacuum Einstein equation.

As a result, it is straightforward to find exact solutionsfor a=&3. One starts with a four-dimensional vacuum

It is also possible to obtain a Kaluza-Klein solution startingwith the charged but massless Kerr-Newmann solution [15].

ROTATlNG DILATON BLACK HOLES 1343

3 =8~[C++C —J ] . (27)

This shows that, in the extremal limit, the area is simplyproportional to the angular momentum: A =8m

~J~. This

clearly demonstrates how the earlier result that the areavanishes for nonrotating black holes is modified by rota-tion. Although the area is nonzero in general, it clearlyhas a different dependence on the physical parametersM, Q,J than the Kerr-Newman solution (17).

Next we consider the surface gravity of the black hole.This is [8]

+(1—v )(m —a )

2m [m ++m2 —a ](28)

If the angular momentum is nonzero, it follows immedi-ately from (28) that, in the extremal limit, the surfacegravity goes to zero. For the nonrotating black hole, thesurface gravity reduces to a=+1—v /4m which agreeswith (13) after using Eq. (26) and setting a=&3. In thiscase the surface gravity clearly diverges in the extrernallimit. Thus, one can view the surface gravity of the ex-tremal black hole (somewhat heuristically) as being givenby a.=5(J).

Finally, we consider the angular velocity of the horizonO. This is defined by the condition that the Killing vec-tor (8/Bt)+Q(B/Btt) is null at the event horizon. Forthe Kaluza-Klein black hole (19), the angular velocity is[8]

0= a+(1—v )

2m [m ++m —a ](29)

This also exhibits an interesting discontinuity. If oneconsiders nonextremal black holes and takes the limit as

and does a simple coordinate transformation r~r+rin (6)—(10). Notice that the extremal limit r ~r+ ofthe unrotating black hole corresponds, from the five-dimensional viewpoint, to boosting the Schwarzschildsolution cross I to the speed of light v= 1 while takingthe unboosted mass m to zero so that the limit is welldefined. In particular, the infinite potential barriers thatarise outside the horizon can be viewed as a result ofboosting the finite barriers outside the Schwarzchild solu-tion.

The first thing to note about the rotating dilaton blackhole (19) is that the causal structure is very similar to theKerr solution. The metric appears singular at the zerosof X and 60. The first turns out to be a true curvaturesingularity at r=0, O=n./2 (the ring singularity) and thesecond is a coordinate effect associated with regular hor-izons. There is an event horizon at r =m ++m —aand a nonsingular inner horizon at r =m —'~ m —a .2 2

Thus, in the presence of rotation, the dilaton does notdestroy the inner horizon. The extremal limit corre-sponds to m =a . (Notice that this condition is indepen-dent of v; an extremal black hole remains extremal underthe boost. ) The extremal limit can be reexpressed asJ =C, where C=m /+1 —v depends on M and Q,but is independent of the angular momentum J. It is easyto show that, for a general black hole, the area of theevent horizon is

dMdt (30)

Thus, a rotating dilaton black hole will radiate away mostof its angular momentum. The area of its event horizonwill decrease and the potential barriers outside the hor-izon will grow.

Does a rotating dilaton black hole, at late times, resem-ble an elementary particle? If the black hole radiatesaway all its angular momentum before it approaches the

the angular momentum goes to zero, then 0 goes to zeroas expected. However, if one considers extremal blackholes and takes the limit as a goes to zero (keeping thephysical mass M fixed), one finds that 0 diverges. This isbecause the horizon is shrinking to zero size as the angu-lar momentum decreases.

We now briefly consider the question of what is the endpoint of Hawking evaporation if one starts with a nonex-tremal black hole in this theory. Let us first consider anonrotating black hole with initial mass M and charge Q.Since the Lagrangian (1) does not include any chargedparticles, it is clear that the charge cannot be radiatedaway. The mass will decrease toward the extremal valueand the temperature will increase. However, the poten-tial barriers outside the black hole will also increase. Itseems likely that the black hole only asymptotically ap-proaches its extremal limit. It is amusing to consider thisfrom the standpoint of the five-dimensional theory. Thefive-dimensional theory is more general than (1) becausemodes depending on the extra dimension correspond tomassive charged fields in the four-dimensional theory.As we have seen, a charged black hole is simply a boostedform of the Schwarzschild solution cross R. If the radiusof the compact direction is much smaller than the wave-length of the Hawking radiation, then the momentum inthis direction cannot change. (This is the five-dimensionalanalogue of the fact that the charge is constant. } Howev-er, the mass will decrease. In order to conserve momen-tum it must speed up. The temperature increases andhence the wavelength of the emitted radiation decreases.This process continues until the wavelength is of orderthe size of the compact direction at which point themomentum can change. However, at this point the po-tential barriers are so large that it is unlikely that theblack hole will be able to completely radiate away itsmomentum.

Now we include angular momentum. There are twobasic facts about the quantum mechanics of rotatingblack holes. The first is that Hawking radiation carriesaway angular momentum. The ratio of angular momen-turn to energy carried away depends on the spin of thematter field and increases with increasing spin [16]. Thesecond fact is that rotating black holes are never quan-tum mechanically stable. Even at zero temperature thereis radiation due to the ergosphere. This spontaneousemission is closely related to the classical superradiance.It occurs only for modes e ' 'e'"~ satisfying tv/n (0,i.e., when the angular velocity of the mode is less than theangular velocity of the black hole. A rough estimate ofthe rate of mass loss due to radiation from the ergosphereis [17]

1344 JAMES H. HORNE AND GARY T. HOROWITZ

extremal limit, then its evolution from that point will bethe same as in the nonrotating case. If instead the blackhole reaches the extremal limit before radiating away itsangular momentum, then its thermal radiation (28) willvanish, but the radiation from the ergosphere (30) will be-come important. As more angular momentum is radiatedaway, the ergosphere radiation will increase, and so willthe potential barriers outside the black hole. Thus, thelate stages of this evolution will resemble the situation de-scribed in [7]. Extremely high potential barriers sur-round a black hole which is radiating rapidly inside thebarriers. In the rotating case the radiation is due to theergosphere and in the nonrotating case the radiation isthermal. Thus, an extremal black hole should behave in asimilar fashion in particle scattering whether it has asmall amount of angular momentum or not. However,one must keep in mind that, in both cases, the semiclassi-cal description breaks down in the late stages of evapora-tion when the horizon becomes very small.

2= r+ds = — 1—

(1 —a )/(1+a )

dt2

(1 r /r)(1 r /r)(1 —a )/(1+a )

2a /(1+a )

+r 1—

2af(r)sin Odt d—P, (31)

IV. SLO%'LY ROTATING BLACK HOLES

It is straightforward to solve Eqs. (3)—(5) to first orderin the angular momentum parameter a. This is becausemost of the metric components depend only on a . Un-fortunately, many of the interesting physical quantitiesalso depend only on a, but we can still extract some use-ful information from the first-order solution.

Let us begin with the unrotating black hole for arbi-trary a, Eq. (6), and consider the effect of adding a smallamount of rotation a to the black hole. We will discardany terms involving a or higher powers in a. Inspectionof the Kerr-Newman and Kaluza-Klein solutions showsthat the only term in the metric that changes to O(a) is

g, &. Similarly, the dilaton does not change to O(a), and

3& is the only component of the vector potential thatchanges to O(a). However, to this order, the change in3 & is uniquely determined by the charge.

Thus, for arbitrary a, we can assume the followingform of the metric:

2(1+ 2)2(1 /r)2a /(1+a )

(1 —a )(1—3a )r

(1 —a )/(1+a )

( 1 +a2)2r2

(1—a )(1—3a )r

(1+a )r

(1 a)r— (33)

In deriving f (r), we have fixed an integration constantarising from the equations of motion so that f ( r )~c /ras r~ ~. This is the desired asymptotic behavior andcompletely determines f (r).

It is straightforward to verify that f (r) agrees with theKerr-Newman solution when a =0, and with theKaluza-Klein solution (19) when a=i/3 [after using (26),shifting r~r+r, and rescaling a by a constant]. Atfirst glance, f (r) appears to diverge in the string limit0, = 1. This is not actually the case, when cz = 1,

rf(r), =2

1 — ln 1—r

2I" + +r r

(34)

Similarly, a = 1/&3 appears singular in the general solu-tion (33), but is also well behaved. Thus, f (r) as given in

Eq. (33) is the correct solution to 0 (a) for all a.We argued earlier that, for a ) 1, a small amount of ro-

tation should produce a large change in the geometryclose to the horizon of a nearly extremal black hole. Wecan see this explicitly from (33). In the extremal limit, asr approaches r+ =r, the second term in f (r) divergesfor o, & 1. Thus, adding even a little rotation drasticallychanges the unrotating solution near the horizon.

What else can we learn from the slowly rotating blackhole? The surface gravity and area of the event horizondo not change to O(a). However, the angular momen-turn J and the magnetic dipole moment p first appear atthis order. The angular momentum is related to theasymptotic form of the metric by

where f (r) is a function to be determined. The dilatonand potential should be

2J . ,g, —— sin 0+0

r(35)

ln 1—1+o:

A, = —,3 = —a sin 0—r'(32)

which gives

a 3 cxJ=—r++ r2 3(]+o.') (36)

Using the linearized form of the equation of motion(3)—(5), we see that Eqs. (31) and (32) are solutions as longas

The magnetic moment is related to the asymptotic formof the vector potential by

46 ROTATING DILATON BLACK HOLES 1345

r

@sin 8r

so that

p=ag .

(37)

(38)

1 —Zd z 2aZ sin (9

8 +1—u

+ (r +a )+a sin 8 sin Odg2 2 ZB2

If we define a parameter g in the usual way by+ dr2+rde2+B-2dx2

0(42)

2M '

then we find

(39) where X, h0, Z, and B were defined previously, with a di-laton field

(43)4a r

g=22(3 a—)r +3(1+a )r+

(40) and antisymmetric tensor field

When a=0, we recover the well known, but still remark-able fact that the Kerr-Newman black hole has g= 2 justlike the electron (up to quantum corrections). For aAO,g still approaches 2 in the limit of small charge. Howev-er, as the charge increases, g decreases to a minimum ofg =2—2a /(a +3) for the extremal black hole. So, forstring theory with a =1, —,

' g 2 and for Kaluza-Kleintheory with a =3, 1 ~ g ~ 2 [13].

V. ROTATING BLACK STRINGS

The Kaluza-Klein black hole was constructed by tak-ing a four-dimensional uncharged black hole, adding anextra dimension, Lorentz boosting, and then using aKaluza-Klein reduction [8]. This procedure is very simi-lar to the procedure recently used to generate axioncharged black string solutions [11]. The only difference is

that, instead of using a Kaluza-Klein reduction in the laststep, one uses a o-model duality transformation [18].This transformation relates solutions to the low-energyaction of string theory: Eq. (1) with a= 1 plus a term in-

volving the three-form H„, (which is the curl of a two-form 8„,). o-model duality is most conveniently de-

scribed in terms of the metric that the string directly cou-ples to. This is equal to the metric appearing in (1) multi-plied by a (dimension dependent) power of e@. In termsof the string metric, the (five-dimensional) low-energysiring action is

S=J d x& ge z —[—R —4(VN) + —,', H ], (41)

where we have set the Maxwell field to zero.To obtain rotating black string solutions to (41), begin

with the Kerr black hole. Since 4=0, this is a solutionto the string equations of motion. Add on an extra Hatdirection x, and Lorentz boost t~(t +ux)/V 1 —v,x~(x+vt)/+1 —v . Now, using the o.-model dualitytransformation, we obtain a five-dimensional rotating,charged black string with metric

8„,=, 8„&=—a sin 81 —u'8' "

1 —v'8~ (44)

ACKNOWLEDGMENTS

We would like to thank D. Wiltshire for bringing [8]and [13—15] to our attention. This work was supportedin part by NSF Grant No. PHY-9008502.

It is straightforward to use the higher-dimensional gen-eralizations of the Kerr solution [19] to derive rotatingblack strings in higher dimensions.

The properties of the rotating black string (42) are verysimilar to the Kaluza-Klein black hole. Indeed, if dsdenotes the Kaluza-Klein metric (19), then the blackstring metric (42) is simply ds =e ds +e dx . Themass per unit length, axion charge per unit length, andangular momentum per unit length are given by(23)—(25). The rotating black string has an event horizonand nonsingular inner horizon at r =m++m —a, anda curvature singularity at r=O, O=n. /2. The surfacegravity of the black string is equal to that of the Kaluza-Klein solution (28).

The extremal limit of the rotating black string is quitedifFerent from the extremal limit of the unrotating blackstring [11]. In the latter case, the extremal limit corre-sponds to the field outside a straight fundamental string[20], and is boost invariant in the x tplane. T-he extremallimit of the rotating black string is not boost invariant,and it is not clear whether it can be interpreted as thefield outside of an excited fundamental string.

Note added. After this paper was submitted for publi-cation, we received two papers discussing rotating dilatonblack holes. The first [K. Shiraishi, Akita Junior CollegeReport No. AJC-HEP-6, 1992 (unpublished)] indepen-dently calculates the solution for slowly rotating blackholes and agrees with our results in Sec. IV. The second[A. Sen, Tata Report No. TIFR/TH/92-20, 1992 (unpub-lished)] finds the rotating charged black-hole solution fora = 1 including the three-form H„„as required by stringtheory. As expected, the extremal limit qualitativelyresembles the Kerr solution rather than the nonrotatingcharged black hole.

1346 JAMES H. HORNE AND GARY T. HOROWITZ 46

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