Rotating dilaton black holes with hair

PHYSICAL REVIEW D 69, 064028 ~2004!

Rotating dilaton black holes with hair

Burkhard KleihausDepartment of Mathematical Physics, University College, Dublin, Belfield, Dublin 4, Ireland

Jutta KunzFachbereich Physik, Universita¨t Oldenburg, Postfach 2503 D-26111 Oldenburg, Germany

Francisco Navarro-Le´rida*Departamento de Fı´sica Teo´rica II, Ciencias Fısicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain

~Received 11 June 2003; published 25 March 2004!

We consider stationary rotating black holes in SU~2! Einstein-Yang-Mills theory, coupled to a dilaton. Theblack holes possess nontrivial non-Abelian electric and magnetic fields outside their regular event horizon.While generic solutions carry no non-Abelian magnetic charge, but non-Abelian electric charge, the presenceof the dilaton field allows also for rotating solutions with no non-Abelian charge at all. As a consequence, thesespecial solutions do not exhibit the generic asymptotic noninteger power falloff of the non-Abelian gauge fieldfunctions. The rotating black hole solutions form sequences, characterized by the winding numbern and thenode numberk of their gauge field functions, tending to embedded Abelian black holes. The stationarynon-Abelian black hole solutions satisfy a mass formula, similar to the Smarr formula, where the dilatoncharge enters instead of the magnetic charge. Introducing a topological charge, we conjecture that black holesolutions in SU~2! Einstein-Yang-Mills-dilaton theory are uniquely characterized by their mass, their angularmomentum, their dilaton charge, their non-Abelian electric charge, and their topological charge.

DOI: 10.1103/PhysRevD.69.064028 PACS number~s!: 04.20.Jb

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I. INTRODUCTION

In Einstein-Maxwell ~EM! theory the unique family ofstationary asymptotically flat black holes comprises thetating Kerr-Newman~KN! and Kerr black holes and thstatic Reissner-Nordstrøm~RN! and Schwarzschild blackholes. EM black holes are uniquely determined by their mM, their angular momentumJ, their electric chargeQ, andtheir magnetic chargeP; i.e., EM black holes have ‘‘no hair’@1,2#.

The EM ‘‘no-hair’’ theorem does not readily generalizetheories with non-Abelian gauge fields coupled to grav@3,4#. The hairy black hole solutions of SU~2! Einstein-Yang-Mills ~EYM! theory possess nontrivial magnetic fields oside their regular event horizon, but carry no magnecharge@3–7#. Their magnetic fields are characterized by twinding numbern and by the node numberk of the gaugefield functions. In the static limit, the solutions with windinnumbern51 are spherically symmetric, whereas the sotions with winding numbern.1 possess only axial symmetry, showing that Israel’s theorem does not generalize to nAbelian theories, either@5#.

In many unified theories, including string theory, dilatoappear. When a dilaton is coupled to EM theory, this hprofound consequences for the black hole solutions.though uncharged Einstein-Maxwell-dilaton~EMD! blackholes correspond to the EM black holes, since the souterm for the dilaton vanishes, charged EMD black hole so

*Present address: Departamento de Fı´sica Atomica, Molecular yNuclear, Ciencias Fı´sicas, Universidad Complutense de MadrE-28040 Madrid, Spain.

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tions possess a nontrivial dilaton field, giving rise to an aditional charge, the dilaton chargeD, and to qualitativelynew features of the black holes. Charged static EMD blahole solutions, for instance, exist for arbitrarily small horizsize @8#, and the surface gravity of ‘‘extremal’’ solutions depends in an essential way on the dilaton coupling constang.Rotating EMD black holes, known exactly only for KaluzaKlein ~KK ! theory withg5) @9,10#, no longer possess thgyromagnetic ratiog52 @11#, the value of KN black holesExtremal charged rotating EMD black holes can possnonzero angular momentum, while their event horizon hzero angular velocity@10#.

Here we consider rotating black hole solutions of SU~2!Einstein-Yang-Mills-dilaton~EYMD! theory @5–7,12#. Forfixed dilaton coupling constantg and winding numbern, theblack hole solutions form sequences, which, with increasnode numberk, tend to limiting solutions. The black holsolutions of a given sequence carry no non-Abelian magncharge, but generically they carry a small non-Abelian eltric charge@6,7,13#. The limiting solutions, in contrast, correspond to embedded Abelian black hole solutions, whcarry no electric charge, but a magnetic charge, equal towinding numbern.

We investigate the physical properties of these black hsolutions. In particular, we consider their global charges,tained from an asymptotic expansion of the fields. Tasymptotic expansion of the gauge field functions genericinvolves noninteger powers of the radial coordinate, depeing on the non-Abelian electric charge of the black holelutions @6#. In the presence of the dilaton, there is, howevalso a special set of solutions, whose non-Abelian eleccharge vanishes@7#. For these, the asymptotic expansion ivolves only integer powers of the radial coordinate. We f

©2004 The American Physical Society28-1

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KLEIHAUS, KUNZ, AND NAVARRO-LERIDA PHYSICAL REVIEW D 69, 064028 ~2004!

ther consider the horizon properties of the non-Abelian blhole solutions, such as their horizon areaA, their surfacegravity ksg, their horizon curvatureK, and their horizon to-pology, and we introduce a topological horizon chargeNH@7,14#.

For the rotating non-Abelian black holes the zeroth lawblack hole mechanics holds@6,7#, as well as a generalizefirst law @15#. The non-Abelian black holes further satisfy thmass formula@7#

M52TS12VJ1D

g12celQ, ~1!

whereT denotes the temperature of the black holes,S theirentropy,V their horizon angular velocity, andcel their hori-zon electrostatic potential. This non-Abelian mass formulasimilar to the Smarr formula@16#. However, instead of themagnetic chargeP, the dilaton chargeD enters. This is cru-cial, since the genuinely non-Abelian black hole solutiocarry magnetic fields, but no magnetic charge. Thus thelaton charge term takes into account the contribution tototal mass from the magnetic fields outside the horizon. Tmass formula holds for all nonperturbatively known blahole solutions of SU~2! EYMD theory@5,6,12#, including therotating generalizations of the static nonspherically symmric non-Abelian black hole solutions@5#. It also holds forembedded Abelian solutions@7,17#.

Concerning the uniqueness conjecture for EYMD blaholes, it is not sufficient to simply replace the magnechargeP by the dilaton chargeD. Black holes in SU~2!EYMD theory are not uniquely characterized by their maM, their angular momentumJ, their dilaton chargeD, andtheir non-Abelian electric chargeQ. Adding as an additionacharge a topological chargeN, however, a new uniquenesconjecture can be formulated@7#: Black holes in SU(2)EYMD theory are uniquely determined by their mass M, thangular momentum J, their dilaton charge D, their noAbelian electric charge Q, and their topological charge N.

In Sec. II we recall the SU~2! EYMD action and the equations of motion. We discuss the stationary ansatz for the mric and gauge and dilaton fields, and we present the bounconditions. In Sec. III we address the properties of the blhole solutions. We briefly present the asymptotic expansat infinity and at the horizon, from which we obtain the glbal charges and the horizon properties, as well as the prothe mass formula. Our numerical results are discussed inIV. In Sec. V we present our conclusions. Appendixes A aB give details of the expansion at infinity and at the horizorespectively. In Appendix C we discuss the energy conditiand algebraic type of the stress-energy tensor in SU~2!EYMD theory.

II. NON-ABELIAN BLACK HOLES

After recalling the SU~2! EYMD action and the generaset of equations of motion, we discuss the ansatz forstationary non-Abelian black hole solutions, employedthe metric, the dilaton field and the gauge field@7#. The an-satz for the metric represents the stationary axially symm

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ric Lewis-Papapetrou metric@18# in isotropic coordinates.The ansatz for the gauge field includes an arbitrary windnumbern @5,7# and satisfies the Ricci circularity and Frobnius conditions@18#. As implied by the boundary conditionsthe stationary axially symmetric black hole solutions areymptotically flat and possess a regular event horizon.

A. SU„2… EYMD action

We consider the SU~2! EYMD action

S5E S R

16pG1LM DA2gd4x, ~2!

with scalar curvatureR and matter LagrangianLM given by

LM521

2]mF]mF2

1

2e2kFTr~FmnFmn!, ~3!

with dilaton field F, field strength tensorFmn5]mAn

2]nAm1 ie@Am ,An#, gauge fieldAm5Ama ta/2, and New-

ton’s constantG, dilaton coupling constantk, and Yang-Millscoupling constante.

Variation of the action with respect to the metric and mter fields leads, respectively, to the Einstein equations

Gmn5Rmn21

2gmnR58pGTmn , ~4!

with stress-energy tensor

Tmn5gmnLM22]LM

]gmn 5]mF]nF21

2gmn]aF]aF

12e2kFTrS FmaFnbgab21

4gmnFabFabD , ~5!

and matter field equations

1

A2g]m~A2g]mF!5ke2kFTr~FmnFmn!, ~6!

1

A2gDm~A2ge2kFFmn!50, ~7!

whereDm5]m1 ie@Am ,•#.

B. Stationary axially symmetric ansatz

The system of partial differential equations~4!, ~6!, and~7! is highly nonlinear and complicated. In order to genersolutions to these equations, one profits from the use of smetries, simplifying the equations.

Here we consider black hole solutions, which are bostationary and axially symmetric. We therefore impose onspacetime the presence of two commuting Killing vecfields j ~asymptotically timelike! and h ~asymptoticallyspacelike!. Since the Killing vector fields commute, we maadopt a system of adapted coordinates—s$t,r ,u,w%—such that

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ROTATING DILATON BLACK HOLES WITH HAIR PHYSICAL REVIEW D 69, 064028 ~2004!

j5] t , h5]w . ~8!

In these coordinates the metric is independent oft andw. Wealso assume that the symmetry axis of the spacetime, thof points whereh50, is regular and satisfies the elementaflatness condition@18#

X,mX,m

4X51, X5hmhm . ~9!

Apart from the symmetry requirement on the met@Ljg5Lhg50—i.e., gmn5gmn(r ,u)], we impose that thematter fields be also symmetric under the spacetime transmations generated byj andh.

This implies, for the dilaton field,

LjF5LhF50, ~10!

so F depends onr andu only.For the gauge potentialA5Amdxm, the concept of gener

alized symmetry@19,20# requires

~LjA!m5DmWj ,

~LhA!m5DmWh , ~11!

whereWj andWh are two compensating su~2!-valued func-tions satisfying

LjWh2LhWj1 ie@Wj ,Wh#50. ~12!

Performing a gauge transformation to setWj50, leavesAandWh independent oft.

To further simplify the system of equations, one can ipose that the Killing fields generate orthogonal 2-surfa~Frobenius condition!

j∧h∧dj5j∧h∧dh50, ~13!

wherej and h are considered as 1-forms. By virtue of thcircularity theorem@21# such a condition may be written iterms of the Ricci tensor

j∧h∧R~j!5j∧h∧R~h!50, ~14!

where„R(v)…m5Rmnvn.The metric can then be written in the Lewis-Papapet

form @22#, which in isotropic coordinates reads

ds252 f dt21m

f@dr21r 2du2#1sin2 ur 2

l

f Fdw2v

rdtG2

,

~15!

wheref, m, l, andv are functions ofr andu only. Note thechange of sign of the functionv with respect to Ref.@6#.

The z axis represents the symmetry axis. The regulacondition along thez axis, Eq.~9!, requires

muu50,p5 l uu50,p . ~16!

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The event horizon of stationary black hole solutionssides at a surface of constant radial coordinate,r 5r H , and ischaracterized by the conditionf (r H ,u)50 @6#. The Killingvector field

x5j1vH

r Hh ~17!

is orthogonal to and null on the horizon@23#. The ergo-sphere, defined as the region in whichjmjm is positive, isbounded by the event horizon and by the surface where

2 f 1sin2 ul

fv250. ~18!

As a result of the Einstein equations~4!, the circularityconditions~14! have consequences on the matter contennamely,

j∧h∧T~j!5j∧h∧T~h!50, ~19!

with „T(v)…m5Tmnvn. However, contrary to the case oAbelian fields, the conditions~19! are not just a consequencof the symmetry requirements, but they give rise to adtional restrictions on the form of the gauge potential.

For the gauge fields we employ a generalized ansatz@7#,which trivially satisfies both symmetry constraints~11! and~12! and circularity conditions~19!:

Amdxm5Cdt1AwS dw2v

rdtD

1S H1

rdr1~12H2!du D tw

n

2e, ~20!

C5B1

t rn

2e1B2

tun

2e, ~21!

Aw52n sinuFH3

t rn

2e1~12H4!

tun

2eG . ~22!

Here the symbolst rn , tu

n , andtwn denote the dot products o

the Cartesian vector of Pauli matrices,tW5(tx ,ty ,tz), withthe spatial unit vectors

eW rn5~sinu cosnw,sinu sinnw,cosu!,

eW un5~cosu cosnw,cosu sinnw,2sinu!,

eWwn5~2sinnw,cosnw,0!, ~23!

respectively. Since the gauge fields windn times around,while the azimuthal anglew covers the full trigonometriccircle once, we refer to the integern as the winding numberof the solutions. The gauge field functionsBi andHi dependon the coordinatesr and u only. @For n51 and k50, thepreviously employed stationary ansatz@6# is obtained,whereas forv5B15B250 the static axially symmetric an

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satz is recovered@5#, including the static spherically symmeric ansatz@12#, for n51 andH15H350, H25H45w(r ),andF5F(r ).]

The ansatz is form invariant under Abelian gauge traformationsU @5,6#:

U5expS i

2tw

nG~r ,u! D . ~24!

With respect to this residual gauge degree of freedomchoose the gauge fixing conditionr ] rH12]uH250 @5,6#.

For the gauge field ansatz, Eqs.~20!–~22!, the compen-sating matrixWh is given by

Wh5ntz

2e. ~25!

We note that by employing the gauge transformationU @24#,

U5expS i

2tznw D , ~26!

one can chooseWh850 @19,25#, leading to the gauge fieldAm8 :

Am8 dxm5C8dt1Aw8 S dw2v

rdtD

1S H1

rdr1~12H2!du D ty

2e, ~27!

C85B1S sinutx

2e1cosu

tz

2eD1B2S cosutx

2e2sinu

tz

2eD2n

tz

2e

v

r, ~28!

Aw852n sinuFH3S sinutx

2e1cosu

tz

2eD1~12H4!S cosu

tx

2e2sinu

tz

2eD G2ntz

2e. ~29!

We further note that when imposing restricted circularconditions by requiringF(j,h)5* F(j,h)50, only Abeliansolutions are possible if asymptotic flatness is assumed@25#.

C. Boundary conditions

1. Dimensionless quantities

For notational simplicity we introduce the dimensionlecoordinatex,

x5e

A4pGr , ~30!

the dimensionless electric gauge field functionsB1 and B2 ,

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B15A4pG

eB1 , B25

A4pG

eB2 , ~31!

the dimensionless dilaton functionf,

f5A4pGF, ~32!

and the dimensionless dilaton coupling constantg,

g51

A4pGk. ~33!

For g51 contact with the low-energy effective action ostring theory is made, whereas in the limitg→0 the dilatondecouples and EYM theory is obtained.

2. Boundary conditions at infinity

To obtain asymptotically flat solutions, we impose on tmetric functions the boundary conditions at infinity:

f ux5`5mux5`5 l ux5`51, vux5`50. ~34!

For the dilaton function we choose

fux5`50, ~35!

since any finite value of the dilaton field at infinity can aways be transformed to zero viaf→f2f(`), x→xe2gf(`).

We further impose that the two electric gauge field funtions Bi vanish asymptotically:

B1ux5`5B2ux5`50.

For magnetically neutral solutions, the gauge field functioHi have to satisfy

H1ux5`5H3ux5`50, H2ux5`5H4ux5`5~21!k,~36!

where the node numberk is defined as the number of nodeof the functionsH2 andH4 along the positive~or negative! zaxis @5,26#. Note that for each node number there is a degerate solution withH2ux5`5H4ux5`52(21)k, related bythe large gauge transformationU5 i t r

n . Under this transfor-mation the functions transform according to

H1→2H1 , H2→2H2 , H3→1H3 , H4→2H4 ,

B1→1B1 , B2→2B222n sinuv

x.

3. Boundary conditions at the horizon

The event horizon of stationary black hole solutionssides at a surface of constant radial coordinate,x5xH , and ischaracterized by the conditionf (xH ,u)50 @6#.

Regularity at the horizon then requires the followinboundary conditions for the metric functions,

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f ux5xH5mux5xH

5 l ux5xH50, vux5xH

5vH5const,~37!

for the dilaton function,

]xfux5xH50, ~38!

and for the ‘‘magnetic’’ gauge field functions,

H1ux5xH50, ]xH2ux5xH

50, ]xH3ux5xH50,

]xH4ux5xH50, ~39!

with the gauge condition]uH150 taken into account@6#.The boundary conditions for the ‘‘electric’’ part of th

gauge potential are obtained from the requirement that

non-Abelian solutions the electrostatic potentialC5xmAmbe constant at the horizon@6,27# ~see Appendix B 1 for de-tails!:

C~r H!5xmAmur 5r H5CH . ~40!

Defining the dimensionless electrostatic potentialc,

c5A4pGC, ~41!

and the dimensionless horizon angular velocityV,

V5vH

xH, ~42!

this yields the boundary conditions~see Appendix B 1!

B1ux5xH5nV cosu, B2ux5xH

52nV sinu. ~43!

With these boundary conditions, the dimensionless horielectrostatic potential reads

cH5nVtz

25cel

tz

2, ~44!

definingcel for the non-Abelian mass formula.

4. Boundary conditions along the axes

The boundary conditions along ther and z axes (u5p/2 and u50) are determined by the symmetries. Thare given by

]u f uu505]umuu505]ul uu505]uvuu5050,

]u f uu5p/25]umuu5p/25]ul uu5p/25]uvuu5p/250, ~45!

]ufuu5050,

]ufuu5p/250, ~46!

B2uu5050, ]uB1uu5050,

H1uu505H3uu5050, ]uH2uu505]uH4uu5050,

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B1uu5p/250, ]uB2uu5p/250,

H1uu5p/25H3uu5p/250,

]uH2uu5p/25]uH4uu5p/250. ~47!

In addition, regularity on thez axis requires condition~16!for the metric functions to be satisfied, and regularity of tenergy density on thez axis requires

H2uu505H4uu50 . ~48!

III. BLACK HOLE PROPERTIES

We derive the properties of the stationary axially symmric black holes from the expansions of their metric and mter functions at infinity and at the horizon. The expansioninfinity yields the global charges of the black holes, thmagnetic moments, and topological charge. Genericallygauge field functions of the black hole solutions show a ninteger power falloff asymptotically, with the exponents dtermined by the non-Abelian electric chargeQ. The expan-sion at the horizon yields the horizon properties, such asarea parameter, the surface gravity, and the horizon defortion. We also introduce a topological charge of the horizWe then give a detailed account of the non-Abelian mformula.

A. Global charges

The massM and the angular momentumJ of the blackhole solutions are obtained from the metric componentsgttandgtw , respectively. The asymptotic expansion for the mric function f,

f 5122M

x1OS 1

x2D ,

yields, for the dimensionless massM,

M51

2lim

x→`

x2]xf , ~49!

where

M5A4pG

eGM . ~50!

Likewise, the asymptotic expansion for the metric funtion v,

v52J

x2 1OS 1

x3D ,

yields, for the dimensionless angular momentumJ,

J51

2lim

x→`

x2v, ~51!

where

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J54p

e2 J. ~52!

The asymptotic expansion for the dilaton functionf,

f52D

x2

gQ2

2x2 1OS 1

x3D ,

yields the dimensionless dilaton chargeD,

D5 limx→`

x2]xf, ~53!

related to the dilaton chargeD via

D51

eD. ~54!

The asymptotic expansion of the gauge field yieldsglobal non-Abelian electromagnetic chargesQ and P. Theasymptotic expansion of the electric gauge field functionsB1

and B2 ,

B15Q cosu

x1OS 1

x2D , B252~21!kQ sinu

x1OS 1

x2D ,

yields the dimensionless non-Abelian electric chargeQ,

Q52 limx→`

x2]x@cosuB12~21!k sinuB2#, ~55!

where

Q5Q

e. ~56!

The boundary conditions of the magnetic gauge field futions guarantee that the dimensionless non-Abelian magnchargeP vanishes.

The non-Abelian global chargesQ and P appear to begauge dependent. In particular, the definition ofQ corre-sponds to rotating to a gauge, where the gauge field comnent c5A4pGC asymptotically only has atz component@6#:

c→ Q

x

tz

2.

Identifying the global non-Abelian electric chargeQ in thisway corresponds to the usual choice@6,20,28#.

We note that the modulus of the non-Abelian electcharge,uQu, corresponds to a gauge-invariant definition fthe non-Abelian electric charge given in Ref.@29#:

QYM51

4p RA(i

~* Fuwi !2du dw5

uQue

, ~57!

where * F represents the dual field strength tensor andintegral is evaluated at spatial infinity.

Likewise, a gauge-invariant definition of the non-Abeliamagnetic charge is given in Ref.@29#:

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r

e

PYM51

4p RA(i

~Fuwi !2du dw5

uPue

, ~58!

where again the integral is evaluated at spatial infinity, yieing P50.

Note that the lowest-order terms in the expansions,needed for the expressions of the global chargesM, J, D, Q,andP, do not involve noninteger powers.

B. Noninteger power falloff

Here we present the lowest-order terms of the expansof the ‘‘magnetic’’ gauge field functionsH1–H4 for windingnumbersn51 and n53. ~For details see Appendix A.!Since forn52 the lowest-order expansion contains nonalytic terms like log(x)/x2 already in the static limit@30#, werefrained from an analysis of the rotating solutions in thcase.

The asymptotic expansion yields, forn51,

H15F2C5

x2 18C4

b21x2~b21!/2

22C2C3~a13!

~a15!Q2 x2~a11!/2Gsinu cosu1oS 1

x2D ,

H25~21!k1C3x2~a21!/21o~x2~a21!/2!,

H35S C2

x2~21!kC3x2~a21!/2D sinu cosu1oS 1

xD ,

H45~21!k1C3x2~a21!/2

1S ~21!kC2

x2C3x2~a21!/2D sin2 u1oS 1

xD , ~59!

whereCi are dimensionless constants.Here a and b determine the noninteger falloff of th

gauge field functions:

a5A924Q2, b5A2524Q2. ~60!

For n53 the lowest-order terms of the expansion involonly integer powers,

H15C5

x2 sin 2u1oS 1

x2D ,

H25~21!k1C5

x2 cos 2u1oS 1

x2D ,

H35C2

xsinu cosu1

~M1gD !C222~21!kC52aMQ

2x2

3sinu cosu1oS 1

x2D ,

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H45~21!k1~21!kC2

xsin2 u1

C5

x2

1~21!k~M1gD !C222~21!kC52aMQ

2x2 sin2 u

1oS 1

x2D , ~61!

but higher-order terms contain noninteger powers involv

e5A4924Q2.To determine whether the generic noninteger falloff is

physical property of the solutions or a gauge artifact,consider the asymptotic behavior of the gauge-invariquantities Tr(FmnFmn) and Tr(* FmnFmn). Inserting the ex-pansions, Eqs.~59! and ~61!, we find

Tr~FmnFmn!52Q2

x4 14~M2gD !Q2

x5 1oS 1

x5D ,

n51, n53, ~62!

and

Tr~ * Fmn Fmn!54C2Q

x5 cosu1oS 1

x5D , n51,

Tr~ * Fmn Fmn!512C2Q

x5 cosu1oS 1

x5D , n53.

Although lowest-order terms do not contain the nonintepowers, they occur in the higher-order terms, indicating tthe noninteger power decay cannot be removed by a gatransformation.

C. Magnetic moment

In Abelian gauge theory, the dimensionless magnetic fiBW A of a magnetic dipole with dipole momentmW A5mz

AeW z isgiven by

BW A5mz

A

x3 ~2 cosueW r1sinueW u!.

To investigate the magnetic moment for the non-Abelblack holes we work in the gauge where asymptoticallyA05(Q/x)tz/21o(1/x). From the asymptotic expansion of thgauge field functions, given in Appendix A, we find, for thmagnetic field for winding numbern51 solutions,

Br52C2

x3 2 cosutz

21br

tr1

2,

Bu52C2

x3 sinutz

21bu

tr1

2,

Bw5bw

tw1

2, ~63!

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ld

n

where the functionsbr ,bu ,bw involve noninteger powers ox and contain the leading terms in the expansion forQÞ0.Since the leading terms decay slower thanx23 for QÞ0, wecannot extract the magnetic moment in general.

In the following we give a suggestion for the definitionthe magnetic moment. The asymptotic form ofA0 suggeststo interprettz/2 as ‘‘electric charge operator’’QPsu(2) inthe given gauge. We then observe that the projection ofBm inthe Q direction,

Br~Q!52

C2

x3 2 cosu1o~1/x3!,

Bu

~Q!52

C2

x3 sinu1o~1/x3!,

Bw~Q!5o~1/x3!, ~64!

corresponds asymptotically to the magnetic field of a mnetic dipolemW 52C2eW z with magnitude

umW u5uC2u. ~65!

We note thatmW is invariant under time-independent gautransformations, since such a gauge transformation rotthe ‘‘electric charge operator’’Q and the magnetic fieldBm

in the same way and leaves the projectionBm(Q) invariant.

A special case arises forn51 andQ50 when the func-tions br ,bu ,bw are of order O(x23). In this case theasymptotic form of the magnetic field reads

Br52C2

x3 2 cosutz

21

C3

x3 2 sinutr

1

21O~1/x4!,

Bu52C2

x3 sinutz

22

C3

x3 cosutr

1

21O~1/x4!,

Bw52C3

x3

tw1

21O~1/x4!. ~66!

Comparison with the magnetic field of a magnetic dipolemW A

in Abelian gauge theory,

BW A52~mW A•¹W !

xW

x3 51

x3 @mzA~2 cosueW r1sinueW u!

1mrA~2 sinueW r2cosueW u!1mw

AeWw#,

suggests the identification of the su~2!-valued magneticmoment:

mx5C3

tx

2,

my5C3

ty

2,

mz52C2

tz

2. ~67!

8-7

-hend

f

l

al

o

r


In the static caseC352C2 , reflecting the spherical symmetry of the solution. For the stationary rotating solutions tspherical symmetry is broken to axial symmetry, correspoing to C3Þ2C2 .

For winding numbern53 the asymptotic expansion othe gauge field functions yields, for the magnetic field,

Br52C2

x3 2 cosutz

21O~1/x4!,

Bu52C2

x3 sinutz

21O~1/x4!,

Bw5O~1/x4!, ~68!

leading again tomW 52C2eW z .

D. Topological charge

We finally introduce a topological chargeN. Let us con-sider a spacelike hypersurfaceS bounded by the horizonH,and a closed surfaceS,S homotopic to the two-dimensionasphereS2, such that the horizon is withinS. On S we intro-duce the su~2!-algebra-valued 2-formFS , corresponding tothe pullback of the Yang-Mills field strength to the surfaceS.Its dual * FS is a su~2!-algebra-valued function onS. Thenormalized function

s5* FS

u* FSu~69!

represents a map from the surfaceS to the 2-sphere in the Liealgebra su~2!. SinceS is homotopic toS2, the degree of themap defines the topological chargeNS :

NS51

4p

2

i ESTr$s ds∧ds%. ~70!

We note thats is not well defined if* FS vanishes at somepoint on S. However, if S approaches the two-dimensionsphereS`

2 at infinity, all singular points are withinS, and wecan define the topological chargeN5NS , in the limit S→S`

2 .In the following we derive the relation between the top

logical chargeN and the winding numbern of the axiallysymmetric black hole solutions. Therefore we considetwo-dimensional sphereS2 with radius xS large enough tocontain all singular points. On this sphere we find

FS5FuwuSdu∧dw ~71!

and its dual

* FS5g221/2FuwuS . ~72!

We decompose* FS as

* FS5* FSrS cosnw

tx

21sinnw

ty

2 D1* FSz tz

2, ~73!

06402

-

-

a

with norm u* FSu:

u* FSu5A~* FSr!21~* FS

z!2. ~74!

The normalized su~2!-valued functions now reads

s5sinQS cosnwtx

21sinnw

ty

2 D1cosQtz

2, ~75!

where

sinQ5* FSr/u* FSu, cosQ5* FS

z/u* FSu.

The degree of the map can now be written as

N51

4p ES~sinQ n]uQ!du dw

52n

2~cosQuu5p2cosQuu50!. ~76!

To evaluate the topological chargeN we express cosQand sinQ in terms of the functionsHi . This yields

sinQ5sinuGr1cosuGu

AGr21Gu

2, cosQ5

cosuGr2sinuGu

AGr21Gu

2,

~77!

where

Gr52@H3,u1H3 cotu1H2H421#,

Gu5@H4,u1cotu~H42H2!2H2H3#.

Symmetry and boundary conditions implyGr uu5p5Gr uu50andGuuu5p5Guuu5050: therefore,

N5nGr

uGr uU

u50

5n sgn~Gr uu50!, ~78!

relating the winding numbern to the topological chargeN.From the asymptotic expansions, Eqs.~59! and ~61!, for n51 andn53, respectively, we find that

Gr uu50→22C2

x.

This leads to

N5n sgn~2C2!.

We note that solutions with winding number2n aregauge equivalent to solutions with winding numbern, butcarry opposite electric charge@31#.

8-8

ho

isa

ed

ey

,he

l

wezoneles,

on

its

-


E. Expansion at the horizon

Expanding the metric and gauge field functions at therizon in powers of

d5x

xH21 ~79!

yields, to lowest order,

f ~d,u!5d2f 2~12d!1O~d4!,

m~d,u!5d2m2~123d!1O~d4!,

l ~d,u!5d2l 2~123d!1O~d4!,

v~d,u!5vH~11d!1O~d2!,

f~d,u!5f01O~d2!,

B1~d,u!5nvH

xHcosu1O~d2!,

B2~d,u!52nvH

xHsinu1O~d2!,

H1~d,u!5dS 121

2d DH111O~d3!,

H2~d,u!5H201O~d2!,

H3~d,u!5H301O~d2!,

H4~d,u!5H401O~d2!. ~80!

The expansion coefficientsf 2 , m2 , l 2 , f0 , H11, H20, H30,andH40 are functions of the variableu. Among these coeffi-cients the following relations hold:

05]um2

m222

]u f 2

f 2, ~81!

H115]uH20. ~82!

Further details of the expansion at the horizon are givenAppendix B.

F. Horizon properties

Let us now discuss the horizon properties of the SU~2!EYMD black hole solutions. The first quantity of interestthe area of the black hole horizon. The dimensionless areA,given by

A52pE0

p

du sinuAl 2m2

f 2xH

2 , ~83!

defines the area parameterxD via @29#

A54pxD2 , ~84!

06402

-

in

and the dimensionless entropyS of the black hole:

S5A

4. ~85!

The surface gravity of the black hole solutions is obtainfrom @23#

ksg2 52

1

4~¹mxn!~¹mxn!, ~86!

with Killing vector x, Eq.~17!. Inserting the expansion at thhorizon, Eqs.~80!, yields the dimensionless surface gravit

ksg5f 2~u!

xHAm2~u!. ~87!

As seen from Eq.~81!, ksg is indeed constant on the horizonas required by the zeroth law of black hole mechanics. Tdimensionless temperatureT of the black hole is proportionato the surface gravity:

T5ksg

2p. ~88!

To obtain a measure for the deformation of the horizoncompare the dimensionless circumference of the horialong the equator,Le , with the dimensionless circumferencof the horizon along a great circle passing through the poLp :

Le5E0

2p

dwAl

fx sinuU

x5xH ,u5p/2

,

Lp52E0

p

duAm

fxU

x5xH ,w5const

. ~89!

We obtain further information about the horizon deformatiby considering its Gaussian curvatureK:

K~u!5Ruwuw

g2, g25guugww2guw

2 . ~90!

We determine the topology of the horizon by computingEuler characteristicxE :

xE51

2p E KAg2 du dw. ~91!

Using the expansion Eqs.~80! in the integrand,

KAg2ux5xH52

]

]u FcosuA l 2

m21

sinu

2Al 2

]

]uA l 2

2

m2G ,

~92!

and the fact thatl 25m2 on the z axis, this yields for theEuler numberxE52, indicating that the horizon has the topology of a 2-sphere.

8-9

sann

tb

trhergutnry

in

t-

lubuheds

nbe

i

ace

illsion


To find the horizon electric chargeQD and the horizonmagnetic chargePD , one can evaluate the integrals, Eq~57! and ~58!, representing dimensionless gauge-invariquantities@29#. Whereas such a definition of the non-Abeliahorizon charges appears adequate in the static case, ipears problematic in the stationary case. For embedded Alian solutions, for instance, one obtains a horizon eleccharge, which differs from the global electric charge. Treason is that when evaluating the horizon electric chaaccording to this prescription, one is taking the absolvalue of the dual field strength tensor. Thus one doesallow for the cancellation, present in a purely Abelian theowhen the integral involves the dual-field strength tensor.

Finally we consider the horizon topological chargeNH ,suggested by Ashtekar@7,14#. It is obtained from Eq.~70!with S5H,

NH51

4p

2

i EHTr$s ds∧ds%.

For the axially symmetric black holes solutions,

NH5n sgn~Gr u~x5xH ,u50!!.

We note that the topological charge defined at infinity,N, andthe horizon topological charge may differ in sign, dependon how oftenGr uu50 changes sign along thez axis.

G. Non-Abelian mass formula

The non-Abelian mass formula, Eq.~1!,

M52TS12VJ1D

g12celQ

is a generalization of the non-Abelian mass formulaM52TS1D/g, obtained previously for static axially symmeric non-Abelian solutions@5,26#, which generically carry nonon-Abelian charges. Rotating non-Abelian black hole sotions in general do carry non-Abelian electric charge,they do not carry non-Abelian magnetic charge. Since tnevertheless carry nontrivial non-Abelian magnetic fielthe Smarr formula@16#

M52TS12VJ1celQ1cmagP ~93!

~where cmag represents the horizon magnetic potential! isbound to fail: The contribution to the mass from the noAbelian magnetic fields outside the horizon would notincluded. In contrast, in the non-Abelian mass formula~1!,this magnetic field contribution to the mass is containedthe dilaton term,D/g.

To derive the non-Abelian mass formula, Eq.~1!, we re-call the general expressions@23# for the total mass,

M5MH11

4pG ESRmnnmjndV, ~94!

and total angular momentum,

06402

.t

ap-e-

ic

eeot,

g

-ty,

-

n

J5JH21

8pG ESRmnnmhndV. ~95!

HereS denotes an asymptotically flat spacelike hypersurfbounded by the horizonH, nm is normal toS with nmnm521, anddV is the natural volume element onS @23#. Thehorizon massMH @23# and horizon angular momentumJHare given by

MH521

8pG EH

1

2«mnrs¹rjsdxmdxn52TS12

vH

r HJH ,

~96!

JH51

16pG EH

1

2«mnrs¹rhsdxmdxn. ~97!

Substituting the horizon massMH in Eq. ~94! and eliminat-ing the horizon angular momentumJH yields, for the totalmassM,

M52TS12vH

r HJ12F 1

8pG ESRmnnmjndV

1vH

r H

1

8pG ESRmnnmhndVG . ~98!

Now we express the Ricci tensor in terms of the Yang-Mand dilaton fields, using the Einstein equations, the definitof the stress-energy tensor, and the Lagrangian:

1

8pGRmn5]mF]nF12e2kFTr~Fm

a Fna!

21

2e2kFTr~FrsFrs!gmn . ~99!

Next we replace the last term in Eq.~99! via the dilatonequation

1

8pGRmn5]mF]nF12e2kFTr~Fm

a Fna!

21

2k

1

A2g]l~A2g]lF!gmn . ~100!

Sincej andh are Killing vector fields and sinceh is tangen-tial to S, we have

jm]mF50, hm]mF50, nmhngmn50, ~101!

and, consequently,

1

8pGRmnnmjn52e2kFTr~Fm

a Fna!nmjn

21

2k

1

A2g]l~A2g]lF!nmjm ,

8-10

ta

s

y

ethte

the

en-

nly

r-

s

atzess


1

8pGRmnnmhn52e2kFTr~Fm

aFna!nmhn.

Noting that

ES

1

A2g]l~A2g]lF!nmjmdV524pD, ~102!

whereD denotes the dilaton charge, this yields, for the tomass,

M52TS12vH

r HJ1

4p

kD14E

SH e2kFTr~Fm

aFna!nmjn

1vH

r He2kFTr~Fm

aFna!nmhnJ dV. ~103!

To evaluate the integral in Eq.~103! we use local coordinate(t,r ,u,w). In these coordinates,

nm52Af g0m, jm5~1,0,0,0!, hm5~0,0,0,1!,

dV51

AfA2g dr du dw,

and we obtain

M22TS22vH

r HJ2

4p

kD

5I

[24ESe2kFTrF S F0m1

vH

r HFwmDF0mGA2g dr du dw,

~104!

defining the integralI.To evaluateI, Eq. ~104!, we make use of the symmetr

relations, Eqs.~11! @20#,

Fm05DmA0 , Fmw5Dm~Aw2Wh!. ~105!

The integral then reads

I54ESe2kFTrF H DmS A01

vH

r H~Aw2Wh! D J F0mG

3A2g dr du dw. ~106!

Adding zero to the above integral, in the form of thgauge field equation of motion for the zero component,covariant derivative then acts on all functions in the ingrand:

I54ESTrFDmH S A01

vH

r H~Aw2Wh! De2kFF0mA2gJ G

3dr du dw. ~107!

06402

l

e-

Since the trace of a commutator vanishes, we replacegauge-covariant derivative by the partial derivative,

I54ESTrF ]mH S A01

vH

r H

~Aw2Wh!D e2kFF0mA2gJ G3dr du dw, ~108!

and employ the divergence theorem. Theu term vanishes,sinceA2g vanishes atu50 andu5p, and thew term van-ishes, since the integrands atw50 and w52p coincide;thus, we are left with

I54E TrF S A01vH

r H~Aw2Wh! De2kFF0rA2gGU

r H

`

du dw.

~109!

We next make use of the fact that the electrostatic pot

tial CuH is constant at the horizon@see Eqs.~40! and ~44!#:

CuH5S A01vH

r HAwD U

H

5vH

r HWh5Cel

tz

2. ~110!

Hence the integrand vanishes at the horizon, and the ocontribution toI comes from infinity.

At infinity the asymptotic expansion yields, to lowest oder,

F0rA2g52Q

esinuS cosu

t rn

22~21!k sinu

tun

2 D 1o~1!,

A05o~1!,

Aw52n

e@12~21!k#sinu

tun

21o~1!. ~111!

HenceA0 does not contribute to the integral andAw contrib-utes for odd node numberk. The integralI,

I5nQ

e2

vH

r HE Tr~cosut r

n2~21!k sinutun!2 sinu du dw

58pnQ

e2

vH

r H

, ~112!

is therefore independent ofk, and the mass formula become

M22TS22vH

r HJ2

4p

kD58pCelQ. ~113!

To complete the proof of the mass formula~1! for genu-inely non-Abelian black holes, subject to the above ansand boundary conditions, we now return to dimensionlvariables. Noting that

TS5A4pG

eGTS, ~114!

8-11

ac

ia

es,w-

0

oun,c-isr

o

ct o

g

ola

s

-

-

re

nd

tivey

hency.ng

en

.

ila-me

lian

the

tontothe

ose

heirnan

ted

er

c-


vH

r H5

e

A4pGV, ~115!

and recalling the definition ofcel , Eq. ~44!, the non-Abelianmass formula is obtained:

M52TS12VJ1D

g12celQ.

The mass formula also holds for embedded Abelian blholes@7,17#.

IV. NUMERICAL RESULTS

We solve the set of 11 coupled nonlinear elliptic partdifferential equations numerically@32#, subject to the aboveboundary conditions, employing compactified dimensionlcoordinates x512(xH /x). The numerical calculationsbased on the Newton-Raphson method, are performedhelp of the programFIDISOL @32#. The equations are discretized on a nonequidistant grid inx and u. Typical gridsused have sizes 100320, covering the integration region< x<1 and 0<u<p/2. ~See @5,6,26# and @32# for furtherdetails on the numerical procedure.!

For a given dilaton coupling constantg, the rotating non-Abelian black hole solutions then depend on the continuparametersxH and vH , representing the isotropic horizoradius and the metric functionv at the horizon, respectivelywhile their ratiovH /xH corresponds to the rotational veloity of the horizon. The solutions further depend on the dcrete parametersn and k, representing the winding numbeand the node number, respectively.

To construct a rotating non-Abelian black hole solutionSU~2! EYMD theory with dilaton coupling constantg andparametersxH , vH , n, andk, we either start from the statiSU~2! EYMD black hole solution possessing the same separameters except forvH50, and then increasevH @5#, orwe start from the corresponding rotating SU~2! EYM blackhole solution and then increaseg @6#.

A. nÄ1 black holes

Rotating non-Abelian black hole solutions with windinnumbern51 have been studied before in EYM theory@6#.Here we study the influence of the presence of the dilatonthe properties of these black hole solutions, with particuemphasis on theQ50 black hole solutions.

1. Global charges

Many features of then51 EYMD black hole solutionsagree with those of then51 EYM solutions, studied before@6#. In particular, when increasingvH from zero, while keep-ing xH fixed, a lower branch of black hole solutions formand extends up to a maximal valuevH

max(xH ,g), where anupper branch bends backwards towardsvH50. This is seenin Figs. 1~a!–1~d!, where the massM, the angular momentum per unit mass,a5J/M , the relative dilaton chargeD/g,and the non-Abelian electric chargeQ are shown as functions of the parametervH for black holes with horizon radius

06402

k

l

s

ith

s

-

f

f

nr

xH50.1, winding numbern51, and node numberk51 forfive values of the dilaton coupling constant,g50, 0.5, 1,),and 3. Forg50 the relative dilaton chargeD/g is extractedfrom the mass formula, Eq.~1!.

The massM @Fig. 1~a!# and the angular momentum peunit massa @Fig. 1~b!# of the non-Abelian solutions increasmonotonically along both branches and diverge withvH

21 inthe limit vH→0 along the upper branch. Both mass aangular momentum become~almost! independent of the di-laton coupling constant along the upper branch. The reladilaton chargeD/g @Fig. 1~c!# decreases monotonicallalong both branches, approaching zero in the limitvH→0.~The curves are discontinued along the upper branch wthe numerical procedure no longer provides high accura!Also, the dilaton charge approaches its limiting value alothe upper branch with a slope~almost! independent of thedilaton coupling constant.

The non-Abelian electric chargeQ @Fig. 1~d!# increasesmonotonically along both branches, wheng,1.15, whereaswhen g.1.15 ~and xH sufficiently small!, it first decreaseson the lower branch until it reaches a minimum and thincreases along both branches. The magnitude ofQ remainsalways small, however. Forg50, Q approaches the finitelimiting valueQlim'0.124 on the upper branch, asvH tendsto zero, independent of the isotropic horizon radiusxH @6#.Apparently, this remains true when the dilaton is coupled

For comparison, we exhibit in Figs. 1~a!–1~c! also themass, the specific angular momentum, and the relative dton charge of embedded Abelian solutions with the sahorizon radius~dotted curves! with Q50 andP51. ~Mass,angular momentum, and dilaton charge of embedded Abesolutions, possessing the same chargeQ as the non-Abeliansolutions andP51, are graphically indistinguishable frommass, angular momentum, and dilaton charge of theQ50solutions, shown. Since the Abeliang50 andg5) solu-tions are known analytically, their global charges alongupper branch are shown up tovH50.) As expected@6#,mass, angular momentum per unit mass, and relative dilacharge of the non-Abelian EYMD solutions are closemass, specific angular momentum, and dilaton charge ofembedded EMD solutions withQ50 andP51. This is true,in particular, for solutions with small horizon radiixH ,where large deviations of these global properties from thof the corresponding Kerr solutions arise@6#. The dilatoncharge of the Abelian solutions, however, approaches tlimiting value D/g50 on the upper branch with a commoslope, different from the common slope of the non-Abelisolutions.

The g dependence of the global charges is demonstrafurther in Figs. 2~a! and 2~b!. Here we exhibit the massM,the angular momentum per unit mass,a5J/M , the relativedilaton chargeD/g, and the non-Abelian electric chargeQfor non-Abelian black hole solutions with winding numbn51, node numberk51, horizon radiusxH50.1, andvH50.020 on the lower branch@Fig. 2~a!# and on the upperbranch @Fig. 2~b!# as functions ofg for 0<g<8. On thelower branchM, a5J/M , D/g, and Q decrease with in-creasingg. In particular, we note that the non-Abelian ele

8-12

d-


FIG. 1. ~a! The dimensionless massM is shown as a function ofvH for EYMD black holes with winding numbern51, node numberk51, horizon radiusxH50.1, for dilaton coupling constantg50, 0.5, 1,), and 3~solid lines!. The dimensionless mass of the corresponing EMD solutions with electric chargeQ50 and magnetic chargeP51 is also shown~dotted lines!. ~b! Same as~a! for the specific angularmomentuma5J/M . ~c! Same as~a! for the relative dilaton chargeD/g. ~d! The electric chargeQ is shown as a function ofvH for the sameset of EYMD black holes shown in~a!.

h-r

.

f r

ckan

o

olu

er

het

-

l

kictiveng-s.u-

tricericnic

d to

lu-

tric chargeQ passes zero atg'1.5. On the upper branconly the relative dilaton chargeD/g decreases with increasing g, whereas the massM and the angular momentum peunit mass,a, increase with increasingg. The non-Abelianelectric chargeQ exhibits a minimum on the upper branch

It is remarkable that the non-Abelian electric chargeQcan change sign in the presence of the dilaton field@7#. Incontrast, in EYM theory the non-Abelian electric chargeQalways has the same sign, depending on the direction otation @6,33#.

Thus we observe the new feature of rotating EYMD blaholes that, for certain values of the dilaton coupling constg and the parametersxH and vH , the non-Abelian electricchargeQ of rotating black hole solutions withn51 andk51 can vanish. Cuts through the parameter space of stions with vanishingQ are exhibited in Fig. 3~a!, where weshowvH as a function ofxH for Q50 solutions with dilatoncoupling constantsg51.5, ), and 2. For fixedg thesecurves divide the parameter space into a region with stions possessing negativeQ and a region with solutions withpositive Q. NegativeQ solutions correspond to parametvalues below the curves.

The maximum of a curve coincides withvHmax(xH ,g). We

note that, for fixedg, the values ofvH to the right of themaximum correspond to solutions on the lower brancwhereas the values to the left correspond to solutions onupper branches. In the limitg→gmin'1.15 the curves degen

06402

o-

t

lu-

-

she

erate to a point, implying that rotatingQ50 solutions existonly abovegmin . This is demonstrated in Fig. 3~b!, where thedependence of the maximal value ofvH together with itslocation xH ong is shown, clearly exhibiting the minimavalue ofg for rotatingQ50 solutions.

TheseQ50 EYMD black holes represent the first blachole solutions, which carry nontrivial non-Abelian electrand magnetic fields and no non-Abelian charge. Perturbastudies@34# previously suggested the existence of rotatinon-AbelianQ50 black hole solutions in EYM theory, satisfying, however, a different set of boundary conditionSuch EYM black hole solutions could not be obtained nmerically @6,20#.

As a consequence of their vanishing non-Abelian eleccharge, these black hole solutions do not exhibit the genasymptotic noninteger power falloff of the non-Abeliagauge field functions@6#. For these solutions, the magnetmoment can be identified according to Eq.~67!, wheremx;C3 and mz;2C2 . The coefficient2C25uC2u is exhib-ited in Fig. 3~c! for the sets of solutions shown in Fig. 3~a!.Note that the right end points of these curves corresponthe static limit.

The corresponding curves forC3 are graphically almostindistinguishable. The difference between2C2 and C3 isdue to the rotation, since in the static caseC21C350. InFig. 3~d! we show this difference for the same set of sotions.

8-13

ns

it

-

re,

-

ysvet to

be

ng

lare

or-g


FIG. 2. ~a! The global chargesM, a, D/g, andQ are shown asfunctions ofg for EYMD black holes with winding numbern51,node numberk51, horizon radiusxH50.1, andvH50.020 on thelower branch~solid lines!. The global charges of the correspondinEMD solutions with electric chargeQ50 and magnetic chargeP51 are also shown~dotted lines!. ~b! Same as~a! for black holesolutions on the upper branch.

06402

Having considered the rotating hairy black hole solutiofor fixed horizon radiusxH as a function ofvH , we nowkeepvH fixed and vary the horizon radius. In Figs. 4~a!–4~d! we show the massM, the angular momentum per unmass,a5J/M , the relative dilaton chargeD/g, and the non-Abelian electric chargeQ of the non-Abelian black hole solutions as functions of the isotropic horizon radiusxH forvH50.040 andg50, 0.5, 1,), and 3.~Solutions along thesteeper branch are numerically difficult to obtain: therefothe D/g and Q curves are discontinued atxH'0.43.) ForEYMD black hole solutions, as for EYM black hole solutions, there is a minimal value of the horizon radiusxH

min(vH),for a given value ofvH . In particular, the limitxH

min→0 isonly reached forvH→0. We note that for KK black holeswith Q50 andP51 the family of solutions characterized b@vH ,xH

min(vH)# tends to the static ‘‘extremal’’ solution axH→0. Anticipating that the non-Abelian black holes behaessentially analogously to the Abelian solutions, we expecfind the static regular EYMD solutions in this limit@6#. Wehave no indication that rotating regular solutions couldreached in this limit@34,35#.

2. Horizon properties

Let us next turn to the horizon properties of the rotatinon-Abelian black holes. In Figs. 5~a!–5~d! we show thearea parameterxD , the surface gravityksg, the deformationof the horizon as quantified by the ratio of equatorial to pocircumferences,Le /Lp , and the Gaussian curvature at thpolesK(0) as functions ofvH , for black hole solutions withhorizon radiusxH50.1, winding numbern51, and nodenumberk51 and for dilaton coupling constantsg50, 0.5, 1,), and 3. Also shown are the horizon properties of the cresponding embedded Abelian black hole solutions withQ50 andP51.

-

FIG. 3. ~a! Cuts through theparameter space ofQ50 blackhole solutions with winding num-ber n51, node numberk51, anddilaton coupling constantg51.5,), and 2. The curves are extrapolated to vH5xH50, as indicatedby the dots.~b! The value of themaximum ofvH and its locationxH are shown as functions ofg forthe Q50 EYMD black hole solu-tions with winding numbern51and node numberk51. ~c! Thecoefficient 2C25uC2u of the di-mensionless magnetic momentmis shown as a function ofxH forthe sets of EYMD back holesshown in~a!. ~d! Same as~c! forC21C3 .

8-14


FIG. 4. ~a! The dimensionless massM is shown as a function ofxH for EYMD black holes with winding numbern51, node numberk51, vH50.040, for dilaton coupling constantg50, 0.5, 1,), and 3.~b! Same as~a! for the specific angular momentuma5J/M . ~c!Same as~a! for the relative dilaton chargeD/g. ~d! Same as~a! for the electric chargeQ.

-pdtherif--

s-eg

exorere

s

u

cl

ig

ds

op-

le.n

the

e, as

.ticsingve atini-o in

ofine ats

sally

The horizon size is quantified by the area parameterxD ,shown in Fig. 5~a!. Starting from the value of the corresponding static non-Abelian black hole solution, the arearameter grows monotonically along both branches andverges along the upper branch. The horizon size ofcorresponding Abelian black hole solutions is slightly largin particular along the lower branch. As illustrated, the dference decreases with increasingg. The difference also decreases with increasing horizon radiusxH .

The surface gravityksg of the non-Abelian black holes ishown in Fig. 5~b!. Starting from the value of the corresponding static non-Abelian black hole solution on the lowbranch in the limitvH→0, it decreases monotonically alonboth branches and reaches zero on the upper branch inlimit vH→0, corresponding to the value assumed bytremal black hole solutions. The surface gravity of the cresponding Abelian black hole solutions is slightly smallin particular along the lower branch. The difference dcreases with increasing horizon radiusxH . We recall that forstatic non-Abelian black holes the surface gravity divergethe limit xH→0 independent ofg @36#, whereas for staticAbelian black holes the limiting value depends ong anddiverges only forg.1, whereas it is finite forg51 and zerofor g,1.

The deformation of the horizon is revealed when measing the circumference of the horizon along the equator,Le ,and the circumference of the horizon along a great cirpassing through the poles,Lp , Eqs. ~89!. The ratioLe /Lpgrows monotonically along both branches, as shown in F

06402

a-i-e,

r

the--,-

in

r-

e

.

5~c!. As vH tends to zero on the upper branch, the ratio tento the valueLe /Lp5p/@&E(1/2)#'1.645, for both non-Abelian and Abelian black holes, independent ofg. Clearly,the matter fields lose their influence on the black hole prerties, in the limitM→`. On the lower branch the ratioLe /Lp assumes the value 1 in the limitvH→0, correspond-ing to the value of a static spherically symmetric black hoOverall, the deformation of the horizon of the non-Abeliablack holes is close to the deformation of the horizon ofcorresponding EMD black holes withQ50 andP51.

For rapidly rotating EYMD and EMD black holes, thGaussian curvature of the horizon can become negativeobserved before for EM black holes@16#. In Fig. 5~d! weshow the Gaussian curvature of the horizon at the poleK(0)as a function ofvH for non-Abelian and Abelian black holesStarting with a positive curvature at the pole of the stasolution, the curvature at the pole decreases with increaangular momentum, crosses zero, and becomes negatisome point along the upper branch. It then reaches a mmum value and starts to increase again, to become zerthe infinite mass limit, whenvH→0.

In Fig. 6 we consider the deformation of the horizonthe non-Abelian black holes in more detail. We exhibitFig. 6~a! the angular dependence of the Gaussian curvaturthe horizon forg51 black hole solutions with horizon radiuxH50.1, winding numbern51, and node numberk51, forseveral values of the angular momentum per unit masa.Whena.0, the Gaussian curvature increases monotonicfrom the pole to the equator. For larger values ofa, the

8-15

D


FIG. 5. ~a! The area parameterxD is shown as a function ofvH for EYMD black holes with winding numbern51, node numberk51, horizon radiusxH50.1, for dilaton coupling constantg50, 0.5, 1,), and 3~solid lines!. The area parameter of the corresponding EMsolutions with electric chargeQ50 and magnetic chargeP51 is also shown~dotted lines!. ~b! Same as~a! for the surface gravityksg. ~c!Same as~a! for the ratio of circumferencesLe /Lp . ~d! Same as~a! for the Gaussian curvature at the polesK(0).

.no

bac-pad

ro

olu

essit

in

en-

m-n-

-

o-

ric

tic

it

Gaussian curvature is negative in the vicinity of the poleFigure 6~b! shows the shape of the horizon of the no

Abelian black holes, obtained from isometric embeddingsthe horizon in Euclidean space@16#. As pointed out in@16#,when the Gaussian curvature becomes negative, the emding cannot be performed completely in Euclidean spwith metric ds25dx21dy21dz2, but the region with negative curvature must be embedded in pseudo-Euclidean swith metric ds25dx21dy22dz2, represented by dashelines in the figure.

3. Limiting Abelian solutions

As seen above, the global charges and the horizon perties of the non-Abelian black hole solutions withn51 andk51 are rather close to those of the embedded EMD stions with Q50 andP51. But also their metric and dilatonfunctions as well as their gauge field functions are imprsively close to their EMD counterparts, except for thogauge field functions which do not vanish in the static lim

With increasing node numberk, the non-Abelian solutionsget increasingly closer to the Abelian solutions, convergpointwise in the limitk→`. In particular, the gauge fieldfunctions H2 and H4 approach zero with increasing nodnumberk in an increasing interval, while their boundary coditions H2(`)5H4(`)5(21)k enforce a value, differingfrom the value of the limiting Abelian solutions,H2

A(`)5H4

A(`)50.

06402

-f

ed-e

ce

p-

-

-e.

g

Convergence of the solutions with increasing node nuber k towards the corresponding EMD solutions is demostrated in Figs. 7~a!–7~e! for the metric functionf, the dilatonfunction f, the gauge field functionsH2 , H3 , and for thefunction xB1 for non-Abelian black hole solutions withxH50.1, vH50.020,n51, k51 – 3, and dilaton coupling constantsg50 and 1. From Fig. 7~e! we conclude that the lim-iting solution has vanishing electric charge.

B. nÌ1 black holes

Here we give for the first time a detailed account of rtating n.1 black hole solutions@7#. Whereas the rotatingn51 black hole solutions reduce to spherically symmetblack hole solutions in the static limit, the rotatingn.1black hole solutions remain axially symmetric in the stalimit @5,7#.

1. Global charges

Many features of then.1 EYMD black hole solutionsagree with those of then51 EYMD solutions. In Figs. 8~a!–8~d! we exhibit the massM, the angular momentum per unmass,a5J/M , the relative dilaton chargeD/g, and the non-Abelian electric chargeQ as functions of the parametervHfor black holes with horizon radiusxH50.1, winding num-bersn51 – 3, and node numberk51 for the dilaton couplingconstants g50, 1, and ). Again, for g50 the

8-16

-

-

nn

in

nga

ds8d

ging

tric

nch

ack

is-be-

aton

-ide

n-

lare

ize

ches

efckIt isck

e

to

llim-

ala-

hib-ing

m

niu


relative dilaton chargeD/g is extracted from the mass formula, Eq.~1!, discussed below.

The massM shown in Fig. 8~a! and the angular momentum per unit mass,a, shown in Fig. 8~b! increase monotoni-cally and diverge on the upper branch in the limitvH→0, asin the n51 case. Whereas the mass on the upper brabecomes~almost! independent both of the coupling constag and the winding numbern already for moderately largevalues forvH , the specific angular momentum becomesdependent only of the coupling constantg, but retains itsdependence on the winding numbern up to considerablysmaller values ofvH .

The relative dilaton charge shown in Fig. 8~c! apparentlyapproaches zero in the limitvH→0, as in the case of then51 black hole solutions. We note, however, that for windinumbern53, the dilaton curve develops a minimum withnegative value ofD and only then bends upwards towarzero. The non-Abelian electric charge exhibited in Fig.increases with increasingn. As in the n51 case,Q appar-ently reaches a limiting value on the upper branch asvH

FIG. 6. ~a! The Gaussian curvature at the horizonK(u) is shownas a function of the angleu for EYMD black holes with windingnumbern51, node numberk51, horizon radiusxH50.1, dilatoncoupling constantg51, for the values of the angular momentuper unit massa50, 0.257, 0.758, and 1.471.~b! The shape of thehorizon is shown for EYMD black holes with winding numbern51, node numberk51, horizon radiusxH50.1, dilaton couplingconstantg51, for the values of the angular momentum per umassa50, 0.76, and 1.47. Solid lines indicate embedding in Eclidean space, dashed lines in pseudo-Euclidean space.

06402

cht

-

→0. This limiting value increases with increasing windinnumber n and appears to be independent of the couplconstantg. ~For small values ofvH on the upper branch, thenumerical error becomes too large to extract the eleccharge and thus the limiting values ofQ.! For n53, theelectric charge develops a maximum on the upper brabefore tending to the limiting value.

Comparing the global charges of these non-Abelian blhole solutions with those of the correspondingQ50 andP5n Abelian black hole solutions, we observe that the dcrepancies between Abelian and non-Abelian solutionscome larger with increasing winding numbern. As in then51 case, the discrepancies decrease with increasing dilcoupling constantg, and with increasing horizon radiusxH .

Trying to find n.1 black holes with vanishing nonAbelian electric charge, we constructed solutions for a wrange of parameters. However, in contrast to then51 case,we did not observe a change of sign ofQ for any set ofsolutions constructed.

2. Horizon properties

Turning next to the horizon properties of the rotating noAbelian n.1 black holes, we exhibit in Figs. 9~a!–9~d! thearea parameterxD , the surface gravityksg, the deformationof the horizon as quantified by the ratio of equatorial to pocircumferences,Le /Lp , and the Gaussian curvature at thpolesK(0) as functions ofvH , for black hole solutions withxH50.1, n51 – 3, andk51 and for dilaton coupling con-stantsg50, 1, and).

As shown in Fig. 9~a!, the horizon size is monotonicallyincreasing along both branches and diverges asvH→0 onthe upper branch. Along the upper branch, the horizon sshows a distinct dependence on the winding numbern, butlittle dependence on the coupling constantg. The g depen-dence increases though with increasingn. The surface grav-ity decreases monotonically along both branches and reaa vanishing value on the upper branch asvH→0, as demon-strated in Fig. 9~b!. For black hole solutions with the samvalue ofg, but different values ofn, we observe crossings othe lower branches. This is in contrast with the Abelian blahole solutions, where analogous crossings do not occur.thus a distinct feature of the non-Abelian nature of the blaholes.

The deformation parameterLe /Lp is shown in Fig. 9~c!.For small values ofvH along the upper branch, we observthat the curves become~rather! independent ofg, but theirslope keeps a distinctn dependence. All curves appearhave the same limiting value though, whenvH→0, indepen-dent ofg andn. ~Unfortunately, the numerical error for smavH along the upper branches is too large to extract the liting value asvH→0.) We expect this common limitingvalue to beLe /Lp51.645, which is the value of the extremKerr black holes, Kerr-Newman black holes, and KaluzKlein black holes.

The Gaussian curvature of the horizon at the poles, exited in Fig. 9~d!, possesses negative values for fast rotatblack holes, also forn.1 black holes. The topology of the

t-

8-17

e


FIG. 7. ~a! The difference of the metric functionf and the metric functionf ` of the limiting solutions is shown as a function of thdimensionless coordinatex for u50 for the EYMD black hole solutions withg50 ~solid lines! andg51 ~dotted lines!, winding numbern51, vH50.020~lower branch!, horizon radiusxH50.1, and node numbersk51 – 3.~b! The same as~a! for the dilaton functionf. ~c! The

same as~a! for the functionH2 (H2`50). ~d! The same as~a! for the functionH3 , but u5p/4. ~e! The same as~a! for the functionxB1 .

e

acanpr

ec

thethe

k

-

horizon of thesen.1 black holes is also that of a 2-spherEq. ~92!.

As expected, these properties of the non-Abelian blhole solutions are rather similar to those of their Abelicounterparts, though the differences become morenounced asn increases.

3. Limiting Abelian solutions

Considering the limit of node numberk→` and fixedwinding numbern as well as fixedg, we observe that theblack hole solutions with winding numbern tend to the cor-responding Abelian black hole solutions with vanishing el

06402

,

k

o-

-

tric chargeQ50 and with magnetic chargeP5n. The con-vergence is observed for the global charges and forhorizon properties, and pointwise convergence is seen formetric, dilaton, and gauge field functions.

C. Mass formula and uniqueness

The numerical stationary axially symmetric EYMD blachole solutions satisfy the mass formula, Eq.~1!, with an ac-curacy of 1023, so do the numerical EMD black hole solutions. EYM black holes are included in the limitg→0, sinceD/g remains finite.

8-18

it


FIG. 8. ~a! The dimensionlessmassM is shown as a function ofvH for EYMD black holes withn51 – 3, k51, xH50.1, and g50, 1, and). ~b! Same as~a! forthe angular momentum per unmass,a. ~c! Same as~a! for therelative chargeD/g. ~d! Same as~a! for the electric chargeQ.

FIG. 9. ~a! The area parameterxD is shown as a function ofvH for EYMD black holes withn51 – 3,k51, xH50.1, andg50, 1, and). ~b! Same as~a! for the surface gravityksg. ~c! Same as~a! for the deformation parameterLe /Lp . ~d! Same as~a! for the Gaussiancurvature at the poleK(0).

064028-19

ia

ce

Eq.uch

ss

tles

g-

nt

calk

n a

ass

g-

ness

u-eir

M

.


We demonstrate the mass formula for the non-Abelblack hole solutions in Figs. 10~a!–10~c!, where we show thefour terms in the mass formula, 2TS, 2VJ, D/g, and2celQ, as well as their sum~denoted byM in the figures! asfunctions of the massM. Black hole solutions with horizonradiusxH50.1, winding numbern51, and node numberk51 and dilaton coupling constantg51 are shown in Fig.10~a!, n51 andk52 solutions in Fig. 10~b!, andn52 andk51 solutions in Fig. 10~c!.

Reconsidering the uniqueness conjecture for EYMD blaholes, one may try to obtain a new uniqueness conjecturonly replacing the magnetic chargeP by the dilaton chargeD

FIG. 10. ~a! Numerical check of the mass formula, Eq.~1!, forn51, k51, xH50.1, g51 black hole solutions. The terms in Eq~1! are plotted versusM: their sum coincides withM, within thenumerical accuracy.~b! Same as~a! for n51 andk52. ~c! Same as~a! for n52 andk51.

06402

n

kby

or the relative dilaton chargeD/g. After all, such a replace-ment gave rise to the new non-Abelian mass formula,~1!. However, in the case of the uniqueness conjecture, sa replacement is not sufficient. Indeed, black holes in SU~2!EYMD theory are not uniquely characterized by their maM, their angular momentumJ, their non-Abelian electricchargeQ, and their dilaton chargeD. This becomes evidenalready in the static case, where genuine EYMD black hohaveJ5Q50, and thus their only charges are the massMand the dilaton chargeD. In Fig. 11 we show the relativedilaton chargeD/g as a function of the massM, for the staticnon-Abelian black hole solutions withn51 – 3 and k51 – 3 for g51. Also shown are the corresponding limitinAbelian solutions withP5n. As seen in the figures, solutions with the same winding numbern and different nodenumberk do not intersect. However, solutions with differewinding numbersn can intersect. The curve ofn53, k51solutions, for instance, intersects the curves of alln52, k.2 solutions.

To remedy this deficiency, we make use of the topologichargeN, Eq. ~70!. Recalling that for the non-Abelian blacholesN5n, Eq. ~78!, and settingN50 for embedded Abe-lian black holes, we can attach to each black hole solutiotopological charge. We then find that all static SU~2! EYMDblack hole solutions are uniquely determined by their mM, their dilaton chargeD ~or the relative dilaton chargeD/g), and their topological chargeN. We note that for agiven winding numberN5n, the dilaton chargeD ~or therelative dilaton chargeD/g) of the non-Abelian solutionsincreases monotonically with increasingk, converging to thedilaton charge of the embedded Abelian solution with manetic chargeP5n.

This observation has led us to suggest a new uniqueconjecture @7#, stating thatblack holes in SU(2) EYMDtheory are uniquely determined by their mass M, their anglar momentum J, their non-Abelian electric charge Q, thdilaton charge D, and their topological charge N. Since therelative dilaton chargeD/g remains finite in the limitg→0, this conjecture may formally be extended to the EY

FIG. 11. The relative dilaton chargeD/g is shown as a functionof the mass for static non-Abelian solutions (g51) and embeddedAbelian solutions withQ50 andP5n.

8-20

efo

c

gn

he

ck

lu-oleedns

vi--

he

allge.-

lu-

s;

ryse-

ifi-irst,ithk

ack

ger

ing

les.in

for

ydntolso

ntorge

w

f


case by replacing the dilaton chargeD by the relative dilatonchargeD/g.

The uniqueness conjecture is supported by all our numcal data. To illustrate the validity of the conjecture alsostationary black holes, we show in Figs. 12~a!–12~c! therelative dilaton chargeD/g for several values of the specifiangular momentuma for black holes with n51 – 3, k51 – 3 (g51). Also shown are the corresponding limitinAbelian solutions withP5n. For instance, we observe iFig. 12~a! that the curve withn51, k52, and a50.25crosses the otherk52 curves and also thek53 curves witha50 and a50.10. However, since it does not cross t

FIG. 12. ~a! The relative dilaton chargeD/g is shown as afunction of the massM for non-Abelian black holes withn51, k51 – 3 (g51), and specific angular momentuma50.25, 0.1, and0. Also shown are the embedded Abelian solutions withQ50 andP51. ~b! Same as~a! for n52 and P52. ~c! Same as~a! for n

53 andP53.

06402

ri-r

k53 curve with the same specific angular momentuma50.25, the conjecture is not violated.

V. CONCLUSIONS

We have constructed nonperturbative rotating hairy blahole solutions of Einstein-Yang-Mills~-dilaton! theory andinvestigated their properties. The rotating black hole sotions emerge from the corresponding static hairy black hsolutions when a small horizon angular velocity is imposvia the boundary conditions. The hairy black hole solutioare labeled by the node numberk and the winding numbernof their gauge field functions. We have extended our preous studies@6# by considering black holes with higher winding numbern and by including a dilaton field with couplingconstantg. We have also provided a detailed analysis of tasymptotic expansion of black hole solutions withn51 andn53.

The generic hairy black hole solutions possess a smnon-Abelian electric charge, but vanishing magnetic charFor fixed winding numbern, the non-Abelian black hole solutions form sequences, labeled by node numberk. In thelimit k→`, they tend to embedded Abelian black hole sotions with magnetic chargeP5n and electric chargeQ50.For g50 the limiting solutions are Kerr-Newman solutionfor gÞ0 they are rotating EMD solutions@9,10,17#. A com-prehensive study of the black hole solutions in EMD theofor general dilaton coupling constant will be presented elwhere@17#.

The presence of the dilaton field has a number of signcant consequences for the hairy black hole solutions. Fthe presence of the dilaton field allows for solutions wvanishing electric charge@7#. Thus there are rotating blachole solutions with no non-Abelian charge at all. TheseQ50 solutions exist in a certain range of parameters for blholes with winding numbern51 and node numberk51.The asymptotic expansion of theseQ50 solutions involvesonly integer powers, in constrast to the generic nonintepower falloff for solutions withQÞ0. We did not find cor-respondingQ50 solutions withn.1 or k.1 when search-ing for them in a wide range of parameters.

Second, the presence of the dilaton is crucial in derivthe mass formula@7# for rotating black holes, involving onlyglobal charges and horizon properties of the black hoHere the derivation of the mass formula was presenteddetail. The mass formula is similar to the Smarr formulaAbelian black holes, but instead of the magnetic chargeP,the dilaton chargeD enters. This is crucial, since the hairblack holes haveP50, but nonvanishing magnetic fields anthus a magnetic contribution to the mass, which is taken iaccount by the dilaton charge term. The mass formula aholds for EMD black holes. Involving the relative dilatochargeD/g, the mass formula may formally be extendedthe EYM and EM cases, since the relative dilaton chaD/g remains finite in the limitg→0.

Third, the presence of the dilaton allows for a neuniqueness conjecture for hairy black holes@7#. Including thehorizon topological chargeN5n, our nonperturbative set o

8-21

eQ

to

isnon

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ea-laot

isle

iga-jor

de

bel-

thefor-

in-


solutions supports the conjecture, thatblack holes in SU(2)EYMD theory are uniquely determined by their mass M, thangular momentum J, their non-Abelian electric chargetheir dilaton charge D, and their topological charge N.Again, since the relative dilaton chargeD/g remains finite inthe limit g→0, this conjecture may formally be extendedthe EYM case by replacing the dilaton chargeD by the rela-tive dilaton chargeD/g.

We have not yet fully addressed a number of furthersues for these rotating black holes. For instance, in themerical part of the paper we have restricted the calculatito small values of the dilaton coupling constantg. For largervalues ofg, new phenomena might arise. We have neitconsidered in any detail the existence of stationary rota‘‘extremal’’ EYMD solutions @6#. Both these issues deservfurther investigation.

Most importantly, we have not addressed the existencrotating regular solutions in EYMD theory. While perturbtive studies indicated the existence of slowly rotating regusolutions @34#, nonperturbative numerical studies did nsupport their existence@6,20#. Independent of the~non!exist-ence of slowly rotating solutions, there might, however, exrapidly rotating branches of regular as well as black ho

06402

ir,

-u-s

rg

of

r

ts

solutions not connected to the static solutions. The investtion of such rapidly rotating solutions remains a machallenge.

ACKNOWLEDGMENTS

F.N.L. has been supported in part by the MinisterioEducacio´n ~Spain! and by DGICYT Project PB98-0772.

APPENDIX A: EXPANSION AT INFINITY

The asymptotic expansion depends on two integers, laing the black hole solutions, the winding numbern, and thenode numberk. We here explicitly show the dependence onkby choosing the boundary conditions according to Eq.~36!.Note, however, that all solutions can be brought to satisfysame asymptotic boundary conditions by a gauge transmation ~see the discussion in Sec. II C!. Here we restrictourselves to winding numbersn51 andn53. The analysisfor n52 seems to be ‘‘prohibitively complicated’’@20# be-cause already in the static case the leading order termsvolve nonanalytic terms; log(x)/x2 @30#.

1. nÄ1 solutions

Generalizing our previous expansion forn51 @6# including the dilaton, we obtain

f 5122M

x1

2M21Q2

x2 1oS 1

x2D ,

m511C1

x2 1Q22M22D222C1

x2 sin2 u1oS 1

x2D ,

l 511C1

x2 1oS 1

x2D ,

v52aM

x2 26aM21C2Q

x3 1oS 1

x3D ,

f52D

x2

gD2

2x2 1C1D24C712g~M2gD !Q2

6x3 1C7

x3 sin2 u1oS 1

x3D ,

B15Q cosu

x2

~M2gD !Q cosu

x2 1F22~21!kC4~b25!

~b21!Qx2~b11!/21

4~21!kC2C3Q

~a23!~a15!x2~a13!/2

1C3

2~a22a28!Q

a~a21!~a12!~a23!x2a1

2~11g2!Q32C1Q14~M2gD !2Q24~21!kC5Q22C6

6x3 Gcosu

1FC6

x3 1C3

2Q

~a12!~a23!x2a2

8~21!kC4Q

~b21!~b15!x2~b11!/22

4~21!kC2C3Q

~a23!~a15!x2~a13!/2Gcos3 u1oS 1

x3D ,

8-22


B252~21!kQ sinu

x1~21!k

~M2gD !Q sinu

x2 2~21!kF C32~a22a28!Q

a~a21!~a12!~a23!x2a

12~11g2!Q32C1Q14~M2gD !2Q12~21!kC5Q22C6

6x3 12~~21!k21!aM

x3 Gsinu

1F2~21!kC6

x3 2~21!kC3

2Q

~a12!~a23!x2a1

4C2C3Q

~a23!~a15!x2~a13!/22

2C4~b25!

~b21!Qx2~b11!/2Gsinu cos2 u1oS 1

x3D ,

H15F2C5

x2 18C4

b21x2~b21!/22

2C2C3~a13!

~a15!Q2 x2~a11!/2Gsinu cosu1oS 1

x2D ,

H25~21!k1C3x2~a21!/21FC5

x2 1C4x2~b21!/21C3~a212a211!

2~a11!S M1gD2

2C2~a13!

~a15!Q2 D x2~a11!/2G1F2

2C5

x2 22C4x2~b21!/21C2C3~a11!~a13!

2~a15!Q2 x2~a11!/2Gsin2 u1oS 1

x2D ,

H35S C2

x2~21!kC3x2~a21!/2D sinu cosu1F2~21!k

C3~a212a211!

2~a11!S M1gD2

2C2~a13!

~a15!Q2 D x2~a11!/2

2~21!kC4x2~b21!/213C3

2

~a11!~a22!x2~a21!2

2~21!kC52C2~M1gD !13aMQ

2x2 Gsinu cosu1oS 1

x2D ,

H45~21!k1C3x2~a21!/21S ~21!kC2

x2C3x2~a21!/2D sin2 u

1FC3~a212a211!

2~a11!S M1gD2

2C2~a13!

~a15!Q2 D x2~a11!/21C4x2~b21!/21C5

x2 G2FC3~a212a211!

2~a11!S M1gD2

2C2~a13!

~a15!Q2 D x2~a11!/21C4x2~b21!/2

23~21!kC3

2

~a11!~a22!x2~a21!1

2C52~21!kC2~M1gD !1~21!k3aMQ

2x2 Gsin2 u1oS 1

x2D , ~A1!

wherea5A924Q2 andb5A2524Q2.

as

-

-

2. QÄ0 solutions „nÄ1,kÄ1…

Let us now consider the asymptotic expansion for the cQ50 and expand the constantsCi in powers ofQ:

C15C101C11Q1O~Q2!,

C25C201C21Q1O~Q2!,

C35C301C31Q1O~Q2!,

C455

4

C20C30

Q2 1C41*

Q1O~Q0!,

06402

e C5521

2

C20C30

Q2

11

4

3~C20C311C21C30!24C41*

Q1O~Q0!,

C651

10

~5C2013C30!C30

Q1O~Q!. ~A2!

The Q dependence ofC4 andC5 is indeed observed numerically, as seen in Figs. 13~a! and 13~b!. For these figureswe first obtainedC20 andC30 from theQ50 solution, yield-ing C205225.014 andC30525.019. The dashed lines corre

8-23

-er


spond to the approximation that only the first term inC4 andC5 is taken using these values forC20 and C30. Then weextractedC4 andC5 from the functionH1 for several valuesof g ~equivalent to several values ofQ, sincexH andvH arefixed!, where we employedC2 and C3 , extracted from thefunctionsH2 andH3 , for the same values ofg. In the figurewe observe good agreement between these points and

06402

the

approximate theoretical curve.We also observe that unlike the pure EYM case@6#, we

may haveC201C30Þ0 when the dilaton is included. Consequently,H3 can approach zero asymptotically with an integpower falloff. Setting C10521/2M0

2, C1150, C2052b,C2150, C305b, C3150, andC41* 50, we obtain the relationsfor the EYM case, reported previously@6#.

3. nÄ3 solutions

For n53 we find,

f 5122M

x1

2M21Q2

x2 1oS 1

x2D ,

m511C1

x2 1Q22M22D222C1

x2 sin2 u1oS 1

x2D ,

l 511C1

x2 1oS 1

x2D ,

v52aM

x2 26aM213C2Q

x3 1oS 1

x3D ,

f52D

x2

gQ2

2x2 1C1D24C712g~M2gD !Q2

6x3 1C7

x3 sin2 u1oS 1

x3D ,

B15Q cosu

x2

~M2gD !Q cosu

x2 1@2~11g2!Q214~M2gD !22C124~21!kC5#Q22C6

6x3 cosu

1C6

x3 cos3 u1oS 1

x3D ,

B252~21!kQ sinu

x1~21!k

~M2gD !Q sinu

x2

2~21!kF @2~11g2!Q214~M2gD !22C112~21!kC5#Q22C6

6x3 26@12~21!k#aM

x3 1C6

x3 cos2 uGsinu1oS 1

x3D ,

H15C5

x2 sin 2u21

2~21!kC9~e21!x2~e21!/2 sinu cosu1oS 1

x3D ,

H25~21!k1C5

x2 cos 2u2~21!kC9S 121

8~e21!2 sin2 u D x2~e21!/21oS 1

x3D ,

H35C2

xsinu cosu1

~M1gD !C222~21!kC52aMQ

2x2 sinu cosu

1FC8

x3 1C9x2~e21!/211

24S 3C2@2~11g2!Q212~M1gD !21C128~21!kC5#

x3

16~M23gD !aMQ230C8

x3 2C9~e13!~e25!x2~e21!/2D sin2 uGsinu cosu1oS 1

x3D ,

8-24


H45~21!k1~21!kC2

xsin2 u1

C5

x2 1~21!k~M1gD !C222~21!kC52aMQ

2x2 sin2 u

1~21!kF2C9x2~e21!/211

24FC9@~e21!218#x2~e21!/21

24@C81~21!kC2C5#

x3

1S 3C2@2~11g2!Q212~M1gD !21C128~21!kC5#16~M23gD !aMQ230C8

x3 2C9~e13!~e25!x2~e21!/2D3sin2 uGsin2 uG1oS 1

x3D , ~A3!

wheree5A4924Q2.

nric

el

s

g

et

it ithe

APPENDIX B: EXPANSION AT THE HORIZON

We here first motivate our choice of boundary conditioat the horizon. Then we give the full expansion of the metdilaton, and gauge field functions at the horizon.

1. Boundary conditions at the horizon

Let us begin by noting that the ansatz of the gauge fihas the property@20#

]wAm5DmWh , ~B1!

with Wh5ntz/2e. The componentsFmw can be expressed a

Fmw5DmW, ~B2!

with

W5Aw2Wh . ~B3!

The functions ofW transform as a scalar doublet under gautransformationsU5exp(iGtw

n/2), Eq. ~24!. Using the defini-tion of W, Eq. ~B3!, we find, for the componentAt of thegauge field,

At5C2nv

r

tz

2e2

v

rW5C2

v

rW, ~B4!

with

C5C2nv

r

tz

2e. ~B5!

Thus the functions ofC also transform as a scalar doublunder gauge transformations.

To discuss the behavior of the solutions at the horizon

convenient to rewriteW andC as

06402

s,

d

e

s

W5SS sinatr

n

2e1cosa

tz

2eD ,

C51

A4pGES sinb

trn

21cosb

tz

2 D , ~B6!

FIG. 13. ~a! The dependence of the asymptotic coefficientC4 onQ is shown for the black hole solutions withn51, k51, xH51, andvH50.040. The dashed line corresponds to the leading term inexpansion ofC4 . The dots correspond to numerical values ofC4 ,extracted for several values ofg. ~b! Same as~a! for C5 .

8-25

din

e

ua

th

lianthe

po-

on-

axi-

theand


wheretrn5cosnwtx1sinnwty anda, b, E, andS are dimen-

sionless functions ofx andu. We now expand the gauge fielfunctions, the dilaton function, and the metric functionspowers ofd5(x2xH)/xH with coefficients depending onu:

H15 (k51

p

H1kdk, H25 (

k50

p

H2kdk, S5 (

k50

p

Skdk,

a5 (k50

p

akdk, E5 (

k50

p

Ekdk, b5 (

k50

p

bkdk,

~B7!

f5 (k50

p

fkdk, v5vH1 (

k51

p

vkdk,

f 5 (k52

p

f kdk, m5 (

k52

p

mkdk, l 5 (

k52

p

l kdk.

~B8!

The coefficients ofH1 andH2 are restricted by the gaugfixing conditionx]xH12]uH250, andH1(xH)50 has beenused.

Substitution into the field equations and the Einstein eqtions yields, to lowest order ind,

S0E0 sin~a02b0!50, E02@]ub02~12H20!#50,

E05const ~B9!

and

H2150, S150, E150, a150, b150, f150,

f150, v15vH . ~B10!

The first set of conditions can be solved byE050⇔CH50. In this case the electrostatic potential is constant athorizon:

CH5nvH

r H

tz

2e. ~B11!

Let us now discuss the second optionE0Þ0 and

a05b01Nap, ]ub0512H20, ~B12!

whereNa is an integer. In this case we find, forp55,

a5Nap1b1O~d6!, 12H25]ub1O~d6!,

H1

x5]xb1O~d5!, ~B13!

06402

-

e

To see that this solution corresponds to an embedded Abesolution we note that under gauge transformations ofform U5exp(iGtw

n/2) the functionsH1 , 12H2 , a, and btransform like

H1→H185H12x]xG,

12H2→12H28512H22]uG,

a→a85a2G,

b→b85b2G, ~B14!

whereas the functionsS and E are invariant. TakingG5byields, for the gauge-transformed dimensionless gaugetential,

Am8 dxm5F S E2~21!NaSv

x Ddt1~~21!NaS1n!dwG tz

2

1O~d5!, ~B15!

which is indeed an embedded Abelian gauge potential~up tothe order of the expansion!.

Hence we argue that it is only possible to have a n

Abelian solution ifCH50. In terms of the functionsB1 andB2 this condition reads

F ~B1 sinu1B2 cosu!tr

n

21~B1 cosu2B2 sinu!

tz

2 GUx5xH

5nvH

xH

tz

2, ~B16!

which yields

B1ux5xH5n

vH

xHcosu, B2ux5xH

52nvH

xHsinu.

~B17!

The boundary conditions for the functionsf, H2 , H3 , andH4 follow from Eqs.~B10! and are given by Eqs.~38!–~39!.

2. General expansion at the horizon

Taking into account the conditions, Eqs.~B17!, we herepresent the expansion of the functions of the stationaryally symmetric black hole solutions at the horizonxH inpowers of d. These expansions can be obtained fromregularity conditions imposed on the Einstein equationsthe matter field equations:

8-26


f ~d,u!5d2f 2X12d1d2

24H S n

xHD 2 f 2

l 2

e2gf0F24 cotu@2~2H30,u112H202 !H302H20H40,u1H40H40,u#

112S H202 ~H30

2 1H402 21!1

~H202H40!21H30

2

sin2 u2(H30

2 1H402 )

12H20~2H30H40,u1H40H30,u!1122H30,u1H30,u2 1H40,u

2 CG22 cotuS 3

f 2,u

f 2

22l 2,u

l 2D 2F3

f 2,u

f 2

l 2,u

l 2

16f 2,uu

f 2

1S l 2,u

l 2D 2

22l 2,uu

l 2

21826S f 2,u

f 2D 2G

224

f 2S 24n sinu

v2

xH

e2gf0@H30B121~12H40!B22#22e2gf0~B122 1B22

2 !

2sin2 uv2

2

xH2 f 2

$xH2 l 212n2f 2e2gf0@H30

2 1~12H40!2#% D J C1O~d5!,

m~d,u!5d2m2H 123d1d2

24F15024l 2,uu

l 212S l 2,u

l 2D 2

13l 2,u

l 2

m2,u

m226

m2,uu

m216S m2,u

m2D 2

26S f 2,u

f 2D 2

12 cotu S 3m2,u

m224

l 2,u

l 2D224f0,u

2 124 sin2 ul 2v2

2

f 22 G J 1O~d5!,

l ~d,u!5d2l 2H 123d1d2

12F S l 2,u

l 2D 2

22l 2,uu

l 217524 cotu

l 2,u

l 2G J 1O~d5!,

v~d,u!5vH~11d!1d2v21O~d4!,

f~d,u!5f02d2

8 H 2 cotuf0,u1f0,u

l 2,u

l 2

12f0,uu

2S n

xHD 2 f 2

l 2

ge2gf0F4 cotu@2H30~2H30,u11!2H20~2H20H301H40,u!1H40H40,u#

12S ~H302 1H40

2 21!H202 12H20~2H30H40,u1H40H30,u!

1H30,u2 22H30,u1H40,u

2 111~H202H40!

2

sin2 u1

H302

sin2 u2~H30

2 1H402 !D G

181

xH2 f 2

ge2gf0$2n sinuxHv2~H30B121~12H40!B22!1xH2 ~B12

2 1B222 !

1n2 sin2 uv22@H30

2 1~12H402 !#%J 1O~d3!

B1~d,u!5nvH

xHcos1d2~12d!B121O~d4!,

B2~d,u!52nvH

xHsinu1d2~12d!B221O~d4!,

H1~d,u!5dS 121

2d DH111O~d3!,

064028-27


H2~d,u!5H201d2

4Fn2

m2

l 2S H20~H30

2 1H402 21!2H30H40,u1H40H30,u

1H202H40

sin2 u2cotu~22H20H301H40,u!D 2~H11,u1H20,uu!G1O~d3!,

H3~d,u!5H302d2

8 F2S 4gf0,u12f 2,u

f 2

2l 2,u

l 2D ~12H40H202H30,u2cotuH30!

22 cotuH20~H202H40!12H30,uu14H20H40,u22S H30

sin2 u2cotuH30,uD

22H30H202 22H40~2H112H20,u!18 sinu

l 2v2

f 22 S xH

nB121sinuv2H30D G1O~d3!,

H4~d,u!5H402d2

8 F S 4gf0,u12f 2,u

f 2

2l 2,u

l 2D @H40,u2H20H302cotu~H202H40!#

1H20~24H30,u12!12@H30~2H112H20,u!1H40,uu2H40H202 #12

H202H40

sin2 u

22 cotu~22H112H40,u1H20H301H20,u!28 sinul 2v2

f 22 S xH

nB221sinuv2~12H40!D G1O~d3!,~B18!

ge

at

its

i-

r

r-

e

s-

sston

whereH20, H30, H40, H11, f 2 , m2 , l 2 , v2 , f0 , B12, andB22 are functions ofu. Relations~81! and ~82! also hold.

APPENDIX C: ENERGY CONDITIONSAND SEGRE TYPE

In this section we revisit the energy conditions for SU~2!EYMD theory, the results being valid for general gaugroup SU(N). We also consider the Segre´ type of this typeof matter.

The dominant energy conditionrequires that ‘‘for everytimelike vectorV, the stress-energy tensor satisfyTmnVmVn

>0 andTmnVn is a nonspacelike vector.’’ Let us show thEq. ~5! satisfies this condition.

Let V be an arbitrary timelike vector. We may defineassociated unit vectorE0 via

V5A2VmVmE0 . ~C1!

By constructing an orthonormal basis withE0 as its timelikeelement@$E0 ,E1 ,E2 ,E3%#, one can reformulate the domnant energy condition as

T~0!~0!>uT~a!~b!u, ;a,b50,1,2,3, ~C2!

where the~•! index indicates the component in the orthonomal basis.

In order to simplify the proof we separate Eq.~5! into twoparts—namely,

Tmn5Tmndil 1Tmn

YM , ~C3!

06402

-

with

Tmndil 5]mF]nF2

1

2gmn]aF]aF[HmHn2

1

2gmnH2,

~C4!

TmnYM52e2kFTrS FmaFnbgab2

1

4gmnFabFabD , ~C5!

where we have definedHm[]mF.A straightforward calculation performed in the orthono

mal basis then yields

T~0!~0!dil >uT~a!~b!

dil u, T~0!~0!YM >uT~a!~b!

YM u, ;a,b50,1,2,3,~C6!

and from Eqs.~C3! and ~C6! we conclude that condition~C2! holds.

Obviously, theweak energy conditionis also satisfied,T(0)(0)>0.

As for thestrong energy condition, it can be formulated asfollows: ‘‘A stress-energy tensor is said to satisfy thstrong energy condition if it obeys the inequality

TmnVmVn>1

2TVmVm ~C7!

for any timelike vectorV ~T stands for the trace of the stresenergy tensor,T5Tm

m).’’The Yang-Mills part of the stress-energy tensor,Tmn

YM , sat-isfies the strong energy condition trivially as it is a traceletensor that satisfies the weak energy condition. The dila

8-28

b

e,

ae

a

trouanues

therheese

ves


part Tmndil also satisfies the strong energy condition as can

easily shown in an orthonormal basis:

Tmndil VmVn>

1

2TdilVmVm. ~C8!

Note thatTdil5T52H2. Thus Eq.~C7! is satisfied by thefull EYMD stress-energy tensor.

Related to the energy conditions is the Segre´ type ~oralgebraic type! of stress-energy tensor@18#. It consists of astudy of the eigenvalue problem of that tensor—namely,

Tmnun5lum. ~C9!

It is well known that a general EM system has an algbraic type A1@~1 1! ~1, 1!#, with two double eigenvaluesopposite in sign.~We are not considering the null case.! Foran arbitrary EYM system the Segre´ type is that of a diago-nalizable general type, A1@1 1 1, 1#. The non-Abelian naturebreaks the degeneracy of the Abelian case.

But the inclusion of the dilaton field also destroys thdegeneracy A1@~1 1! ~1, 1!# in the Abelian case, since theigenvalues for a general EMD solution satisfy

l11l252H2,

l31l450, ~C10!

with H defined as before. It is no surprise then to findgeneric type for the EYMD solutions.

If we restrict ourselves to Eqs.~15! and ~20!–~22!, wemay write the eigenvalues in the form

l151

2@~T t

t 1T ww !1A~T t

t 2T ww !214T t

w T wt #,

l251

2@~T t

t 1T ww !2A~T t

t 2T ww !214T t

w T wt #,

hend

06402

e

-

t

l351

2@~T r

r 1T uu !1A~T r

r 2T uu !214T r

u T ur #,

l451

2@~T r

r 1T uu !2A~T r

r 2T uu !214T r

u T ur #.

~C11!

Note that this is just a consequence of the Lewis-Papapeform of the metric and the Einstein equations. By usingorthonormal basis it is easy to show that all the eigenvalare real for EMYD solutions.

The explicit expressions of these eigenvalues are racomplicated. To conclude this section we show tasymptotic behavior of the dimensionless version of theigenvalues (n51):

l152D22Q2

2x4 1M ~D222Q2!

x5 1oS 1

x5D ,

l252D21Q2

2x4 1M ~D212Q2!22gDQ2

x5 1oS 1

x5D ,

l35D22Q2

2x4 2M ~D222Q2!

x5 1oS 1

x5D ,

l452D22Q2

2x4 1M ~D222Q2!

x5 1oS 1

x5D . ~C12!

Obviously, we observe that for charged EYM solutions (D50), the algebraic type of the stress-energy tensor behaasymptotically as that of a pure EM system.

D

.

@1# W. Israel, Commun. Math. Phys.8, 245 ~1968!; D. C. Robin-son, Phys. Rev. Lett.34, 905 ~1975!; P. Mazur, J. Phys. A15,3173 ~1982!.

@2# M. Heusler, Black Hole Uniqueness Theorems~CambrigdeUniversity Press, Cambridge, England, 1996!.

@3# M. S. Volkov and D. V. Galt’sov, Sov. J. Nucl. Phys.51, 747~1990!; P. Bizon, Phys. Rev. Lett.64, 2844 ~1990!; H. P.Kunzle and A. K. M. Masoud-ul-Alam, J. Math. Phys.31, 928~1990!.

@4# See, e.g., M. S. Volkov and D. V. Gal’tsov, Phys. Rep.319, 1~1999!; D. V. Gal’tsov, hep-th/0112038, in Proceedings of t16th International Conference on General Relativity aGravitation, Durban, South Africa, 2001.

@5# B. Kleihaus and J. Kunz, Phys. Rev. Lett.79, 1595 ~1997!;Phys. Rev. D57, 6138~1998!.

@6# B. Kleihaus and J. Kunz, Phys. Rev. Lett.86, 3704~2001!; B.Kleihaus, J. Kunz, and F. Navarro-Le´rida, Phys. Rev. D66,104001~2002!.

@7# B. Kleihaus, J. Kunz, and F. Navarro-Le´rida, Phys. Rev. Lett.90, 171101~2003!.

@8# G. W. Gibbons and K. Maeda, Nucl. Phys.B298, 741 ~1988!;D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev.43, 3140~1991!.

@9# G. W. Gibbons and K. Maeda, Ann. Phys.~N.Y.! 167, 201~1986!; V. Frolov, A. Zelnikov, and U. Bleyer, Ann. Phys~Leipzig! 44, 371 ~1987!.

@10# D. Rasheed, Nucl. Phys.B454, 379 ~1995!.@11# J. H. Horne and G. T. Horowitz, Phys. Rev. D46, 1340~1992!.@12# E. E. Donets and D. V. Gal’tsov, Phys. Lett. B302, 411~1993!;

G. Lavrelashvili and D. Maison, Nucl. Phys.B410, 407~1993!.

@13# M. S. Volkov and N. Straumann, Phys. Rev. Lett.79, 1428~1997!.

@14# A. Ashtekar~private communication!.@15# M. Heusler and N. Straumann, Class. Quantum Grav.10, 1299

~1993!; Phys. Lett. B315, 55 ~1993!.

8-29

,

. D

fr-

v,,

larof


@16# L. Smarr, Phys. Rev. Lett.30, 71 ~1973!; Phys. Rev. D7, 289~1973!.

@17# B. Kleihaus, J. Kunz, and F. Navarro-Le´rida Phys. Rev. D~tobe published!; also ~in preparation!.

@18# D. Kramer, H. Stephani, E. Herlt, and M. MacCallum,ExactSolutions of Einstein’s Field Equations~Cambridge UniversityPress, Cambridge, England 1980!

@19# P. Forga`cs and N. S. Manton, Commun. Math. Phys.72, 15~1980!.

@20# J. J. van der Bij and E. Radu, Int. J. Mod. Phys. A17, 1477~2002!.

@21# W. Kundt and M. Tru¨mper, Z. Phys.192, 419 ~1966!.@22# A. Papapetrou, Ann. Inst. Henri Poincare, Sect. A4, 83 ~1966!.@23# R. M. Wald, General Relativity~University of Chicago Press

Chicago, 1984!.@24# Alternatively, the gauge transformation U

5exp(iuty/2)exp(iwtz/2) yields the gauge potential 2eAm8 dxm

5(B1tz1B2tx)dt2n sinu@H3tz1(12H4)tx#@dw2(v/r)dt#1@(H1 /r)dr1(12H2)du#ty2tydu2tz cosudw1tx sinu dw.

@25# F. J. Chinea and F. Navarro-Le´rida, Phys. Rev. D65, 064010~2002!.

@26# B. Kleihaus and J. Kunz, Phys. Rev. Lett.78, 2527 ~1997!;Phys. Rev. D57, 834 ~1998!.

@27# A. Ashtekar, Ch. Beetle, and J. Lewandowski, Phys. Rev64, 044016~2001!.

06402

@28# L. F. Abbott and S. Deser, Phys. Lett.116B, 259 ~1982!; J. D.E. Creighton and R. B. Mann, Phys. Rev. D52, 4569~1995!.

@29# A. Corichi and D. Sudarsky, Phys. Rev. D61, 101501~2000!;A. Corichi, U. Nucamendi, and D. Sudarsky,ibid. 62, 044046~2000!.

@30# B. Kleihaus and J. Kunz, Phys. Rev. D61, 107503~2000!.@31# The gauge transformation isU5 ity .@32# W. Schonauer and R. Weiß, J. Comput. Appl. Math.27, 279

~1989!; M. Schauder, R. Weiß, and W. Scho¨nauer, TheCADSOL

Program Package, Universita¨t Karlsruhe, Interner Bericht Nr.46/92, 1992.

@33# The perturbative calculations of@13# indicate a sign change oQ for n51 k52 black hole solutions: however, the nonpertubative calculations do not show such a change of sign.

@34# O. Brodbeck, M. Heusler, N. Straumann, and M. S. VolkoPhys. Rev. Lett.79, 4310~1997!; O. Brodbeck and M. HeuslerPhys. Rev. D56, 6278~1997!.

@35# This is to be expected in EYM theory, since globally regurotating solutions would have to satisfy a different setboundary conditions at infinity@34#. Nonperturbative numeri-cal analysis, however, suggests@20# that no such globally regu-lar rotating solutions exist.

@36# B. Kleihaus, J. Kunz, and A. Sood, Phys. Rev. D54, 5070~1996!.

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Rotating dilaton black holes with hair