PHYSICAL REVIEW D, VOLUME 63, 084002

Non-Abelian Einstein-Born-Infeld black holes

Marion Wirschins, Abha Sood, and Jutta KunzFachbereich Physik, Universita¨t Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany

~Received 12 July 2000; published 1 March 2001!

We construct regular and black hole solutions in SU~2! Einstein-Born-Infeld theory. These solutions havemany features in common with the corresponding SU~2! Einstein-Yang-Mills solutions. In particular, se-quences of neutral non-Abelian solutions possess magnetically charged limiting solutions, related to embeddedAbelian solutions. Thermodynamic properties of the black hole solutions are addressed.

DOI: 10.1103/PhysRevD.63.084002 PACS number~s!: 04.20.Jb, 04.40.Nr, 04.70.Bw

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I. INTRODUCTION

Nonlinear electrodynamics was proposed in the 1930remove the singularities associated with charged pointparticles @1#. More recently such nonlinear theories weconsidered in order to remove the singularites associwith charged black holes@2–4#.

Among the nonlinear theories of electrodynamics, BoInfeld ~BI! theory @1# is distinguished, since BI type actionarise in many different contexts in superstring theory@5,6#.The non-Abelian generalization of BI theory yields an amguity in defining the Lagrangian from the Lie-algebra valufields. While the superstring context favors a symmetriztrace @7#, the resulting Lagrangian is so far only knownperturbative expansion@8#. However, the ordinary tracestructure has also been suggested@9#.

Motivated by this strong renewed interest in BI and noAbelian BI theory, recently nonperturbative non-Abeliansolutions were studied, both in flat and in curved spaceparticular, non-Abelian BI monopoles and dyons exist in flspace@10# as well as in curved space@11# together withdyonic BI black holes.

Likewise one expects regular and black hole solutionspure SU~2! Einstein-Born-Infeld~EBI! theory, representingthe BI generalizations of the regular Bartnik-McKinnon slutions@12# and their hairy black hole counterparts@13#. Thisexpectation is further nurtured by the observation that SU~2!BI theory possesses even in flat space a sequence of resolutions@14#.

Here we construct these SU~2! EBI regular and black holesolutions, employing the ordinary trace as in@14#. The set ofEBI equations depends essentially on one parameterg,composed of the coupling constants of the theory. The stions are labeled by the node numbern of the gauge fieldfunction. We construct solutions up to node numbern55and discuss the limiting solutions, obtained forn→`, forvarious values of the horizon radiusxH and the parameterg.We also address thermodynamic properties of the blacksolutions.

II. SU„2… EBI EQUATIONS OF MOTION

We consider the SU~2! EBI action

S5SG1SM5E LGA2gd4x1E LMA2gd4x ~1!

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~2!

LM5b2S 12A111

2b2Fmn

a Famn21

16b4~Fmn

a Famn!2D ,

field strength tensor

Fmna 5]mAn

a2]nAma 1eeabcAm

b Anc, ~3!

gauge coupling constante, gravitational constantG and BIparameterb.

To construct static spherically symmetric regular ablack hole solutions, we employ Schwarzschild-like coornates and adopt the spherically symmetric metric

ds25gmndxmdxn

52A2Ndt21N21dr21r 2~du21sin2udf2!, ~4!

with the metric functionsA(r ) and

N~r !5122m~r !

r. ~5!

The static spherically symmetric and purely magnetic ansfor the gauge fieldAm is

A0a50, Ai

a5eaikrk12w~r !

er2. ~6!

For purely magnetic configurationsFmnFmn50.With the ansatz~4!–~6! we find the static action

S}E drH A

2 F11NS 112r S A8

A1

N8

2ND D G1a2b2r 2AF12A11

2Nw82

b2e2r 21

~12w2!2

b2e2r 4 G J , ~7!

wherea254pG.Introducing dimensionless coordinates and a dimens

less mass function

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MARION WIRSCHINS, ABHA SOOD, AND JUTTA KUNZ PHYSICAL REVIEW D63 084002

x5Aebr , m5Aebm ~8!

as well as the dimensionless parameter

g5a2b

e, ~9!

we obtain the set of equations of motion

m852gx2@12R#, ~10!

A8

A5g

2w82

xR , ~11!

S ANw8

R D 85

Aw~w221!

x2R , ~12!

where

R5A112Nw82

x21

~12w2!2

x4. ~13!

In Eq. ~12! the metric functionA can be eliminated by meanof Eq. ~11!.

We consider only asymptotically flat solutions, where tmetric functionsA andm both approach a constant at infinity. Here we choose

A~`!51. ~14!

For magnetically neutral solutions the gauge field configution approach a vacuum configuration at infinity

w~`!561. ~15!

Globally regular solutions satisfy at the origin the boundaconditions

m~0!50, w~0!51, ~16!

whereas black hole solutions with a regular event horizonxH satisfy there

m~xH!5xH

2, N8w8UxH

5w~w221!

x2 UxH

. ~17!

III. EMBEDDED ABELIAN BI SOLUTIONS

Before discussing the non-Abelian BI solutions, letbriefly recall the embedded Abelian BI solutions@15,2,16#.With constant functionsA andw,

A51, w50, ~18!

they carry one unit of magnetic charge, and their mass fution satisfies

m85g~Ax4112x2!, ~19!

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or upon integration

m~x!5m~0!1gE0

x

~Ax84112x82!dx8. ~20!

Their massm` is thus given by

m`5m~0!1gp

32

3G2 34

. ~21!

The solutions are classified according to the integratconstantm(0) @16#. For m(0).0 black hole solutions withone nondegenerate horizonxH are obtained. Form(0)50black hole solutions with one nondegenerate horizonxH areobtained forg. 1

2 , otherwise the solutions possess no hozon. Form(0),0 black hole solutions with two nondegenerate horizons, extremal black hole solutions with one degerate horizon, or solutions with no horizons are obtainsimilarly to the Reissner-Nordstro”m ~RN! case.

For black hole solutions with event horizon atxH52m(xH) the integration constantm(0) is given by

m~0!5xH

22gE

0

xH~Ax4112x2!dx. ~22!

Since extremal black hole solutions satisfy

m8~xH!5m~xH!

xH5

1

2, ~23!

Eq. ~19! yields for the degenerate horizon of extremal blaholes

xHex5Ag2

1

4g. ~24!

Thus we find a critical valuegcr512 , where xH,cr

ex 50 andmcr(0)50. Finally, the Hawking temperatureT of the blackholes is given by@16#

T51

4pxH@122g~AxH

4 112xH2 !#. ~25!

IV. REGULAR SOLUTIONS

Recently, a sequence of non-Abelian regular BI solutiowas found in flat space by Gal’tsov and Kerner@14#, labeledby the node numbern of the gauge field functionw. We nowconsider this sequence of BI solutions in the presencegravity, i.e., for finiteg ~and thus finitea), and compare thenon-Abelian regular BI solutions to the regular EinsteYang-Mills ~EYM! solutions@12#.

With increasing g the regular BI solutions evolvesmoothly from the corresponding flat space BI solutions.us first consider the sequence of regular BI solutions forgcr .In Fig. 1 we show the gauge field functionw of the regularBI solutions with node numbersn51, 3, and 5. With in-creasing node numbern the gauge field functionw ap-proaches the value zero of the~limiting! Abelian BI gauge

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NON-ABELIAN EINSTEIN-BORN-INFELD BLACK HOLES PHYSICAL REVIEW D63 084002

field function. Because of the boundary conditions this cvergence is nonuniform.

The metric functionN of these BI solutions is shown inFig. 2, together with the metric functionN of the extremalAbelian BI solution withxH

ex50, and one unit of magneticcharge. Clearly, with increasingn the metric function of thenon-Abelian regular BI solution converges nonuniformlythe metric function of the extremal Abelian solution.

In Fig. 3 we show the charge functionP(x)

P2~x!52x

g@m`2m~x!#, ~26!

obtained from an expansion of the Abelian mass functiongeneral chargeP, for these regular BI solutions together withe constant chargeP51 of the extremal Abelian BI solution with xH

ex50.Furthermore, forg5gcr the masses of the non-Abelia

regular BI solutions converge exponentially to the massthe extremal Abelian BI solution withxH

ex50 and unit mag-netic charge, which we therefore refer to as the limitinglution of this sequence.

Let us now considergÞgcr . For g,gcr the limiting so-lution of the sequence of non-Abelian regular BI solutions

FIG. 1. The gauge field functionw is shown as a function of thedimensionless coordinatex for the regular BI solutions with nodenumbersn51, 3, and 5 forg5gcr .

FIG. 2. The metric functionN is shown as a function of thedimensionless coordinatex for the regular BI solutions with nodenumbersn51, 3, and 5 forg5gcr . Also shown is the metricfunction N for the extremal Abelian BI solution withxH

ex50 andunit magnetic charge.

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an Abelian BI solution without horizon, with unit magnetcharge, and with integration constantm(0)50.

For g.gcr we must distinguish two spatial regions,x,xH

ex @given in Eq.~24!#, andx.xHex. Only in the regionx

.xHex the metric functionN of the sequence of non-Abelia

regular BI solutions converges to the metric function of textremal Abelian BI black hole solution with horizonxH

ex andunit magnetic charge. Forx,xH

ex it converges to a non-singular limiting function. This is demonstrated in Fig.where the metric functionN is shown forn55, g50.01,g51, andg5100, together with the metric functionN ofthe corresponding limiting Abelian BI solutions.

Thus the non-Abelian regular BI solutions are very simlar to their EYM counterparts. In particular, the sequenceneutral SU~2! EYM solutions also possesses a limitincharged solution, which for radiusx>1 is the extremal em-bedded RN solution with magnetic chargeP51 @17,18#.

V. BI BLACK HOLE SOLUTIONS

Imposing the boundary conditions Eq.~17! leads to non-Abelian BI black hole solutions with a regular horizon. Liktheir non-Abelian EYM counterparts, non-Abelian BI blac

FIG. 3. The charge functionP2 is shown as a function of thedimensionless coordinatex for the regular BI solutions with nodenumbersn51, 3, and 5 forg5gcr . Also shown is the constanfunction P51 of the Abelian BI solution.

FIG. 4. The metric functionN is shown as a function of thedimensionless coordinatex for the regular BI solutions with nodenumbern55 for g50.01, g51, andg5100. Also shown are themetric functionsN of the corresponding limiting Abelian BI solutions.

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MARION WIRSCHINS, ABHA SOOD, AND JUTTA KUNZ PHYSICAL REVIEW D63 084002

hole solutions exist for an arbitrary value of the horizondius xH . In both cases the sequences of neutral non-AbeBI black hole solutions possess limiting solutions with umagnetic charge.

For g,gcr the limiting solution of the non-Abelian Bblack hole solutions is always the Abelian BI black hosolution with the same horizon. The same holds true fog5gcr .

For g.gcr and xH,xHex the limiting solution is the ex-

tremal Abelian BI black hole solution with horizonxHex only

in the regionx.xHex, but differs from it in the regionxH

,x,xHex. For xH.xH

ex the limiting solution is always theAbelian BI black hole solution with the same horizon. Thisdemonstrated in Fig. 5 forg51 and horizon radiixH50.1,0.2, 0.5, and 10.

The temperature of the non-Abelian BI black holes is otained from@19#

T51

4pAN8. ~27!

In Fig. 6 we show the inverse temperature of the noAbelian BI black hole solutions with node numbersn51, 3,and 5 as a function of their mass forg50.01 andg51. Alsoshown is the inverse temperature of the corresponding liming Abelian BI solutions. Again, with increasingn rapid con-

FIG. 5. The metric functionN is shown as a function of thedimensionless coordinatex for the BI black hole solutions withnode numbern55 for g51 and horizon radiixH50.1, 0.2, 0.5,and 10. Also shown are the metric functionsN of the correspondinglimiting Abelian BI solutions.

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vergence towards the limiting values is observed, anagously to the EYM and EYM-dilaton case@18,20#.

Further details will be given elsewhere@21#.

VI. CONCLUSIONS

We have constructed sequences of regular and blacksolutions in SU~2! EBI theory. The solutions are labeled bthe node numbern. These sequences of non-Abelian neutsolutions possess limiting solutions, corresponding~at leastin the outer spatial region! to Abelian BI solutions with unitmagnetic charge. These features are similar to thoseserved for non-Abelian EYM and EYM-dilaton regular anblack hole solutions@13,17,18,20#. This similarity is also ob-served for the Hawking temperature.

By generalizing the framework of isolated horizonsnon-Abelian gauge theories@22#, recently new results wereobtained for EYM black hole solutions. In particular, notrivial relations between the masses of EYM black holes aregular EYM solutions were found, and a general testing bfor the instability of non-Abelian black holes was sugges@22#. Application of such considerations to non-Abelian Bblack holes appears to be interesting.

Note added. After completion of this work similar resultswere reported in@23#.

FIG. 6. The inverse temperature 1/T is shown as a function ofthe dimensionless massm for the non-Abelian BI black hole solutions with node numbersn51, 3, and 5 forg50.01 andg51.Also shown is the inverse temperature for the corresponding liming Abelian BI solutions.~The g50.01 curves cannot be graphcally resolved in the figure.!

. D

ys.

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