Nutty dyons

b

ions ap-ropertieslian mag-rge may

present-regular

lizations,-ergeshe glob-toon

Physics Letters B 615 (2005) 1–13

www.elsevier.com/locate/physlet

Nutty dyons

Yves Brihayea, Eugen Radub

a Physique-Mathématique, Universite de Mons-Hainaut, Mons, Belgiumb Department of Mathematical Physics, National University of Ireland Maynooth, Maynooth, Ireland

Received 25 February 2005; received in revised form 6 April 2005; accepted 7 April 2005

Available online 15 April 2005

Editor: N. Glover

Abstract

We argue that the Einstein–Yang–Mills–Higgs theory presents nontrivial solutions with a NUT charge. These solutproach asymptotically the Taub–NUT spacetime and generalize the known dyon black hole configurations. The main pof the solutions and the differences with respect to the asymptotically flat case are discussed. We find that a non-Abenetic monopole placed in the field of gravitational dyon necessarily acquires an electric field, while the magnetic chatake arbitrary values. 2005 Elsevier B.V. All rights reserved.

1. Introduction

A feature of certain gauge theories is that they admit classical solutions which are interpreted as reing magnetic monopoles. For non-Abelian gauge fields interacting with a Higgs scalar, there exist evenconfigurations with a finite mass, as proven by the famous ’t Hooft–Polyakov solution[1]. Typically, the mag-netic monopoles admit also electrically charged generalizations—so-called dyons, the Julia–Zee solution[2] of theSU(2)-Higgs theory possibly being the best known case. These solutions admits also gravitating generaboth regular and black hole solutions being considered in the literature (see[3] for a general review of this topics). In SU(2)-Einstein–Yang–Mills–Higgs (EYMH) theory, a branch of globally regular gravitating dyons emsmoothly from the corresponding flat space solutions. The non-Abelian black hole solutions emerge from tally regular configurations, when a finite regular event horizon radius is imposed[4,5]. These solutions ceaseexist beyond some maximal value of the coupling constantα (which is proportional to the ratio of the vector mesmass and Planck mass).

E-mail address:[email protected](Y. Brihaye).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2005.04.016

http://www.elsevier.com/locate/physletb

mailto:[email protected]

2 Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13

es of thertant in

quationsexampleehildted

al

s

etered byic timerequired

agneticnding

urationsssionheialstatic

ck holen thee super-al massitegeneral

the fieldn

Higgs

otion of

It has been speculated that such configurations might have played an important role in the early stagevolution of the Universe. Also, various analyses indicate that the monopole and dyon solutions are impoquantum theories.

Since general relativity shares many similarities with gauge theories, one may ask whether Einstein’s epresent solutions that would be the gravitational analogous of the magnetic monopoles and dyons. The firstof such a solution was found in 1963 by Newman, Unti and Tamburino (NUT)[6,7]. This metric has becomrenowned for being “a counterexample to almost anything”[8] and represents a generalization of the Schwarzscvacuum solution[9] (see[10] for a simple derivation of this metric and historical review). It is usually interpreas describing a gravitational dyon with both ordinary and magnetic mass.1 The NUT charge which plays a durole to ordinary mass, in the same way that electric and magnetic charges are dual within Maxwell theory[11]. Bycontinuing the NUT solution through its horizon one arrives in the Taub universe[7], which may be interpreted aa homogeneous, nonisotropic cosmology with the spatial topologyS3.

As discussed by many authors (see, e.g.,[13,14]), the presence of magnetic-type mass (the NUT paramn) introduces a “Dirac-string singularity” in the metric (but no curvature singularity). This can be removappropriate identifications and changes in the topology of the spacetime manifold, which imply a periodcoordinate. Moreover, the metric is not asymptotically flat in the usual sense although it does obey thefall-off conditions.

A large number of papers have been written investigating the properties of the gravitational analogs of mmonopoles[15,16], the vacuum Taub–NUT solution being generalized in different directions. The correspoconfiguration in the Einstein–Maxwell theory has been found in 1964 by Brill[17]. This Abelian solution hasbeen generalized for the matter content of the low-energy string theory, a number of NUT-charged configbeing exhibited in the literature (see, e.g.,[18] for a recent example and a large set of references). A discuof the non-Abelian counterparts of the Brill solution is presented in[19]. These configurations generalize twell-known SU(2)-Einstein–Yang–Mills hairy black hole solutions[20], presenting, as a new feature, a nontrivelectric potential. However, the “no global non-Abelian charges” results found for asymptotically flat EYMconfigurations[21] are still valid in this case, too.

Here we present arguments for the existence of NUT-charged generalizations of the known EYMH blasolutions[4,5]. Apart from the interesting question of finding the properties of a Yang–Mills–Higgs dyon ifield of a gravitational dyon, there are a number of other reasons to consider this type of solutions. In somsymmetric theories, closure under duality forces us to consider NUT-charged solutions. Furthermore, dusolutions play an important role in Euclidean quantum gravity[22] and therefore cannot be discarded in spof their causal pathologies. Also, by considering this type of asymptotics, one may hope to attain morefeatures of gravitating non-Abelian dyons.

The Letter is structured as follows: in the next section we present the general framework and analyseequations and boundary conditions. In Section3 we present our numerical results. We conclude with Sectio4,where our results are summarized.

2. General framework and equations of motion

2.1. Action principle

The action for a gravitating non-Abelian SU(2) gauge field coupled to a triplet Higgs field with vanishingself-coupling is

1 Note that the Taub–NUT spacetime plays also an important role outside general relativity. For example, the asymptotic mmonopoles in (super-)Yang–Mills theories corresponds to the geodesic motion in a Euclideanized Taub-NUT background[12]. However, thesedevelopments are outside the interest of this work.

Y. Brihaye, E. Radu / Physics Letters B 615 (2005) 1–13 3

g the

mmetric

d to thee vacuumoatchith theassefinds also

he

bys.

(1)S =∫ √−g d4x

(R

16πG− 1

2Tr

(FµνF

µν) − 1

4Tr

(DµΦDµΦ

)),

with Newton’s constantG. The field strength tensor is given byFµν = ∂µAν − ∂νAµ − ie[Aµ,Aν], with Dµ =∂µ − ie[Aµ, ] being the covariant derivative ande the Yang–Mills coupling constant.

Varying the action(1) with respect togµν , Aµ andΦ we have the field equations

(2)

Rµν − 1

2gµνR = 8πGTµν,

1√−gDµ

(√−gFµν) = 1

4ie

[Φ,DνΦ

],

1√−gDµ

(√−gDµΦ) = 0,

where the stress-energy tensor is

(3)Tµν = 2 Tr

FµαFνβgαβ − 1

4gµνFαβFαβ

+ Tr

1

2DµΦDνΦ − 1

4gµνDαΦDαΦ

.

2.2. Metric ansatz and symmetries

We consider NUT-charged spacetimes whose metric can be written locally in the form

(4)ds2 = dr2

N(r)+ P 2(r)

(dθ2 + sin2 θ dϕ2) − N(r)σ 2(r)

(dt + 4nsin2

(θ

2

)dϕ

)2

,

the NUT parametern being defined as usually in terms of the coefficient appearing in the differentialdt +4nsin2(θ/2) dϕ. Hereθ andϕ are the standard angles parametrizing anS2 with ranges 0 θ π , 0 ϕ 2π .

Apart from the Killing vectorK0 = ∂t , this line element possesses three more Killing vectors characterizinNUT symmetries

K1 = sinϕ∂θ + cosϕ cotθ∂ϕ + 2ncosϕ tanθ

2∂t ,

K2 = cosϕ∂θ − sinϕ cotθ∂ϕ − 2nsinϕ tanθ

2∂t ,

(5)K3 = ∂ϕ − 2n∂t .

These Killing vectors form a subgroup with the same structure constants that are obeyed by spherically sysolutions[Ka,Kb] = εabcKc.

Thensin2(θ/2) term in the metric means that a small loop around thez-axis does not shrink to zero atθ = π .This singularity can be regarded as the analogue of a Dirac string in electrodynamics and is not relateusual degeneracies of spherical coordinates on the two-sphere. This problem was first encountered in thNUT metric. One way to deal with this singularity has been proposed by Misner[8]. His argument holds alsindependently of the precise functional form ofN andσ . In this construction, one considers one coordinate pin which the string runs off to infinity along the north axis. A new coordinate system can then be found wstring running off to infinity along the south axis witht ′ = t + 4nϕ, the string becoming an artifact resulting frompoor choice of coordinates. It is clear that thet coordinate is also periodic with period 8πn and essentially becomean Euler angle coordinate onS3. Thus an observer with(r, θ,ϕ) = const follows a closed timelike curve. Thelines cannot be removed by going to a covering space and there are no reasonable spacelike surface. Onethat surfaces of constant radius have the topology of a three-sphere, in which there is a Hopf fibration of tS1 oftime over the spatialS2 [8].

Therefore forn different from zero, the metric structure(4) generically shares the same troubles exhibitedthe vacuum Taub–NUT gravitational field[23], and the solutions cannot be interpreted properly as black hole


acetimee trans-

etdesthe

ic

ction

is

on the

2.3. Matter fields ansatz

While the Higgs field is given by the usual form

(6)Φ = φτ3,

the computation of the appropriate SU(2) connection compatible with the Killing symmetries(5) is a more in-volved task. This can be done by applying the standard rule for calculating the gauge potentials for any spgroup[24,25]. According to Forgacs and Manton, a gauge field admit a spacetime symmetry if the spacetimformation of the potential can be compensated by a gauge transformation[24] LKi

Aµ = DµWi , whereL standsfor the Lie derivative.

Taking into account the symmetries of the line element(4) we find the general form

A = 1

2e

(dt + 4nsin2

(θ

2

)dϕ

)u(r)τ3 + ν(r)τ3 dr + (

ω(r)τ1 + ω(r)τ2)dθ

(7)+ [cosθτ3 + (

ω(r)τ2 − ω(r)τ1)sinθ

]dϕ

.

This gauge connection remains invariant under a residual U(1) gauge symmetry which can be used to sν = 0.Also, because the variablesω andω appear completely symmetrically in the EYMH system, the two amplitumust be proportional and we can always setω = 0 (after a suitable gauge transformation). Thus, similar ton = 0 case, the gauge potential is described by two functionsω(r) andu(r) which we shall refer to as magnetand electric potential, respectively.

2.4. Field equations and known solutions

Within the above ansatz, the classical equations of motion can be derived from the following reduced a

S =∫

dr dt

[1

8πG

(σ(1− NP ′2 − PP ′N ′) + 2P ′(σNP )′ + n2σ 3N

P 2

)

(8)−(

1

e2

(Nσω′2 + σ(ω2 − 1+ 2nu)2

2P 2− P 2u′2

2σ 2− ω2u2

σN

)+ 1

2σNP 2φ′2 + σω2φ2

)],

where the prime denotes the derivative with respect to the radial coordinater .At this point, we fix the metric gauge by choosingP(r) = √

r2 + n2, which allows a straightforward analysof the relation with the Abelian configurations.

Dimensionless quantities are obtained by considering the rescalingsr → r/(ηe), φ → φη, n → n/(ηe), u →ηeu (whereη is the asymptotic magnitude of the Higgs field). As a result, the field equations depend onlycoupling constantα = √

4πGη.The EYMH equations reduce to the following system of five nonlinear differential equations

rN ′ = 1− N + n2N

P 2

(3σ 2 − 1

)

− 2α2(

Nω′2 + 1

2P 2

(ω2 − 1+ 2nu

)2 + P 2u′2

2σ 2+ ω2u2

σ 2N+ 1

2NP 2φ′2 + ω2φ2

),

σ ′ = n2σ(1− σ 2)

rP 2+ α2σ

r

(P 2φ′2 + 2ω′2 + 2ω2u2

σ 2N2

),

(Nσω′)′ = σω

((ω2 − 1+ 2nu) + φ2 − u2 )

,
P 2 σ 2N


es of theal Brill,

tion

catedgular

chosenill be an

(NσP 2φ′)′ = 2σω2φ,

(9)

(P 2u′

σ

)′= 2ω2u

σN− 2nσ

P 2

(ω2 − 1+ 2nu

).

Two explicit solution of the above equations are well known. The vacuum Taub–NUT one corresponds to

(10)ω(r) = ±1, u(r) = 0, σ (r) = 1, φ(r) = 1, N(r) = 1− 2(Mr + n2)

r2 + n2.

The U(1) Brill solution[17] has the form

ω(r) = 0, u(r) = u0 + nQm − Qer

r2 + n2, σ (r) = 1, φ(r) = 1,

(11)N(r) = 1− 2(Mr + n2)

r2 + n2+ α2(Q2

e + Q2m)

4(r2 + n2),

and describes a gravitating Abelian dyon with a massM , electric chargeQe and magnetic chargeQm ≡ 1− 2u0n,u0 being an arbitrary constant, corresponding to the asymptotic value of the electric potential.

It can be stressed that the Brill solution possesses two, one or zero horizons, according to the valufree parametersQe, M , u(∞). In the same way as in the case of Reissner–Nordström solutions, the extremsolution can be defined as the solutions with a degenerate horizon atr = r0. This gives the following conditionsfixing M andr0

(12)r0 = M, M2 + n2 − α2

4

(Q2

e + Q2m

) = 0.

As we will see later, it is convenient to further specify the arbitrary constantu(∞) in such a way theu(r0) = 0,this implying

(13)1− Qm

2n+ nQm − MQe

M2 + n2= 0,

which fixesQm and leavesQe as the only remaining free parameter. In the following we will refer to this soluas to the extremal Brill solution. As far as we could see, it is not possible to expressM andQm in a closed formdepending on(α,n,Qe), but the solution can be constructed numerically.

2.5. Boundary conditions

We want the metric(4) to describe a nonsingular, asymptotically NUT spacetime outside an horizon loat r = rh. HereN(rh) = 0 is only a coordinate singularity where all curvature invariants are finite. A nonsinextension across this null surface can be found just as at the event horizon of a black hole. If the time isto be periodic, as discussed above, this surface would not be a global event horizon, although it would stapparent horizon. The regularity assumption implies that all curvature invariants atr = rh are finite.

The corresponding expansion asr → rh is

N(r) = N1(r − rh) + O(r − rh)2, σ (r) = σh + σ1(r − rh) + O(r − rh)

2,

ω(r) = ωh + ω1(r − rh) + O(r − rh)2, u(r) = u1(r − rh) + u2(r − rh)

2 + O(r − rh)3,

(14)φ(r) = φh + φ1(r − rh) + O(r − rh)2,

whereP 2h = r2

h + n2 and

N1 = 1(

1− 2α2 (ω2h − 1)2

2+ 1 u2

1P2h

2+ ωhφ

2h

),

rh 2Ph2 σh


nnottricing the

ntly setgnetic

ich toymptotic

rge

merical

e values

nctionsll

yting any

σ1 = n2σh(1− σ 2h )

rhP2h

+ α2σh

rh

(P 2

h φ21 + 2ω2

1 + 2ω21u

21

σ 2hN2

1

),

(15)ω1 = ωh

N1

(ω2

h − 1

P 2h

+ φ2h

), u2 = σ1u1

2σh

− nσ 2h (ω2

h − 1)

P 4h

+ u1ω2h

N1P2h

− u1rh

P 2h

, φ1 = 2ω2hφh

N1P2h

,

σh, u1, ωh, φh being arbitrary parameters.The analysis of the field equations asr → ∞ gives the following expression in terms of the constantsc, u0, Qe,

φ1, M

N(r) ∼ 1− 2M

r− 2n2 − α2(φ2

1 + (1− 2nu20)

2 + Q2e)

r2+ M(2n2 + α2φ2

1)

r3+ · · · ,

σ ∼ 1− α2φ21

2r2− 4α2φ2

1M

3r3+ · · · , ω(r) ∼ ce

−√

1−u20r + · · · ,

(16)φ ∼ 1− φ1

r+ φ1M

r2+ · · · , u(r) ∼ u0 − Qe

r+ n(1− 2nu0)

r2− Qe(6n2 + α2φ2

1)

6r3+ · · · .

Note that similar to then = 0 asymptotically flat case, the magnitude of the electric potential at infinity caexceed that of the Higgs field,|u0| < 1.2 The constantM appearing in the asymptotic expansion of the mefunction N(r) can be interpreted as the total mass of solutions (this can be proven rigorously by applygeneral formalism proposed in[28]). Note thatM andn are unrelated on a classical level.

Also, no purely monopole solution can exist for a nonvanishing NUT charge (i.e., one cannot consisteu = 0 unlessω = ±1, in which case the vacuum Taub–NUT solution is recovered). Thus, a non-Abelian mamonopole placed in the field of gravitational dyon necessarily acquires an electric field.

We close this section by remarking that the definition of the non-Abelian charges is less clear forn = 0. Althoughwe may still define a ’t Hooft field strength tensor, in the absence of a nontrivial two-sphere at infinity on whintegrate, the only reasonable definition the non-Abelian magnetic and electric charges is in terms of the asbehavior of the gauge field. By analogy to the asymptotically flat case,Qe andQm are defined fromF (3)

tr Qe/r2

andF(3)θφ Qm sinθ (a similar problem occurs for an U(1) field[27]). Thus, sinceQm = 1 − 2nu0, the usual

quantization relation for the magnetic charge is lost forn = 0, which is a consequence of the pathological lascale structure of a NUT-charged spacetime.

3. Numerical results

Although an analytic or approximate solution appears to be intractable, we present in this section nuarguments that the known EYMH black hole solutions can be extended to include a NUT parameter.

The equations of motion(9) have been solved for a large set of the parameters (α,n,Qe, rh), looking for solu-tions interpolating between the asymptotics(14)and(16). NUT-charged solutions are found for anyn = 0 EYMHdyonic black hole configuration by slowly increasing the parametern (since the transformationn → −n leavesthe field equations unchanged except for the sign of the electric potential, we consider here only positivof n). As expected, these configurations have many features in common with then = 0 solutions discussed in[4];they also present new features that we will pointed out in the discussion. Typical profiles for the metric fuN(r) andσ(r) and for the electric potentialu(r) are presented inFig. 1, for a dyonic black hole solution as weas for two NUT-charged solutions. The gauge functionω(r) and the Higgs scalarφ(r) interpolates monotonicallbetween some constant values on the event horizon and zero respectively one at infinity, without presenlocal extremum (seeFig. 3).

2 This depends on the asymptotic structure of the spacetime. For example, in an anti-de Sitter spacetime,u may take arbitrary values[26].
0


rs. If we

te inton

hapes in theave

Tme-,

tead

c

Fig. 1. The functionsN(r), σ(r) andu(r) are plotted for three typical solutions for the same values of(rh,Qe,α).

The domain of existence of the non-Abelian nutty dyons can be determined in the space of parametefix the electric chargeQe of the solution, then there likely exist a volumeVQ in the parameter space of(α,n, rh)

inside which non-Abelian solutions exist and on the side of which they become singular and/or bifurcaAbelian solution of the type of the Brill solution. Forn = 0 the domain of the(α, rh) plane where non-Abeliasolutions exist was determined in[5] for Qe = 0 and in[4] for Qe = 0.

The determination ofVQ is of course a huge task. In this Letter, we will not attempt to determine the sof VQ accurately but rather attempt to sketch it by analyzing the pattern of solutions on some generic linespace of parameters. For definiteness we setQe = 0.2 in our numerical analysis, although nontrivial solutions hbeen found also for other values of the electric charge.

3.1. n varying

First, we have integrated the system of equations(9) with fixed values forα, rh andQe and increased the NUchargen. Our values here areα = 1.0, rh = 0.2 andQe = 0.2 corresponding to a generic values for the paraters (the correspondingn = 0 gravitating dyon was constructed in[4]). As far as the functionu(r) is concernedthere exists a main difference between the casen = 0 andn = 0. Indeed in the casen = 0 this function behaveasymptotically likeu(r) ∼ u0 + Qe/r + O(1/r4) while in the presence of a NUT charge the behaviour is insu(r) ∼ u0 + Qe/r + K/r2, where, as seen from(16), the constantK increases withn. Thus, whenn becomeslarge, it becomes more difficult to construct numerical solutions with a good enough accuracy,3 for a given valueof the electric chargeQe.

The effect of increasingn apparently depend strongly of the valueα. Forα small (typicallyα 1) the patterncan be summarized by the following points: (i) No local extrema ofN(r) are found for small enough values ofn.For largern, the functionN(r) develops a local maximum and also a local minimum, sayNM andNm at someintermediate,n-depending values ofr . For n large enough, we haveNmax > 1. No local minimum ofN persistfor large enoughn, the minimum ofN(r) (Nm = 1) being attained asr → ∞. (ii) The second metric functionσ(r) still remains monotonically increasing but the valueσ(rh) diminishes whenn increases. (iii) The asymptoti

3 To integrate the equations, we used the differential equation solver COLSYS which involves a Newton–Raphson method[29].


e

c-mmarize

where

Fig. 2. The values of the parametersM , σ(rh), Nm, NM , φ(rh), ω(rh) andu(∞) = u0 are shown as a function ofn for solutions withrh = 0.2,Qe = 0.2 and two different values of the coupling constantα.

valueu(∞) also decreases for increasingn. With the values choosen, we haveu(∞) ≈ 0.189 forn = 0; we find.u(∞) = 0 for n ≈ 0.25 and negative values for largern.

These effects are illustrated onFig. 2a. On this figure we have set 0< n < 2 but we noticed no significant changof the behaviour for larger values ofn.

For larger values ofα (typically α 2) the situation is quite different, namely: (i) The functionN(r) possesesboth a local minimum and a local maximum. (ii) The valuesw(rh) andφ(rh) increase withn and approach respetively zero and one, suggesting that the solution approaches an Abelian Brill solution. These results are suonFig. 2b for α = 2. However, due to numerical difficulties, we could not determine properly the value ofn wherethe bifurcation occur. The statement of a bifurcation into a Brill solution is confirmed in the next subsectionα is varying.


ln

ues

e of

enal

lian

e

lue

l

Fig. 3. The metric functionsN(r), σ(r) and the matter functionsω(r), φ(r) andu(r) are shown as a function ofr for fixed values of(rh,Qe,n)

and three different values ofα. The functionsN(r) andu(r) of the corresponding extremal Brill solution (withω(r) = 0, h(r) = 1) are alsoexhibited.

Nevertheless, it seems that there are two possible patterns forn → ∞: for values ofα smaller than a criticavalueα, solutions with large values ofn seem to occur, while forα > α, the solutions bifurcate into a Brill solutio(for Qe = 0.2 we findα ∼ 1.5). The occurence of these two patterns is reminiscent to the case ofn = 0 gravitatingdyons.

Note also that, as shown in these plots, the mass parameterM takes negative values for large enough valof n. This is not a surprise, since something similar happens already for the U(1) Brill solution(11).

3.2. α varying

We now discuss the behaviour of the solutions for a varyingα and the other parameters fixed. In absenca NUT charge it is know[4,5] that non-Abelian dyonic black hole exist forrh ∈ [0,

√3+ 4Q2

e/2]. For fixedQe

and rh and increasingα they bifurcates into an extremal Reissner–Nordström solution atα ∼ αc. The valueαc

depends of course onrh andQe. For rh 1 the valueαc ≈ 1.4 is found numerically and depends weakly onQe.For rh ∼ √

3+ 4Q2e/2 we haveαc ≈ √

(3+ 4Q2e)/(1+ Q2

e)/2.For n > 0 we see (e.g., onFig. 3) that the local maximum characterizing the functionN(r) of a nutty solution

(at least for large enough values of the NUT chargen) progressively disappears in favor of a local minimum whα increases. This minimum appears far outside the event horizonr = rh and becomes deeper. In fact, the minimvalueNm approaches zero whenα tends to a critical value, sayαc(n,Qe, rh). If we denote byrm the value ofthe radial variable whereN(rm) = 0 (with rm > rh) our numerical results strongly indicate that the non-Abesolution converges into an extremal Brill solution on the intervalr ∈ [rm,∞) for α → αc.

Indeed, the matter functions’ profilesu,w,φ and the metric functionsσ,N all approaches the profiles of thcorresponding extremal Brill solution with the sameαc,Qe,n. This result is illustrated onFig. 3for n = 1, rh = 0.2andQe = 0.2; in this case, we findαc ≈ 2.35 but we believe that the result holds for generic values of(n, rh,Qe).The determination of the critical valueαc(n, rh,Qe) is not aimed in this Letter. However, it seems that the vaαc depends weakly ofn, for example, we findαc ≈ 2.22 forn ∈ [3,4].

Nevertheless we can conclude that nutty dyons exist on a finite interval ofα and bifurcate into extremal Brilsolutions forα = α .
c


ingral

on

fare,

told

charge.ceres.nly

ossiblee cross

Fig. 4. The profiles of the functionsN(r), σ(r) andu(r) are represented forQe = 0.2, n = 0.5, α = 1 and three different values ofrh.

3.3. rh varying

In the case of gravitating dyonic black holes with event horizonrh, the solutions approach the correspondregular gravitating solution on the interval]rh,∞[ when the limitrh → 0 is considered. It is therefore a natuquestion to investigate how nutty-dyons behave in the same limit.

Considering this problem for a few generic values of(α,n) we reach the conclusion that, in the limitrh → 0,the nutty dyon becomes singular atr = 0 because the valueσ(rh) tends to zero. This situation is illustratedFig. 4where the functionsN(r), σ(r) andu(r) are plotted for three different values ofrh andα = 1, n = 0.5.

Remarkably, this figure reveals that the functionsσ(r) andu(r) are rather independant ofrh (it is also truefor w(r), φ(r) which are not represented) while the functionN(r) indeed involves nontrivially withrh. Note alsothat for the metric gauge choiceP(r) = √

r2 + n2, the area of two-spheredΩ2 = P 2(r)(dθ2 + sin2 θ dϕ2) doesnot vanish atr = 0. However, by choosing a Schwarzschild gauge choiceP(r) = r , a straightforward analysis othe corresponding field equations (which can easily be derived from(8)) implies that it is not possible to takeconsistent set of boundary conditions atr = 0 without introducing a curvature singularity at that point. Therefono globally regular EYMH solutions are found forn = 0.

The determination of the domain of nutty dyons for fixed(α,n,Qe) and increasing the horizon radiusrh is verylikely an involved problem. Already in the casen = 0, discussed in[4] the numerical analysis reveals several (upthree) branches of solutions on some definite intervals of the parameterrh. We believe that similar patterns couoccur forn > 0 but their analysis is out of the scope of this Letter.

4. Further remarks

In this work we have analysed the basic properties of gravitating YMH system in the presence of a NUTWe have found that despite the existence of a number of similarities to then = 0 case (for example, the presenof a maximal value of the coupling constantα), the NUT-charged solutions exhibits some new qualitative featu

The static nature of an = 0 spherically symmetric gravitating non-Abelian solution implies that it can oproduce a “gravitoelectric” field. There both non-Abelian monopole and dyon black hole solutions are pto exist, with a well defined zero event horizon radius limit. For a nozero NUT charge, the existence of th


nathe totalhorizon

s, for

iggsat leastns (thegu-tating

moree genericcharge

ty”rge is.etric-)

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metric termgϕt shows that the solutions have also a “gravitomagnetic” field. The termgϕt does not produce aergoregion but it will induce an effect similar to the dragging of inertial frames[30]. In this case we have found thonly non-Abelian dyons are possible to exist and the usual magnetic charge quantization relation is lost. Tmass of these solutions may be negative and the configurations do not survive in the limit of zero eventradius.

A discussion of possible generalizations of this work should start with the radially excited nutty dyonwhich the gauge functionω(r) possesses nodes. These configurations are very likely to exist, continuing forn > 0the excited configurations discussed in[4]. Also, in our analysis, to simplify the general picture, we set the HpotentialV (φ) to zero. We expect to find the same qualitative results for a nonvanishing scalar potential (if the parameters are not to large). It would be a challenge to construct axially symmetric NUT-charged dyocorrespondingn = 0 monopoles are discussed in[31]). Such dyon solutions would present a nonvanishing anlar momentum, generalizing the Abelian Kerr-Newman–NUT configurations (a set of asymptotically flat rosolutions have been considered recently in[32]).

Similar to the casen = 0, the solutions discussed in this work can also be generalized by including ageneral matter content. However, we expect that these more general configurations will present the samproperties discussed in this work. This may be important, since there are many indications that the NUTis an important ingredient in low energy string theory[27], conclusion enhanced by the discovery of “dualitransformations which relate superficially very different configurations. In many situations, if the NUT chanot included in the study, some symmetries of the system remain unnoticed (see, e.g.,[33] for such an example)Therefore, we may expect the NUT charge to play a crucial role in the duality properties of a (supersymmtheory presenting gravitating non-Abelian dyons.

Unfortunately, the pathology of closed timelike curves is not special to the vacuum Taub–NUT solutiafflicts all solutions of Einstein equations solutions with “dual” mass in general[28]. This condition emerges onlfrom the asymptotic form of the fields, and is completely insensitive to the precise details of the naturesource, or the precise nature of the theory of gravity at short distances where general relativity may be expbreak down[23]. This a causal behavior precludes the nutty dyons solutions discussed in this Letter havinclassically and implies a number of pathological properties of these configurations.

Nevertheless, there are various features suggesting that the Euclidean version of NUT-charged solutan important role in quantum gravity[22]. For example, the entropy of such solutions generically do not othe simple “quarter-area law”. As usual, a positive-definite metric is found by considering in(4) the analyticalcontinuationt → it , n → in, which givesP 2(r) = r2 − n2. In this case, the absence of conical singularities atroot rh of the functionN(r) imposes a periodicity in the Euclidean time coordinate

(17)β = 4π

N ′(rh)σ (rh),

which should be equal with the one to remove the Dirac stringβ = 8πn. In the usual approach, the solutionparameters must be restricted such that the fixed point set of the Killing vector∂t is regular at the radial positior = rh. We find in this way two types of regular solutions, “bolts” (with arbitraryrh = rb > n) or “nuts” (rh = n),depending on whether the fixed point set is of dimension two or zero (see[13] for a discussion of these solutionin the vacuum case and[34] for a recent generalization with anti-de Sitter asymptotics).

We expect that the Euclidean nutty dyons will present some new features as compared to the Lorentziterparts. For example, globally regular solutions may exist in this case, sincer = rh corresponds to the origiof the coordinate system (note also that the SU(2) Yang–Mills system is known to present self-dual soluthe background of a vacuum Taub–NUT instanton[35]). In the absence of closed form solutions, the properof these non-self dual EYMH solutions cannot be predicted directly from those of the Lorentzian configurHowever, similar to the Lorentzian case, they can be studied in a systematic way, by using both analytnumerical arguments. For example, the magnitude of the electric potential at infinity of the Euclidean solu


int

pport.003/390

matics,

mbridge,

not restricted. Also, the conditionβ = 8πn impliesN ′(rh)σ (rh) = 2n and introduces a supplementary constraon the matter functions asr → rh. A study of such solutions may be important in a quantum gravity context.

Acknowledgements

E.R. thanks D.H. Tchrakian for useful discussions. Y.B. is grateful to the Belgian FNRS for financial suThe work of E.R. is carried out in the framework of Enterprise-Ireland Basic Science Research Project SC/2of Enterprise-Ireland.

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Nutty dyons