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Magnetically Charged Black Holes and Their Stability

arXiv:gr-qc/9212009v1 14 Dec 1992UWThPh-1992-63ESI-1992-1November 6, 2018Magnetically Charged Black Holes and Their StabilityPeter C. AichelburgInstitut f¨ur Theoretische PhysikUniversit¨at WienandPiotr Bizon*International Erwin Schr¨odinger Institute for Mathematical PhysicsA-1090 Vienna, AustriaAbstractWe study magnetically charged black holes in the Einstein-Yang-Mills-Higgs theory in thelimit of infinitely strong coupling of the Higgs field. Using mixed analytical and numerical meth-ods we give a complete description of static spherically symmetric black hole solutions, bothabelian and nonabelian.

In particular, we find a new class of extremal nonabelian solutions.We show that all nonabelian solutions are stable against linear radial perturbations. The im-plications of our results for the semiclassical evolution of magnetically charged black holes arediscussed.

*) On leave of absence from Institute of Physics, Jagellonian University, Cracow, Poland.

11IntroductionIt has long been thought that the only static black hole solution in spontaneously brokengauge theories coupled to gravity (Einstein-Yang-Mills-Higgs (EYMH) theories) is the Reissner-Nordstrøm (RN) solution, with covariantly constant Higgs field and the electromagnetic fieldtrivially embedded in a nonabelian gauge group [1]. This belief, based on no-hair properties ofblack holes, was put in doubt by the discovery of essentially nonabelian black holes in EYMtheory (so called colored black holes) [2], because it became clear that the uniqueness propertiesof black holes in Einstein-Maxwell theory [3] are lost when the gauge field is nonabelian.

Thissuggested that there might also exist nontrivial (i.e. different than RN) black holes when theHiggs field is included.

Indeed, it was found recently that in the EYMH theory there are spher-ically symmetric magnetically charged black holes which are asymptotically indistinguishablefrom the RN solution but carry nontrivial YM and Higgs fields outside the horizon [4,5]. Theseessentially nonabelian solutions may be viewed as black holes inside ’t Hooft-Polyakov magneticmonopoles.

Because the EYMH field equations are rather complicated (even for spherical sym-metry), one does not have yet a complete picture of these solutions. In particular, one would liketo know how the solutions depend on external parameters (like coupling constants)1, whetherthere are solutions with degenerate horizon, and, most important, whether the solutions arestable.The aim of this paper is to give answers to these questions within a model obtained from theEYMH system by taking the limit of infinitely strong coupling of the Higgs field.

In this limit theHiggs field is effectively “frozen” in its vacuum expectation value. Thus one degree of freedomis reduced which simplifies considerably the analysis of solutions.

This model is interesting onits own for testing ideas about black holes and as we will argue has the additional virtue thatmany of its features carry over to the full EYMH theory (albeit there are also some importantdifferences).The paper is organized as follows. In the next section we define the model and derive thespherically symmetric field equations.In Section 3 we consider static black hole solutions.

The analysis of asymptotic behaviourof solutions at infinity and at the horizon allows us to obtain information about possible global1This question was examined in the Prasad-Sommerfield limit in Ref. [5].

2behaviour of solutions. We show that black hole solutions may exist only if their parameterssatisfy certain inequalities.

In particular, we derive necessary conditions for the existence ofextremal black holes.In Section 4 we describe numerical results and discuss the qualitative properties of solutions.The bifurcation structure of solutions as functions of external parameters is discussed in detail.We also comment on the status of no-hair conjecture in our model.Section 5 is devoted to stability analysis. We derive the pulsation equation governing theevolution of small radial perturbations about a static solution and solve it numerically.

Specialemphasis is placed on the relation between stability properties and the bifurcation structure ofsolutions. Using this relation we conjecture about the stability of black holes in the full EYMHtheory.Finally, in Section 6 we summarize our results and touch upon the question of evolution ofblack holes in the model due to the Hawking radiation.2Field EquationsThe Einstein-Yang-Mills-Higgs (EYMH) theory is defined by the actionS =Zd4x √−g116πGR + Lmatter(1)withLmatter = −116π F 2 −12(Dφ)2 −λ(φ2 −v2)2(2)F = dA + e[A, A](3)Dφ = dφ + e[A, φ](4)where the YM connection A and the Higgs field φ take values in the Lie algebra of the gaugegroup G. Here we consider G = SU(2).There are four dimensional parameters in the theory: Newton’s constant G, the gauge cou-pling constant e, the Higgs coupling constant λ and the vacuum expectation value of the Higgsfield v. Out of these parameters one can form two scales of length:"√Ge#= 1ev= length.

3The ratio of these two scalesα =√G v(5)plays a rˆole of the effective coupling constant in the model. The second dimensionless parameterβ =√λe(6)measures the strength of the Higgs field coupling.

In this paper we consider in detail the caseβ = ∞. In this limit the Higgs field is “frozen” in its vacuum expectation value v.We are interested in spherically symmetric configurations.

It is convenient to parametrizethe metric in the following wayds2 = −B2Ndt2 + N −1dr2 + r2(dϑ2 + sin2 ϑdϕ2)(7)where B and N are functions of (t, r).We assume that the electric part of the YM field vanishes. Then the purely magnetic spher-ically symmetric su(2) YM connection can be written, in the abelian gauge, aseA = wτ1dϑ + (cot ϑτ3 + wτ2) sin ϑdϕ(8)where τi (i = 1, 2, 3) are Pauli matrices and w is a function of (t, r).The spherically symmetric Ansatz for the Higgs field in the abelian gauge isφ = vτ3h(t, r).

(9)As we said we concentrate on the limit β = ∞, which implies h(r, t) ≡1. Note that althoughthe Higgs field is constant, its covariant derivative is nonzero and generates a mass term for theYM field:(Dφ)2 = v2w2.The YM curvature is given byeF = ˙wdt ∧Ω+ w′dr ∧Ω−(1 −w2)τ3dϑ ∧sin ϑdϕ(10)where (˙= ∂t, ′ = ∂r) andΩ= τ1dϑ + τ2 sin ϑdϕ.

4From the EYMH Lagrangian (1) one derives the following components (in orthonormal frame)of the stress energy tensor Tabρ ≡14π Tbtbt =1e2r2 (Nw′2 + B−2N −1 ˙w2) + (1 −w2)22e2r4+ v2 w2r2(11)τ ≡−14πTbrbr = −1e2r2 (Nw′2 + B−2N −1 ˙w2) + (1 −w2)22e2r4+ v2 w2r2(12)σ ≡14π Tbtbr = 2B−11e2r2 ˙ww′. (13)(In what follows we do not make use of the other nonzero components Tbϑbϑ = Tbϕbϕ.

)The Einstein equations reduce to the systemN ′ = 1r(1 −N) −2Grρ(14)B′ = GrBN −1(ρ −τ)(15)˙N = −2GrBNσ. (16)The remaining bϑbϑ-component of the Einstein equations is equivalent to the YM equation−(B−1N −1 ˙w)˙+ (BNw′)′ + 1r2Bw(1 −w2) −Be2v2w = 0.

(17)For G = 0 the equations (14-16) are solved either by the Minkowski spacetime N = B = 1, orby the Schwarzschild spacetime B = 1, N = 1 −2m/r, and then the Eq. (17) reduces to theYMH equation on the given background.

For v = 0, the system (14 – 17) reduces to the EYMequations.3Static SolutionsIn this section we wish to consider static black hole solutions of the system (14 – 17). In termsof the dimensionless variable x =e√Gr the static equations areN ′ = 1x(1 −N) −2x(Nw′2 + (1 −w2)22x2+ α2w2)(18)B′ = 2xBw′2(19)(NBw′)′ + B 1x2 w(1 −w2 −α2x2) = 0.

(20)

5Note that the function B can be eliminated from Eq. (20) by using Eq.

(19). This system ofequations has been studied previously by Breitenlohner et al.

[5] who found asymptotically flatsolutions with naked singularity at x = 0. This singularity is an artefact of the limit β = ∞,since then there is no symmetry restoration at the origin and the last term in the expression(11) for the energy density diverges at r = 0.

In the black hole case the singularity is hiddeninside the horizon.We will consider solutions of Eqs. (18 – 20) in the region x ∈[xH, ∞), where xH is theradius of the (outermost) horizon.

The boundary conditions at x = xH areN(xH) = 0,N ′(xH) ≥0,(21)and the functions w, w′, B are assumed to be finite.At infinity we impose asymptotic flatness conditions which are ensured byN(∞) = 1,w(∞) = w′(∞) = 0,B(∞) finite. (22)One explicit solution satisfying these boundary conditions is well known.

This is the Reissner-Nordstrøm (RN) solutionw = 0,B = 1,N = 1 −2mx + 1x2(23)where mass m ≥1 (mass is measured in units 1/√G e). The corresponding YM curvature isF = −τ3dϑ ∧sin ϑdϕshowing that it is an abelian solution describing a black hole with unit magnetic charge (theunit of charge is 1/e).It turns out that the system (14-16) admits also nonabelian solutions which we have foundnumerically.

Before discussing the numerical results we want to make some elementary observa-tions about the global behaviour of solutions satisfying the boundary conditions (21) and (22).In what follows we assume that α is nonzero.First, consider the behaviour of solutions at infinity. The asymptotic solution of Eqs.

(18 –20) isw≃e−αxN≃1 −2mx + 1x2 + O(e−2αx)B≃1 + O(e−2αx)

6hence for large x the solutions are very well approximated by the RN solution (23).Second, notice that w has no maxima for w > 1 and no minima for w < −1, which followsimmediately from Eq. (20).

Thus, if w once leaves the region w ∈(−1, 1), it cannot reenterit. Actually, w stays within this region for all x ≥xH, because w(xH) < 1 (without loss ofgenerality we may assume that w(xH) > 0, because there is a reflection invariance w →−w).To see this, consider Eq.

(20) at x = xH. Assuming that w′′(xH) is finite one hasN ′w′ + 1x2 w(1 −w2 −α2x2)x=xH= 0.

(24)If w(xH) ≥1, then w′(xH) > 0, but since there are no maxima for w > 1 the condition w(∞) = 0cannot be fulfilled. Thus w(xH) < 1.

Next one can show that w has no positive minima andnegative maxima. Suppose that there is a positive minimum at some x0 > xH.

Then Eq. (20)implies that the function f(x) = 1 −w2 −α2x2 is negative at x0.

For x ≥x0, f(x) decreases(because w′ ≥0), hence there cannot exist a maximum of w for x > x0, so again the conditionw(∞) = 0 cannot be met. One can repeat this argument to show that w′(xH) < 0, becauseif w′(xH) > 0, then f(xH) < 0, as follows from Eq.

(20). Finally, notice that for x > 1/α,w cannot have any extrema because then f(x) < 0, hence there are no positive maxima andnegative minima (whereas other extrema we have already excluded).To summarize, we have shown that if there exits a solution of Eqs.

(18 – 20), satisfying theboundary conditions (21-22), then the function w stays in the region w ∈(−1, 1) and eithermonotonically tends to zero or oscillates around w = 0 for x < 1/α and then monotonicallygoes to zero.Now, we will show that black hole solutions may exist only if the parameters α and xHsatisfy certain inequalities. Let b ≡w(xH).

We have shown above that w′(xH) < 0 for b > 0,hence from (24) we obtain1 −b2x2H≥α2(25)which implies the necessary condition for the existence of a black hole solutionαxH ≤1. (26)As we shall see in the next section this is not a sufficient condition.For xH < 1 we can improve the inequality (26).

From Eq. (18) we haveN ′(xH) = 1xH−(1 −b2)x3H−2α2b2xH(27)

7hence(1 −b2)2 −x2H + 2α2x2Hb2 ≤0. (28)This inequality has a real solution for b only if the discriminant14∆= (α4x2H −2α2 + 1)x2H(29)is nonnegative, which is always satisfied for xH ≥1 but for xH < 1 this implies the additionalconditionα2 ≤α2max = 1x2H(1 −q1 −x2H).

(30)It turns out from numerical results that the inequality (30) is also a sufficient condition for theexistence of black holes. Note that when xH →0 then αmax →1/√2, which is the upper boundfor “regular” solutions found by Breitenlohner et al.

[5].Finally, let us see whether there may exist extremal black holes in the model. By extremalwe mean a solution with degenerate horizon, i.e.

N ′(xH) = 0. For such solutions the inequalities(25) and (28) are saturated.

Eliminating b2 we obtain the conditionα4x2H −2α2 + 1 = 0(31)which is equivalent to ∆= 0 and is solved by α = αmax given by (30) provided that xH ≤1. Letus point out that the necessary conditions for the existence of extremal black hole solution inEYMH theories were derived some time ago by Hajicek [6].

In the terminology of Hajicek, Eq. (31) is a special case of the zeroth-order condition.

In the next section we will find numericallythe extremal solutions satisfying (31).4Numerical ResultsThe formal power-series expansion of a solution near the horizon isw(x)=b +∞Xk=11k!w(k)(xH)(x −xH)k,N(x)=∞Xk=11k!N (k)(xH)(x −xH)k.(32)All coefficients in the above series are determined, through recurrence relations, by b, in particu-lar the expressions for w′(xH) and N ′(xH) are given by Eqs. (24) and (27).

Thus this expansion

8(assuming that its radius of convergence is nonzero) defines a one-parameter family of localsolutions labelled by the initial value b. We use a standard numerical procedure, called shoot-ing method, to find such values of b for which the local solution extends to a global solutionsatisfying the asymptotic boundary conditions (22).

Note that b is not arbitrary but, as followsfrom Eqs. (25) and (28), must lie in the interval1 −α2x2H −xHqα4x2H −2α2 + 1 ≤b2 ≤1 −α2x2H.

(33)We find that for every xH, there is a maximal value αmax(xH), such that for α > αmax(xH) thereare no solutions, while for α ≤αmax there is exactly one solution for which w is monotonicallydecreasing. This solution we call fundamental, in contrast to solutions with oscillating w whichwe call excitations.

In what follows we restrict our attention to fundamental solutions. At theend we will briefly describe excitations.The numerical results fo several values of xH are summarized in Table 1.

When xH ≤1,then αmax(xH) is given by Eq. (30).

For xH > 1, αmax(xH) is displayed in Table 2 (in this casethe analytical bound given by Eq. (26) is not sharp).

In Fig. 1 we plot w(x) for xH = 2 andseveral values of α.Table 1:Shooting parameter b and mass m (in units 1/√G e) as functions of α for xH = 0.5,xH = 1, and xH = 2.xH = 0.5xH = 1xH = 2αbmαbmαbm0.010.999930.2680.010.999750.5170.010.999071.0180.20.982960.5720.10.981910.6630.10.93171.1470.40.947820.8090.50.630990.98050.20.71421.22860.60.9230280.9540.740.18520.999370.260.430741.2480.70.9283590.9900.90.008680.999990.280.24141.24980.730.9305340.9960.288670.0121.25

9Table 2:Maximal value of α as a function of xH for xH ≥1.xHαmax111.0000010.8221.010.7441.10.6051.50.39820.288100.055200.027On the basis of analytical and numerical results we have a pretty clear picture of the qual-itative behaviour of solutions. Let us discuss in more detail how the solutions depend on theparameters α and xH.

First consider the limit α →0. In our units this corresponds to v →0.In this limit a solution on any finite region outside the horizon is well approximated by theSchwarzschild solutionw = 1,N = 1 −xHx ,with mass m = xH/2.

However, for large x and small but nonzero α the term α2x2 in Eqs. (18)and (20), becomes dominant and asymptotically the solution tends to the RN solution.Next, consider the limit α →αmax(xH).

In this case the behaviour of solutions dependson whether xH is less or greater than one. For xH ≤1, the interval of allowed values ofb, given by Eq.

(33), shrinks to zero as α goes to αmax, and for α = αmax, b is uniquelydetermined by xH. The corresponding limiting solution describes an extremal black hole whichis essentially nonabelian.

For xH ≥1, when α goes to αmax the solution tends to the RNsolution and for α = αmax coalesce with it. Thus for given xH ≥1, the point α = αmax(xH)is a bifurcation point: for α > αmax there is only one solution (RN), while for α ≤αmax thesecond (nonabelian) solution appears.

The bifurcation diagram in the plane (α, xH) is graphedin Fig. 2.

The nonabelian solutions exist only in the region below the curve ABC. Along thecurve AB, the nonabelian solutions are extremal.

Along the curve BC the nonabelian and RNsolutions coalesce. The RN solutions exist for xH ≥1 and do not depend on α (for xH = 1 theRN solution is extremal).

10The bifurcation of solutions is also shown in Fig. 3, where the mass m as a function of α isgraphed for given xH ≥1.

Notice that the mass of the RN solution mRN = 12(xH + 1xH ) is largerthan the mass of the nonabelian solution with the same xH. This suggests that for α < αmax,where there are two distinct solutions with the same radius of the horizon xH, the RN solutionis unstable.

We will show in the next section that this is indeed the case.Let us comment on the status of the no-hair conjecture in our model. The no-hair conjecture(in its strong version) states that stationary black hole solutions, within a given model, areuniquely determined by global charges defined as surface integrals at spatial infinity such asmass, angular momentum and electric or magnetic charge.

In our case, all solutions are staticand have unit magnetic charge, so the only global parameter by which the solutions may differat infinity is their mass. For α ≥1 there is only one solution for given mass m, namely theRN solution, hence the strong no-hair conjecture is valid.

However, for α < 1 the situationis different. This is illustrated in Table 3 which shows how the masses of nonabelian and RNsolutions depend on xH for given α.Table 3:Masses of nonabelian and RN black holes as functions of xH for α = 0.1xHmNAmRN0.10.2260.20.27410.6630.7521.1471.2552.5992.65.552.865082.865095.5952.8868652.88686563.0833384.0625For the RN solution xH ≥1, so its mass mRN = 12(xH + 1xH ) is bounded from below by 0.75.At xmaxH(α) the nonabelian and RN solutions coalesce, so the maximal mass of the nonabeliansolution is equal to mRN(xmaxH).

For given α < 1 and 0.75 ≤m < mRN(xmaxH) there are two

11distinct (i.e. RN and nonabelian) solutions with the same mass2.

This violates the strong no-hairconjecture.We will show in the next section that in the region where two distinct solutions coexist, onlyone of them is stable. Thus, in our model, a weak no-hair conjecture holds, i.e.

global chargesdetermine uniquely the stable black hole solution. However, as we will argue, in the full EYMHtheory even the weak no-hair conjecture is violated.Finally, let us briefly consider solutions for which w is not monotonic.

Such solutions maybe viewed as excitations of the fundamental solutions described above. Their existence in theEYMH model was first noticed in Ref.

5. To understand the existence of excitations it is usefulto consider the limit α →0.

Then, the Eqs. (18 – 20) become the EYM equations, whichhave a countable family of so-called colored black holes [2].

These solutions are labelled byan integer n = number of nodes of w, and in contrast to the α ̸= 0 case, they have zeromagnetic charge (because w(∞) = ±1). For sufficiently small α, the excitations may be viewedas singular perturbations (in α) of colored black holes.

As long as αx ≪1, the excitationsare well approximated by colored black hole solutions (with the same number of nodes for w),but for large x the term α2x2 in Eqs. (18) and (19) becomes dominant and forces w to decayexponentially.

This is shown in Fig. 5, where the n = 1 excitation is graphed for two values ofα and compared to the n = 1 colored black hole solution.It is rather difficult to find numerically the excitations with large n, however we expect thatfor given xH and α ̸= 0, the number of excitations is finite (in contrast to the α = 0 case, wherethere are infinitely many solutions).

Our expectation is based on the monotonicity of solutionsfor x < 1/α, proven in Section 3. The existence of excitations with arbitrarily large n would beinconsistent with this property, because, in analogy to the colored black holes, the location ofnodes is expected to extend to infinity as n increases.The bifurcation structure of excitations is similar to that for the fundamental solution.

Forgiven xH ≥1 there is a decreasing sequence {α0max, α1max, . .

.} of bifurcation points such thatat αnmax the n-th excitation bifurcates from the RN branch (by zeroth excitation we mean thefundamental solution).

As we will discuss in the next section this picture is closely related tothe stability properties of the RN solution.2For sufficiently small α the excitations appear so there are even more than two solutions with the same mass.

125Stability AnalysisIn this section we address the issue of linear stability of the black hole solutions described above.To that purpose we have to study the evolution of linear perturbations about the equilibriumconfigurations. Since we do not expect nonspherical instability (and since, admittedly, the anal-ysis of nonspherical perturbations would be extremely complicated), we restrict our study toradial perturbations.

For radial perturbation the stability analysis is relatively simple, becausethe spherically symmetric gravitational field has no dynamical degrees of freedom and thereforethe perturbations of metric coefficients are determined by the perturbations of matter fields.This was explicitly demonstrated for the EYM system by Straumann and Zhou [7] who derivedthe pulsation equations governing the evolution of radial normal modes of the YM field. Belowwe repeat their derivation with a slight modification due to the presence of the mass term inEqs.

(14 – 17).We define the functions a and b by ea ≡BN and eb ≡N and write the perturbed fields asw(r, t)=w0(r) + δw(r, t)a(r, t)=a0(r) + δa(r, t)(34)b(r, t)=b0(r) + δb(r, t)where (w0, a0, b0) is a static solution. We insert the expressions (34) into Eqs.

(14 – 17) andkeep first order terms in the perturbations. Hereafter we use dimensionless variablesτ =e√Gtandx =e√Grand we omit the subscript 0 for static solutions.

From (16) we obtainδ˙b = −4xw′δ ˙w. (35)The asymptotically flat solution of this equation isδb = −4xw′δw.

(36)Linearization of Eq. (15) yieldsδa′ −δb′ = 4xw′δw′.

(37)Thus, using (36), we obtainδa′ = 4x2 w′δw −4xw′′δw. (38)

13Multiplying Eq. (17) by e−a and linearizing we get−e−2aδ ¨w + e−a(eaδw′)′ + δa′w′ −δbe−bw 1x2 (1 −w2) −α2+ e−b 1x2 (1 −3w2) −α2δw = 0.

(39)Now, we insert (36) and (38) into (39) and make the Ansatzδw = eiωτξ(x)(40)to obtain the eigenmode equation−ea(eaξ′)′ + Uξ = ω2ξ(41)wheree−aU = −4x2 ea(1 + a′x)w′2 −8xea−bww′ 1 −w2x2−α2!−ea−b 1x2 (1 −3w2) −α2. (42)The potential U is a smooth bounded function which vanishes at xH and tends to α2 for x →∞.One can introduce the tortoise radial coordinate to transform Eq.

(41) into the one-dimensional Schr¨odinger equation. However we will not do so, because the numerical analysisis easier when one uses the coordinate x.A static solution (w, a, b) is stable if there are no integrable eigenmodes ξ with negative ω2.To check this we have applied a (slightly modified) rule of nodes for Sturm-Liouville systems[8], which states that the number of negative eigenmodes is equal to the number of nodes of thezero eigenvalue solution.

Namely we have considered Eq. (41) with negative ω2 and looked howthe solution satisfying ξ(xH) = 0 and ξ′(xH) > 0 behaves as ω2 →0−.

We have found that thefunction ξ has no nodes, when U is determined by the fundamental nonabelian solution (for allallowed values of the parameters). Thus we conclude that the fundamental nonabelian solutionsare stable.

When U is determined by the n-th excitation the function ξ seems to have exactlyn nodes (we have checked this up to n = 2), hence the n-th excitation has n unstable modes.For ω2 = 0 and α →αnmax the function ξ, after making n oscillations, tends to zero at infinitywhich signals the existence of static perturbations (bifurcation points).One can apply the same method to examine the stability of the RN solution. This was doneby Lee et al.

[9] in the full EYMH theory. They showed that the pulsation equation for theHiggs field (which in the case of RN solution decouples from the pulsation equation for the YMfield) has no unstable modes.

Thus the question of stability of the RN solution does not depend

14on the parameter β (strength of the Higgs coupling) and reduces for every β to the Eq. (41),where nowU = ea−b α2x2 −1x2.

(43)For given xH ≥1, the existence of negative eigenmodes in this potential depends on α. Forα > αmax(xH) there are no negative eigenvalues, hence the RN solution is stable. For α <αmax(xH), the RN is unstable.

Equivalently, one can say that for given α, the RN is stable ifxH > xmaxH(α), otherwise it is unstable.We have found numerically that when α decreases the RN solution picks up additionalunstable modes. That is, there is a decreasing sequence {αn}, where α0 = αmax, such thatin the interval αn+1 < α < αn there are exactly n unstable modes (we have checked this upto n = 4).

This is consistent with the fact that the RN solution has infinitely many unstablemodes for α = 0, as follows easily from (43). We are convinced that the sequence {αn} coincideswith the sequence {αnmax} of the bifurcation points at which the n-th excitation appears.

Dueto highly unstable behaviour of solutions near αnmax we were able to verify this assertion withsufficient numerical accuracy only up to n = 1. However, as long as there are no other bifurcationpoints (as we believe), our conclusion is based on the general theorem of the bifurcation theorywhich says that if the operator governing small fluctuations if self-adjoint (hence its eigenvaluesare real), then at the bifurcation point one eigenvalue passes through zero, whereas elsewherethe eigenvalues cannot change sign [10].One could have anticipated the stability properties of solutions from a mere comparison ofmasses.

In the region of the plane (α, xH) where two distinct solutions coexist (with the samexH), the solution with lower mass (i.e. nonabelian) is stable, while the solution with higher mass(i.e.

RN) is unstable. However, it should be emphasized that in more complicated situationsthe naive comparison of masses is not conclusive for stability.

Actually, this happens in the fullEYMH theory. To see this, let us consider the bifurcation diagram (m, α) for xH > 1 in the fullEYMH theory.

As follows from the results of Ref. 5 in the Prasad-Sommerfield limit and ourpreliminary results for finite β, this diagram is more complicated than that shown in Fig.

3. Wesketch it in Fig.

5.The horizontal line in Fig. 5 represents the mass of the RN solution.

There are two branchesof (fundamental) nonabelian solutions: the upper branch AB and the lower branch CB. Thesetwo branches merge at the bifurcation point B, which corresponds to αmax(xH).

There is a

15second bifurcation point A, corresponding to some α0(xH), where the upper branch mergeswith the RN solution. One can use these facts to infer the stability properties of solutions.

Aswe have pointed out above the existence of a bifurcation point is a necessary condition for thetransition between stability and instability. However, it is not a sufficient condition, since it isonly when the lowest eigenvalue passes through zero that the stability changes.

Therefore, ifthere are two branches which merge at the bifurcation point and one of them is stable then theother one will have exactly one unstable mode. Applying this reasoning in the present context,we see that if the lower branch is stable for small α (what we expect), then the whole lowerbranch must be stable, whereas the upper one is unstable with exactly one negative mode.

By asimilar argument the RN solution becomes unstable at α0. Therefore, in the region between α0and αmax there are two stable solutions and one unstable.

Note that in this region there is norelation between stability and mass. In particular the point in Fig.

5 at which the lower branchcrosses the horizontal line is not related to change in stability (contrary to the suggestion inRef. 5).The analogous bistability region exists in the plane (m, xH) for fixed α.

This means that forgiven mass m there are two distinct stable black hole solutions which violates the weak no-hairconjecture [11].6DiscussionWe have analyzed static spherically black hole solutions in the strong Higgs coupling limit ofthe EYMH theory. We have found that the spectrum of solutions depends on the value of thedimensionless parameter α characterizing the model.

Our results may be summarized as follows.For α > 1 there is only one branch of solutions, namely the RN family (parametrized by theradius of the horizon rH ≥√G/e). For α < 1 there appears a second (fundamental) branch ofessentially nonabelian solutions.

The radius of the horizon of these solutions is confined to theinterval rminH (α) ≤rH ≤rmaxH(α). The upper bound rmaxH(α) is a bifurcation point at which thetwo branches merge.

The lower bound rminH (α) is nonzero only for 1/√2 < α < 1. The solutionswith rH = rminH (α) > 0 are extremal in the sense that they have a degenerate event horizonand therefore their Hawking temperature vanishes.

The analysis of linear radial perturbationsabout the solutions shows that the nonabelian solutions are stable whereas the RN solution

16changes stability at the bifurcation point, i.e. it is stable for rH > rmaxH(α) and unstable forrH < rmaxH(α).

For sufficiently small values of α there exist also other branches of nonabeliansolutions, which may be viewed as excitations of the fundamental solution. All excitations areunstable.These results have interesting implications for the fate of evaporating black holes in ourmodel.

Consider the RN black hole with unit magnetic charge and large mass. There are threedifferent scenarios of its evolution due to the Hawking radiation depending on the value of α.For α > 1 the situation is the same as in the Einstein-Maxwell theory.

The RN black hole emitsthermal radiation and loses mass. When the horizon shrinks to rH =√G/e (extremal limit),the temperature drops to zero and the evaporation stops.For α < 1, the RN black hole contracts to rH = rmaxH(α) where it becomes classicallyunstable.

The further evolution proceeds along the classically stable nonabelian branch anddepends on whether α is smaller or greater than 1/√2. For α > 1/√2, the temperature dropsto zero when the horizon shrinks to rminH (α) (extremal limit) and the black hole settles downas an extremal nonabelian solution.

For α < 1/√2, the temperature grows as the black holecontracts, so the black hole evaporates completely leaving behind a magnetic monopole remnant.This last scenario was first suggested by Lee et al. [6] as a possible consequence of the instabilityof RN solution.AcknowledgementsP.B.

would like to acknowledge the hospitality of the Aspen Center of Physics where part ofthis work was carried out. This work was supported in part by the Fundaci´on Federico.

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[3] For a review, see P. O. Mazur, in Proceedings of the 11th International Conference onGeneral Relativity and Gravitation, ed. M. A. H. MacCallum (Cambridge University Press,1987).

[4] K. Lee et al., Phys. Rev.

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[11] To our knowledge, the violation of the weak no-hair conjecture was first observed in theEinstein-Skyrme model by P. Bizon and T. Chmaj, Gravitating Skyrmions, University ofVienna preprint, UWThPh-1992-23, to be published in Phys. Lett.

B.Figure captionsFig.1 The solution w(x) for xH = 2 and α = 0.01 (solid line), α = 0.2 (dashed line), andα = 0.288 (dotted line).

18Fig.2 The bifurcation diagram in the (α, xH) plane. Excitations are not included.Fig.3 Mass m (in units 1/√G e) of the fundamental nonabelian solution as a function of α forxH = 2.

The mass of the RN solution is mRN = 1.25.Fig.4 The n = 1 excitations for xH = 1. The dashed line represents the n = 1 colored blackhole solution.Fig.5 The sketch of the bifurcation structure of black hole solutions with xH ≥1 in full EYMHtheory.

Excitations are not included.

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