We construct finite mass, asymptotically flat black hole solutions in d=4
Einstein-Yang-Mills theory augmented with higher order curvature terms of the
gauge field. They possess non-Abelian hair in addition to Coulomb electric
charge, and, below some non-zero critical temperature, they are
thermodynamically preferred over the Reissner-Nordstrom solution. Our results
indicate the existence of hairy non-Abelian black holes which are stable under
linear, spherically symmetric perturbations.Comment: 8 pages, 3 figure

A. Messiah

D. H. Tchrakian

Eugen Radu

J. Polchinski

M. S. Volkov

English

arXiv

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Physical Review D

Stable black hole solutions with non-Abelian fields

We construct finite mass, asymptotically flat black hole solutions in d = 4 Einstein– Yang-Mills theory augmented with higher order curvature terms of the gauge field. They possess non-Abelian hair in addition to Coulomb electric charge, and, below some non-zero critical temperature, they are thermodynamically preferred over the Reissner-Nordström solution. Our results indicate the existence of hairy non-Abelian black holes which are stable under linear, spherically symmetric perturbations

Radu, Eugen

Tchrakian, D. H.

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arXiv:1111.0418v1  [gr-qc]  2 Nov 2011Stable black hole solutions with non-Abelian fieldsEugen Radu† and D. H. Tchrakian⋆⋄†Institut fu¨r Physik, Universita¨t Oldenburg, D-26111 Oldenburg, Germany⋆Department of Computer Science, National University of Ireland Maynooth, Maynooth, Ireland⋄School of Theoretical Physics – DIAS, 10 Burlington Road, Dublin 4, IrelandNovember 3, 2011AbstractWe construct finite mass, asymptotically flat black hole solutions in d = 4 Einstein–Yang-Mills theory augmented with higher order curvature terms of the gauge field. Theypossess non-Abelian hair in addition to Coulomb electric charge, and, below some non-zerocritical temperature, they are thermodynamically preferred over the Reissner-Nordstro¨msolution. Our results indicate the existence of hairy non-Abelian black holes which arestable under linear, spherically symmetric perturbations.Introduction.– In recent years it has been realized that the electrically charged Reissner-Nordstro¨m (RN) black hole, when considered as solution of a more general theory, may becomeunstable to forming hair at low temperatures. This has lead to the discovery of some holographicmodels for condensed matter systems, and, in particular, to a gravitational description ofsuperconductivity (see [1] for a review).The case of Einstein-Yang-Mills (EYM) model with negative cosmological constant Λ ind = 4 spacetime dimensions provides an interesting illustration of these aspects. As shown in[2], there is a second order phase transition between the RN–anti-de Sitter solutions, whichare preferred at high temperatures, and symmetry breaking non-Abelian black holes, which arepreferred at low temperatures. In [2], Λ plays an essential role; although electrically chargedhairy black holes do exist also in a Minkowski spacetime background [3], they have ratherdifferent properties as compared to the anti-de Sitter (AdS) solutions in [2]. In particular theydo not emerge as perturbation of the RN black holes, and, similar to the well-known d = 4asymptotically flat, purely magnetic EYM solutions [4], are also perturbatively unstable.However, one might take the view that in the strong coupling regime the (Λ = 0) EYMtheory is incomplete. Perhaps the simplest possibility to describe this situation is to supplementthe action of the EYM model with higher order curvature terms, for both gravitational andgauge field sectors. As discussed e.g. in [5], the inclusion of (string theory inspired-) correctionsto the gravity action does not lead to qualitatively new features. By contrast, we will here arguethat the situation is different vis a vis the inclusion of higher order Yang-Mills (YM) curvatureterms. This possibility has been overlooked so far in the literature. The first relevant order in1this case is the fourth, in which case the most general such density added to the Lagrangianconsists of the four terms,Ls = c1Tr {FµνFρσF µνF ρσ}+ c2Tr {FµνF µνFρσF ρσ} (1)+ c3Tr{FτνFµτFµλFλν}+ c4Tr{FµνFνρFρλFλµ},with some constant coefficients ci. A particularly priviledged such combination, which weadopted here, is that with c1 = c2 = −4c3, c4 = 0. In that case, Ls features only the secondpower of any “velocity field” and is a causal density just like the Gauss-Bonnet term in gravity[6] or the Skyrme [7] term of the O(4) sigma model. With this specific choice of the constantsci, the Lagrange density (1) is nothing else than the trace of the square of the 4−form cur-vature Fµνρσ = {Fµ[ν , Fρσ]}. This is the second member of the YM hierarchy [8], providinga natural generalization of the usual YM model. A convenient way to express this system isTr{(FµνF˜µν)2}, where a tilde denotes the Hodge dual.Notwithstanding our specific choice for the constants ci in (1), we have verified that forcertain other choices, some salient features of the solutions discussed in this work, in particularthe instability of the RN black hole, persist.The model.– Ignoring for simplicity other possible corrections, we consider the followingaction for the modelS =∫d4x√−g[14R− 12Tr {FµνF µν}+ 3τ2Tr{(FµνF˜µν)2}], (2)(here we have set 4πG/e2 = 1, such that the only parameter of the theory is τ).In what follows, we shall prove that the presence of the last term in (2) leads to an instabilityof the RN black hole, together with the occurance of stable black holes with non-Abelian hairoutside the horizon. We shall restrict attention to the following spherically symmetric Ansatz:ds2 =dr2N+ r2(dθ2 + sin2 θdφ2)−Nσ2dt2, (3)where N, σ are functions of r and t in general. The minimal gauge group for which the super-position of a Coulomb field and a non-Abelian hair is not forbidden by the ’baldness’ theorems[9] is SU(3). Then, as in the τ = 0 case in [3], we shall restrict to an SU(2)× U(1) truncationof the SU(3) group, the general spherically symmetric ansatz for the gauge potential beingA ={(νT3 + UT8)dr + (wT1 + w˜T2)dθ + ((wT2 − w˜T1) sin θ + cos θT3) dφ+ (vT3 + V T8)dt}, (4)where ν, w, w˜, v and U, V are functions of (r, t) and Ti are the standard generators of the SU(3)Lie algebra.For static solutions, one can set the functions ν, w˜, v and U to zero without any loss ofgenerality, resulting in the equationsm′ = Nw′2 +(1− w2)22r2+r2V ′22σ2+ τ(1− w2)2V ′2r2σ2, σ′ =2σrw′2,w′′ + (N ′N+σ′σ)w′ +w(1− w2)r2N+2τ(w2 − 1)V ′2r2Nσ2= 0, (5)2together with the first integral for the electric potential,V ′ = Qσr2(1 +2τ(1− w2)2r4)−1, (6)with Q an arbitrary constant.The RN solution corresponds to a vanishing SU(2) field, w(r) = ±1, and m(r) = M − Q22r,σ = 1, V (r) = Φ − Q/r. This solution has an outer event horizon at rh = M +√M2 −Q2,which becomes extremal for Q =M .Solutions with nonzero magnetic gauge fields should also exist. However, one can see that,for Q 6= 0, the first integral (6) excludes the existence of particle-like configurations with aregular origin. Thus, the only physically interesting solutions of this model describe blackholes, with an event horizon at r = rh > 0, located at the largest root of N(rh) = 0. Theregularity conditions at the horizon imply the following series expansion therem(r) =rh2+m1(r − rh) + . . . , σ(r) = σh + 2σhw21rh(r − rh) + . . . ,w(r) = wh + w1(r − rh) + . . . , V (r) = v1(r − rh) + . . . , (7)where v1 =Qr2hσhr4h+2τ(1−w2h)2, m1 =σ2h(1−w2h)2+v21(r4h+2τ(1−w2h)2)2rhσ2h, w1 =(σ2h−2τv21)wh(w2h−1)(1−2m1)rhσ2h. It is alsostraightforward to show that the requirement of finite energy implies the following asymptoticbehavior at large rm(r) = M − Q22r+ . . . , σ(r) = 1− J22r4+ . . . , w(r) = ±1 + Jr+ . . . , V (r) = Φ− Qr+ . . . . (8)Once the parameters σh, wh and J, M, Q are specified, all other coefficients in (7), (8) canbe computed order by order. M and Q correspond to the mass and electric charge of thesolutions; other quantities of interest are the Hawking temperature TH =14piσ(rh)N′(rh), theentropy S = AH4= πr2h and the chemical potential Φ. J here is an order parameter describingthe deviation from the Abelian solution.The results.– The solutions of the equations (5), (6) interpolating between the asymptotics(7), (8) were constructed numerically, by employing a shooting strategy. Finite mass, non-Abelian black holes exist for any τ ≥ 0. For given Q, rh, τ , the solutions are found for discretevalues of the parameter wh, labeled by the number of nodes, n, of the magnetic YM potentialw(r). To simplify the picture, in this work we have restricted attention to solutions with amonotonic behavior of the magnetic gauge potential w(r) (and thus n = 0, 1 only).In characterizing the non-Abelian configurations, it is convenient to introduce the quantityκ =τQ2. (9)For 0 ≤ κ < 1/2, the solutions can be thought of as nonlinear superpositions of the RN and thepurely magnetic SU(2) black holes in [10]. In particular, they are unstable since the magneticgauge potential has always n ≥ 1 nodes. Moreover, their free energy F = M − THS is alwaysgreater than that of the RN configuration with the same TH and Q.3 0 0.25 0.5 0.75 1 0  0.5  1  1.5  2  2.5  3log10(r)event horizonm(r)V(r)w(r)+1σ(r)τ=2 Q=1.05 rh=1 0 0.04 0.08 0.12 0  0.5  1  1.5  2  UoddUevenFigure 1: The profiles of a typical non-Abelian nodeless solution are shown together with thecorresponding potentials for the perturbation equations (12), (14) below.The picture is very different for κ ≥ 1/2. In this case, for given κ, we notice the existenceof a set of non-Abelian solutions emerging as perturbations of the RN black holes, for a criticalvalue of the charge to mass ratio. This instability of the Abelian configuration is found withinthe Ansatz (4), for values of the magnetic gauge potential w(r) close to the vacuum values ±1everywhere, w(r) = ±1 + ǫW (r). The perturbation W (r) starts from some nonzero value atthe horizon and vanishes at infinity, being a solution of the linear equation(NW ′)′ − (1− 2τQ2r4)2Wr2= 0 , (10)where N = (1 − rhr)(1 − Q2rhr). The second term in this equation can be seen as an effectivemass term µ2 for W near the horizon, with µ2 ∼ 1 − 2κ(Q/rh)4. We have found that for anyκ ≥ 1/2 (i.e. µ2 < 0), an instability occurs for a critical value of the mass to charge ratio ofthe RN solution. An approximate value of this ratio is found by using an asymptotic matchingexpansion for the approximate solutions of (10) at the horizon and at infinity, the resultQM=2√6√1 +√1 + 48κ7 +√1 + 48κ(11)providing good agreement with the numerical data.This unstability signals the emergence of a symmetry breaking branch of the non-Abeliansolution bifurcating from the RN black holes. In contrast to the solutions with κ < 1/2, herewe notice the existence of a fundamental branch of solutions without nodes in the magneticgauge function w(r), see e.g. the typical profile shown in Fig. 1.Among the solutions bifurcating from RN black holes, those with 1/2 ≤ κ ≤ 2 are of specialinterest, sharing some striking similarity with the picture found for the EYM-AdS system in4 0 20 40 60 80 0  0.015  0.03  0.045  0.06  0.075a HtHRNκ=0κ=1 κ=1.815κ=3.125κ=8κ=50 0 10 20 0  0.01  0.02  κ=0.89κ=1.28κ=1.81κ=2RNFigure 2: The scaled horizon area aH = AH/Q2 is plotted vs. the scaled temperature tH = THQfor several values of the ratio κ = τ/Q2.[2]. A plot of the horizon area as a function of the temperature reveals the existence of a singlebranch of non-Abelian solutions, which exist below a critical temperature T(c)H only (the rangeof the scaled temperature tH = THQ being 0 < tH < 0.0203). In a canonical ensemble, the non-Abelian black holes within this range of κ exist for a finite interval of rh (i.e. of the entropy) only.These solutions always possess a positive specific heat, the Hawking temperature vanishing for aminimal value r(min)h =√2τ√√2κ− 1 of the event horizon radius. As r → r(min)h , an extremalnon-Abelian black hole solution with a regular horizon is approached, the charge to mass ratioof this configurations being always greater than one. Furthermore, it turns out that the freeenergy of a RN solution is always larger than the free energy of a non-Abelian solution with thesame temperature and electric charge. Therefore, for 1/2 ≤ κ ≤ 2 these non-Abelian black holesare preferred. The difference of free energies scales like (T(c)H − TH)2 near the transition point,signaling that this is a second order phase transition, while J ∼√1− TH/T (c)H . Moreover, forthe same values of the mass and electric charge, the RN solution has a smaller event horizonradius (and thus a smaller entropy), than the non-Abelian black hole.The picture is somehow different for κ > 2. Again, one finds a single branch of solutionsemerging as a perturbation of the RN black hole, nodeless configurations existing also in thiscase. However, these solutions exist for values of the temperature greater than the critical valueT(c)H and are thermally unstable.Some of these features are shown in Fig. 2 where we plot the scaled horizon area aH =AH/Q2 as a function of the scaled temperature tH = THQ for several values of the ratio τ/Q2.The branch of RN solutions is also shown there. In Fig. 3 the scaled free energy f = F/Qis plotted as a function of the scaled temperature tH for several values of κ. The inlet thereshows the behavior of the parameter J which enters the asymptotic behavior of the magneticpotential, as a function of the ratio TH/T(c)H .5 0.8 0.9 1 1.1 1.2 1.3 0  0.01  0.02  0.03  0.04  0.05ftHRNκ=0κ=2κ=50κ=3.125 0.04 0 0  0.5  1 TH/TH(c)J/Qκ=0.888κ=1.28κ=1.814Figure 3: The scaled free energy f = F/Q is plotted vs. the scaled temperature tH = THQfor several values of κ = τ/Q2. The inlet shows the behavior of the order parameter J whichenters the asymptotics of the magnetic potential.Existence of stable solutions.– An outstanding question now is whether the F 4 termleads also to stable non-Abelian black holes. The fact that, for a range of κ, we have foundnodeless solutions which are thermodynamically favored over the RN black holes suggests apositive answer to this question.For simplicity, we consider linear, spherically symmetric perturbations only. Even in thiscase, the analysis is highly involved and the details will be presented elsewhere. Here we brieflyoutline just the key features.As usual, all field variables are written as the sum of the static equilibrium solution whosestability we are investigating and a time dependent perturbation. Choosing a gauge suchthat U = v = 0, one finds that the fluctuations decouple in two groups. δw(r, t), δV (r, t),δσ(r, t) and δm(r, t) form even-parity perturbations, whereas δw˜(r, t) and δν(r, t) form odd-parity perturbations. The linearized equations imply that δσ(r, t), δV (r, t) and δm(r, t) aredetermined by δw(r, t) = w1(r)e−iΩt, leading to a single Schro¨dinger equation− d2w1dρ2+ Ueven(ρ)w1 = Ω2w1, (12)(where dr/dρ = Nσ) with a potentialUeven =Nσr2[(1− 2τV′2σ2)(3w2 − 1 + 8ww′r(w2 − 1)− 4w′2(1− w2)2r2)(13)+4w′2(2(1− w2)2r2+r2V ′2σ2− 1)+32τ 2w2(1− w2)2V ′2σ2r4(1 + 2τr4(1− w2)2)],which is a regular function in the entire range −∞ < ρ < ∞. The corresponding analysisfor the odd sector is much more evolved. After much algebra, the perturbation equations for6δw˜(r, t) = w˜1(r)e−iΩt and δν(r, t) = ν1(r)e−iΩt can be cast in the form− d2Ψdρ∗2+ Uodd(ρ∗)Ψ = Ω2Ψ, (14)where Ψ = [(1+6τ(1−w2)2/r4)ν1−12τ(w2−1)w′/r4w˜1]F1, dr/dρ∗ = Nσ/(1+12τNw′2/(r2(1+6τ(1 − w2)2/r4)))1/2 and ν1(r) = F2Ψ + F3Ψ′. The potential Uodd and the functions Fi havecomplicated expressions depending on the equilibrium functions m, σ, w and V , with F1 > 0.For nodeless solutions, Uodd and Fi are regular in the entire range −∞ < ρ∗ <∞.For both equations (12) and (14), the potential vanishes near the horizon and at infinity.Then standard results [11] imply that there are no negative eigenvalues for Ω2 (and hence nounstable modes) if the potentials Ueven and Uodd are everywhere positive.Our results indicate that this is indeed the case for some of the solutions with κ > 1/2, seee.g. the inlet in Fig. 1. Interestingly, approaching the extremality appears to imply genericallypositive values for the potentials in (12), (14).Therefore we conclude that, in contrast to all other known d = 4 asymptotically flat hairyblack holes with non-Abelian gauge fields only [4], at least some of our solutions here are linearlystable.Further remarks.– We close with some remarks on the generality of the results in thiswork. First, we remark that the F 2 and F 4 terms in (2) are the pieces which also enter theLagrangian of the non-Abelian Born-Infeld theory [12] describing the low energy dynamics ofD−branes. Although the equations of motion of that model coupled with gravity are morecomplicated and do not admit the RN black hole as a solution, it is natural to expect that thepicture we have found here will share some similarities with the results in that case.Non-Abelian fields featuring both the F 2 and F 4 terms appear also in the higher loop cor-rections to the action of the d = 10 heterotic string [13]. However, a generalization in thatframework of the solutions considered here is not an easy task, due to the occurrence there ofa variety of other fields. Previous work in this direction [5] indicates that the solutions with astandard F 2 term only, share the basic features of the EYM black holes in [10], in particularbeing unstable. It appears however, that inclusion of higher order gauge field curvature termscould lead to a very different situation (for example, we have found rather similar results tothose discussed above, for a generalization of (2) with an extra dilaton field, a Gauss-Bonnetterm and gauge group SO(5)). Thus we expect the following picture to be generic: the F 4term introduces a supplementary interaction between electric and magnetic fields, which, forsome range of the parameters, implies a tachyonic mass for the vacuum perturbations of thenon-Abelian magnetic fields around the Abelian solutions. Then the Abelian gauge symmetryis spontaneously broken near a black hole horizon for some critical value of the charge to massratio, with the appearance of a condensate of magnetic non-Abelian gauge fields there. Thepossible relevance of these aspects in providing analogies to phenomena observed in condensedmatter physics is yet to be explored.Acknowledgements. This work is carried out in the framework of Science Foundation Ireland(SFI) project RFP07-330PHY. E.R. gratefully acknowledges support by the DFG.7References[1] G. T. Horowitz, arXiv:1002.1722 [hep-th].[2] S. S. Gubser, Phys. Rev. Lett. 101 (2008) 191601 [arXiv:0803.3483 [hep-th]].[3] D. V. Galtsov and M. S. Volkov, Phys. Lett. B 274 (1992) 173.[4] M. S. Volkov and D. V. Gal’tsov, Phys. Rept. 319 (1999) 1 [arXiv:hep-th/9810070].[5] E. E. Donets and D. V. Gal’tsov, Phys. Lett. B 352 (1995) 261 [arXiv:hep-th/9503092];P. Kanti and E. Winstanley, Phys. Rev. D 61 (2000) 084032 [arXiv:gr-qc/9910069].[6] B. Zwiebach, Phys. Lett. B156 (1985) 315.[7] T. H. R. Skyrme, Nucl. Phys. 31 (1962) 556.[8] D. H. Tchrakian, Phys. Lett. B 150 (1985) 360.[9] D. V. Galtsov and A. A. Ershov, Phys. Lett. A 138 (1989) 160;P. Bizon, O. T. Popp, Class. Quant. Grav. 9 (1992) 193.[10] M. S. Volkov and D. V. Galtsov, JETP Lett. 50 (1989) 346;H. P. Kuenzle and A. K. Masood- ul- Alam, J. Math. Phys. 31 (1990) 928;P. Bizon, Phys. Rev. Lett. 64 (1990) 2844.[11] A. Messiah, Quantum Mechanics, North-Holland, Amsterdam (1962).[12] A. A. Tseytlin, Nucl. Phys. B 501 (1997) 41 [arXiv:hep-th/9701125].[13] J. Polchinski, ‘String theory. Vol. 2: Superstring theory and beyond’, Cambridge, CambridgeUniversity Press, (1998).8

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Stable black hole solutions with non-Abelian fields

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