Reissner-Nordstr\"om black holes with non-Abelian hair

We consider $d\geqslant 4$ Einstein--(extended-)Yang-Mills theory, where the gauge sector is augmented by higher order terms. Linearizing the (extended) Yang-Mills equations on the background of the electric Reissner-Nordstr\"om (RN) black hole, we show the existence of normalizable zero modes, dubbed non-Abelian magnetic stationary clouds. The non-linear realization of these clouds bifurcates the RN family into a branch of static, spherically symmetric, electrically charged and asymptotically flat black holes with non-Abelian hair. Generically, the hairy black holes are thermodynamically preferred over the RN solution, which, in this model, becomes unstable against the formation of non-Abelian hair, for sufficiently large values of the electric charge.


Introduction
According to the uniqueness theorems [1], the Kerr-Newman solution [2] is the most general asymptotically flat, non-singular (on and outside the event horizon), single black hole (BH) solution in electro-vacuum General Relativity (GR). Such theorems have shaped the appealing worldview that the multitude of BHs in the Cosmos is well described by the Kerr metric [3], when near equilibrium (assuming their electric charge is negligible). This worldview, however, relies deeply on another ingredient: the celebrated BH perturbation theory, developed in the 1970s. This framework allowed establishing that uncharged rotating [charged non-rotating] BHs, described by the Kerr [Reissner-Nordström (RN)] metric, are stable against vacuum [4] [electro-vacuum [5,6]] linear perturbations, in a mode analysis. 1 A follow-up question is if Kerr or RN BHs are still stable when other matter fields are considered (beyond electro-vacuum). The relevance of this question is illustrated by the superradiant instability of Kerr BHs, triggered by massive bosonic fields, which are most commonly taken to be scalar or Proca fields (see [10] for a review). In the presence of appropriate seeds of these fields, the instability develops, extracting rotational energy and angular momentum from the BH, that piles up into a bosonic cloud around the horizon. For a single superradiant mode, this growth stops when the BH's angular velocity decreases sufficiently to synchronise with the phase angular velocity of the superradiant mode [11], forming a Kerr BH with synchronised bosonic hair [12][13][14][15]. The existence of these new BH solutions, that bifurcate from the Kerr solution, could be inferred prior to this dynamical analysis, by analysing the linearised bosonic wave equation around the Kerr BH and observing the existence of zero modes, dubbed stationary bosonic clouds, precisely at the threshold of the unstable modes [12,[16][17][18][19][20][21]. The hairy BHs are the non-linear realisation of these zero modes [22] and they are thermodynamically favoured over Kerr BHs [12].
The above considerations for the Kerr case can be extended to the generic Kerr-Newman case: the superradiant instability exists [23], there are zero-modes [17,18] and Kerr-Newman BHs with scalar hair have been constructed [24]. But the same does not apply in the particular case of the RN BH. Even though superradiant scattering exists around RN BHs, these are not afflicted by the superradiant instability of massive bosonic fields. 2 As such, massive bosonic hair does not grow around the asymptotically flat RN BH, in contrast to the Kerr case, and in agreement with known no-hair theorems [31]. The purpose of this paper is to point out that in the presence of a class of non-Abelian fields, it is possible to grow hair around the asymptotically flat RN BH, and the process resembles the aforementioned discussion of the Kerr superradiant instability.
BHs with non-Abelian hair were initially discovered in d = 4 Einstein-Yang-Mills (EYM) SU(2) theory [33]. These so called coloured BHs are asymptotically flat and their global YM charge is completely screened, endowing them with a single global "charge" -the ADM mass. Since for a given value of the mass there can be infinitely many different solutions, the no-hair conjecture is violated. This discovery triggered an extensive search for hairy BHs in various other models -see [32,35,40,41] for reviews. 3 The coloured BHs in [33] are, however, unstable against spherical linear perturbations within the EYM model [42,43]. This instability can be (partly) attributed to the fact that they possess a purely magnetic gauge field and to the absence of a global YM charge (a.k.a 'baldness' theorem [44][45][46]). This also signifies that the (EYM embedded, Abelian) RN BHs and the coloured BHs in [33] form disconnected 'branches' of the EYM SU(2) model. 4 Analogous coloured BHs, with a purely magnetic field, perturbatively unstable and with a solitonic limit, exist in more than four spacetime dimensions [48][49][50]. For d > 4, however, a Derrick-type virial argument implies that no finite mass solutions can be found in standard EYM theory. The path taken in [48][49][50], was to extend the YM action with particular higher order terms which yield a YM version of Lovelock's gravity [51]. The resulting Einsteinextended -YM (EeYM) model [53] has the desirable property that equations of motion are still second order and Ostrogradsky instabilities [52] are avoided.
In d 4 EeYM, it turns out to be possible to circumvent the no-go results in [44][45][46] and obtain coloured, electrically charged BHs continuously connected to the RN solution (embedded in this model) [54]. Moreover, as we shall show: 1) the RN solution becomes unstable against eYM perturbations; 2) threshold eYM linear perturbations correspond to static eYM clouds on the RN background (as test fields); and 3) the non-linear realisation of these clouds corresponds to coloured, electrically charged BH solutions in EeYM theory, which are thermodynamically favoured over the RN BHs. Thus, we argue, eYM "matter" triggers an instability of RN BHs that parallels the familiar superradiant instability of Kerr BHs, likely leading to a similar outcome: the dynamical formation of a RN BH with non-Abelian hair, of the type we present below.
where G, that will be set to unity, is Newton's constant and τ (p) are a set of P -input constants whose values are not constrained a priori.
Apart from providing a natural YM counterparts to Lovelock gravity and its mathematical elegance, another reason of interest for this EeYM model is the occurrence of such F (2p) 2 terms in non-Abelian Born-Infeld theory [56] or in the higher loop corrections to the d = 10 heterotic string low energy effective action [57]. Here, however, we adopt a 'phenomenological' viewpoint and choose the basic action (1) primarily for the purpose of identifying the new features induced by such terms.
The field equations are obtained by varying the action (1) with respect to the field variables g µν and A µ where we define the p-stress tensor pertaining to each term in the matter Lagrangian as The solutions reported herein are spherically symmetric, obtained with the metric Ansatz the function m(r) being related to the local mass-energy density up to some d−dependent factor. r, t are the radial and time coordinate, respectively, while dΩ 2 d−2 is the line element of a unit sphere. The choice of gauge group compatible with the symmetries of the line element (5) is somewhat flexible. In this work we choose to employ chiral representations, with a SO(d + 1) gauge group. Then a spherically symmetric gauge field Ansatz is [58,59]: 5 Note the analogy with the corresponding expression for the Gauss-Bonnet density in Lovelock gravity.
Σ ij being the chiral representation matrices of SO(d − 1), and Σ d,d+1 of the SO(2), subalgebras in SO(d + 1), while x i are the usual Cartesian coordinates, being related to the spherical coordinates in (5) as in flat space.

The equations and known solutions
Plugging (5) and (3) into the equations of motion (2) results in 6 where we use the shorthand notation The d-dimensional RN BH is a solution of the model for a purely electric field, in which case only the p = 1 YM term in (1) contributes. It has: with M 0 and q two integration constants fixing the mass and electric charge, where V (d−2) is the area of the unit (d − 2)-sphere. These solutions possess an outer event horizon at r = r h , with r h the largest root of the equation N(r) = 0, a condition which imposes an upper bound on the charge parameter, q q Saturating this condition results in an extremal BH.
The electrically uncharged, coloured BHs of [33,[48][49][50] are a second set of solutions. They possess nontrivial magnetic field on and outside the horizon, while the electric field vanishes V (r) = 0. Their mass is finite being the only global charge, since the YM fields leave no imprint at infinity. These solutions do not trivialize in the limit of zero horizon size, becoming gravitating non-Abelian solitons.
The d = 4 BHs in [47] are yet another set of solutions of the above equations, being found for τ (2) = 0 and an SO(3) × U(1) gauge group. They possess a nonzero magnetic field, and approach the limit (12) in the far field. Similar to Ref. [33], the magnetic potential w(r) possesses at least a node, with the absence of solutions where it becomes infinitesimally small.
We also note that the eqs. (7)- (10) are not affected by the transformation: while σ and w remain unchanged. Thus, in this way one can always take an arbitrary positive value for one of the constants τ (p) . In this work this symmetry is used to set τ (1) = (d − 2)/2. Also, to simplify some relations, we shall introduce τ ≡ ((d − 2)/8) 2 τ 2 /τ 3 1 .

Zero and unstable EeYM modes on the RN BH
In contrast to the EYM model in [47], the presence of a p > 1 term in the EeYM action leads to a direct interaction between the electric and magnetic fields, a feature which holds already in the d = 4 version of the model. This, as we shall see, makes the RN BH unstable when considered as a solution of the full model. At the threshold of the unstable modes, a set of zero modes appear, as we now show. Let us investigate the existence of a perturbative solution around the RN BH background, with w(r) = ±1 + ǫw 1 (r) + . . . (ǫ being a small parameter). Similar perturbative expression are written also for m, σ and V ; however, to lowest order, the equation for w 1 (r) decouples, taking the simple form with N = 1−2m (RN ) /r d−3 . Observe that only the p = 1 and p = 2 terms enter this equation; other terms only start to contribute at higher order in perturbation theory. 7 The second term in (15) can be seen as providing an effective mass term, for the gauge potential perturbation w 1 . This mass term becomes strictly positive for large r, µ 2 (ef f ) → 1 while it possesses no definite sign near the horizon. In fact, µ 2 (ef f ) becomes negative for large enough values of the electric charge, and this turns out to be a necessary condition for the existence of w 1 solutions with the correct asymptotic behaviour. 8 Requiring µ 2 (ef f ) < 0 at the horizon, together with the existence of a horizon (which puts an upper bound on the electric charge), actually implies the existence of a maximal value of the electric charge for a given τ , if one wishes normalizable zero mode perturbations to exist: For charges smaller than Q (max) , we have found that the equation (15)  Following the terminology for scalar fields, [12,16,18], these configurations with infinitesimally small magnetic fields are dubbed non-Abelian stationary clouds around RN BHs. The corresponding subset of RN BHs span an existence line in the parameter space of solutions. 9 This set is shown below, in Figures 3,4 (the blue dotted line); the plotted results are for d = 4, 5 but a similar picture has been found for d = 6, . . . , 9 and we expect a similar pattern to occur for any d.
Even though eq. (15) does not appear to be solvable in terms of known functions, an approximate expression of the solutions can be found by using the method of matched asymptotic expansions. For example, for d = 4, the solution in the near horizon region [w where b and J are free parameters. These approximate solutions together with their first derivatives are matched at some intermediate point, which results in the constraint 3r 4 h − Q 2 (r 2 h + 4τ ) = 0. This condition can be expressed as a relation between the event horizon 8 Multiplying by w 1 the eq. (15), results in the equivalent form Normalizable modes have w 1 vanishing at infinity. Then, integrating this equation between horizon and infinity, the left hand side vanishes and the first term in the right hand side is strictly positive. Thus, one finds that µ 2 (ef f ) is necessarily negative for some range of r. 9 A rigorous existence proof for the existence of solutions of the eqs. (15) for a number d = 4 of spacetime dimensions can be found in [60]. area and the electric charge of the RN BHs on the existence line: a result which provides a good agreement with the numerical data 10 . The RN solutions supporting these zero modes or marginally stable mode separate different domains of dynamical stability in the parameter space. We have investigated this issue for the (physically most interesting) d = 4 case. Starting with a more general Ansatz than (5), (6) with a dependence of both time and radial coordinate, which includes more gauge potentials and an extra g rt metric component, one considers again fluctuations around the RN BH, with a value of non-Abelian magnetic gauge potential close to the vacuum value everywhere, w = ±1 + ǫw 1 (r)e Ωt + . . . , and real Ω. Again, it turns out that, to lowest order in ǫ, the coupled equations separate, w 1 (r) being a solution of the equation where we have introduced a new 'tortoise' coordinate ρ defined by dr dρ = N, such that the horizon appears at ρ → −∞.
This eigenvalue problem has been solved by assuming again that w 1 is finite everywhere and vanishes at infinity. Restricting again to the fundamental mode, we display in Fig. 1 the square of the frequency as a function of the mass parameter M for several values of Q. One notices that, given Q, the RN BH becomes unstable for all values of M below a critical value, or equivalently, when the horizon is sufficiently small. Also, the solutions with Ω 2 → 0 corresponds precisely to the d = 4 existence line discussed above.

The hairy BH solutions
The instability of the RN solution found above can be viewed as an indication for the existence of a new branch of BH solutions within the EeYM model, having nontrivial magnetic non-Abelian fields outside the horizon, and continuously connected to the RN solution. 11 This is confirmed by numerical results obtained for 4 d 8, that we now illustrate.
Let us start our discussion by noticing that the total derivative structure of the equation for the electric potential (10) allows treating the value of the electric charge as an input parameter. However, the same equation excludes the existence of particle-like configurations with a regular origin and Q = 0. Thus, the only physically interesting solutions of this model describe BHs, with an event horizon at r = r h > 0, located at the largest root of N(r h ) = 0. The metric and the gauge field must be regular at the horizon, which, in the non-extremal case implies an approximate solution around r = r h of the form all coefficients being determined by w h and σ h . It is also straightforward to show that the requirement of finite energy implies the following asymptotic behavior at large r  (21) is an order parameter describing the deviation from the Abelian RN solution.
The solutions of the field equations interpolating between the asymptotics (20), (21) were constructed numerically, by employing a shooting strategy. For a F (2) 2 + F (4) 2 model (the only case shown here), the input parameters are Q, r h and τ . Then the solutions are found for discrete values of the parameter w h , labeled by the number of nodes, n, of the magnetic YM potential w(r). However, as mentioned above, in this work we shall restrict to the fundamental set of solutions which possess a monotonic behavior of the magnetic gauge potential w(r).
The profile of a typical d = 5 solution is shown in Figure 2 (a similar pattern has been found for other spacetime dimensions). This figure shows that, for the same values of the mass and electric charge, the RN solution has a smaller event horizon radius (and thus a 11 This branching off of a family of solutions at the onset of an instability is a recurrent pattern in BH physics. Examples include the Gregory-Laflamme instability of d 5 black strings [61,62], the d = 4 BHs with sinchronyzed hair discussed in the Introduction or the bumpy BHs in d 6 dimensions [63,64]. This special behaviour can be partially understood by studying the near horizon limit of the extremal hairy BHs. The condition of extremality implies N(r) = N 2 (r − r h ) 2 + . . . , as r → r h , while the expansion of w(r), σ(r) and V (r) is similar to that in (20). Then, restricting for simplicity to a F (2) 2 + F (4) 2 model, eqs. (7)-(10) reduce to two algebraic After eliminating the w(r h ) parameter, one finds 15 that the extremal BHs satisfy the following 14 As expected, the near horizon structure of the extremal hairy solutions can be extended to a full AdS 2 × S d−2 exact solution of the field equations, their properties being essentially fixed by (22). 15 Although one can write a general (Q, A H )-relation valid for d 6, its expression is very complicated; however, one finds Q → 0 as A H → 0.
charge-area relations: Therefore, the d = 4 extremal hairy BHs are special, stopping to exist for a minimal value of Q = τ /2, where the horizon area vanishes. As seen in Figures 3, 4, the set of critical solutions connect this point with the limiting configuration with vanishing (scaled) quantities.
No similar solutions are found for d > 4, since the limit Q → 0 is allowed in that case. Let us remark that the domains of existence for RN BHs and hairy BHs overlap in a region, see Figures 3,4. Therein, we have found that the free energy F = M − T H A H /4 of a hairy solution is lower than that of the RN configurations with the same values for temperature and electric charge. Finally, we notice the existence of overcharged non-Abelian solutions, i.e with electric charge to mass ration greater than unity, which do not possess RN counterparts (e.g. for d = 5 those between the extremal RN and the extremal hairy BH lines). These solutions cannot arise dynamically from the instability of RN BHs.

Further remarks
The paradigmatic coloured BHs are disconnected from the RN solution and are unstable against linear perturbations. 16 By considering a simple model with higher order curvature terms of the gauge field (dubbed EeYM model), we have constructed here a qualitatively different set of electrically charged, coloured BHs. The extended YM terms can provide a tachyonic mass for the eYM magnetic perturbations around the embedded RN BH. This leads to the existence of unstable modes. At the threshold of the unstable spectrum lies a zero mode, whose non-linear realization is the family of hairy BHs. The similarity with the more familiar superradiant instability of Kerr BHs is clear, and, as in that case, we expect a dynamical evolution to drive the unstable modes into forming condensate of non-Abelian magnetic field around the RN BH, and saturating when an appropriate hairy BH forms.
We remark that, for d = 5, a rather similar picture is found when considering instead solutions in a Einstein-Yang-Mills-Chern-Simons model [67], the Chern-Simons term providing an alternative to the higher order curvature terms of the YM hierarchy employed here. Again, the hairy BHs emerge as perturbations of the RN solution, being thermodynamically favoured over the Abelian configurations.
As a possible avenue for future research, it would be interesting to consider the stability of the hairy solutions in this work. Since they maximize the entropy for given global charges, we expect them to be stable. This is indeed confirmed by the d = 4 results reported in [54], which were found, however, for an SU(3) gauge group. The corresponding problem in the SO(d + 1) case appears to be more challenging and we leave it for future study.
Let us close by remarking on some similarities with yet another class of solutions: the coloured, electrically charged BHs in Anti-de Sitter (AdS) spacetime. As found in [68], the RN-AdS BH becomes unstable when considered as a solution of the pure EYM theory, the 'box'-type behaviour of the AdS spacetime provinding the mechanism for the appearance of a magnetic non-Abelian cloud close to the horizon. Similarly to the situation here, this feature occurs for a particular set of RN-AdS configurations which form an existence line in the parameter space. Again, the hairy BHs are the nonlinear realization of the non-Abelian clouds. Their study via gauge/gravity duality has received a considerable attention in the literature (see e.g. the review [69]) leading to models of holographic superconductors. It would be interesting to explore the possibility that, despite the different asymptotic structure, the hairy BHs in this work could also provide connections to phenomena observed in condensed matter physics.