Higher dimensional Yang-Mills black holes in third order Lovelock gravity

By employing the higher (N\TEXTsymbol{>}5) dimensional version of the Wu-Yang Ansatz we obtain magnetically charged new black hole solutions in the Einstein-Yang-Mills-Lovelock (EYML) theory with second ($\alpha_{2}$) and third ($\alpha_{3}$)order parameters. These parameters, where $\alpha_{2}$ is also known as the Gauss-Bonnet parameter, modify the horizons (and the resulting thermodynamical properties) of the black holes. It is shown also that asymptotically ($r\to \infty $), these parameters contribute to an effective cosmological constant -without cosmological constant- so that the solution behaves de-Sitter (Anti de-Sitter) like.


I. INTRODUCTION
As the requirements of string theory/brane world cosmology, higher dimensional (N > 4) space times have been extensively investigated during the recent decades. Extensions of N = 4 Einstein's gravity has already gained enough momentum from different perspectives.
General relativity admits local black hole solutions as well as global cosmological solutions such as de Sitter (dS) and anti de-Sitter (AdS), which are important from the field theory (i.e., AdS/CFT) correspondence point of view. Once a pure gravity solution (with or without a cosmological constant) is found the next (routine) step has been to search for the corresponding Einstein-Maxwell (EM) solution with the inclusion of electromagnetic fields.
As a result EM dS (AdS) space times have all been obtained and investigated to great extend. One more extension from a different viewpoint, which is fashionable nowadays, is to consider extra terms in the Einstein-Hillbert action such as the ones considered by Lovelock in higher dimensions to obtain solutions, both black holes and cosmological [1]. The added extra terms in the action have the advantage, as they should, that they don't give rise to higher order field equations. It is known that in N = 4, the second order (also known as Gauss-Bonnet) Lovelock Lagrangian becomes trivial unless coupled with non-trivial sources such as non-minimal scalar fields [9]. In higher dimensions (N ≥ 5), however coupling with electromagnetic fields proved fruitful and gave rise to interesting black hole/cosmological solutions [2]. In this regard, Brihaye, et al, have worked on particle-like solutions of EYM fields coupled with GB gravity in N-dimensional spherically symmetric spacetime [5]. To have a non-trivial theory with the higher order Lovelock Lagrangian with third order parameter on the other hand, we need the dimensionality of our space time to be N ≥ 7. In this paper we shall follow similar steps, to extend the results of electromagnetic fields to the Yang-Mills (YM) fields with gauge group SO(N − 1). As expected, going from Maxwell to YM constitutes a highly non-trivial process originating from the inherent non-linearity of the latter. To cope with this difficulty we employ a particular YM Ansatz solution which was familiar for a long time to the high energy physics community. This is the Wu-Yang Ansatz, which was originally introduced in N = 4 field theory [3,4]. Recently we have generalized this Ansatz to N = 5, in the Einstein-Gauss-Bonnet (EGB) theory and obtained a new Einstein-Yang-Mills-Gauss-Bonnet (EYMGB) black hole [5]. By a similar line of thought we wish to extend those results further to N > 5 and also within the context (for N ≥ 7) of third order Lovelock gravity. Our results show that both the second (α 2 ) and third (α 3 ) order parameters modify the EYM black holes as well as their formation significantly. For instance in the g tt term the gauge charge term comes with the opposite sign and fixed power of 1 r 2 , which is unprecedented in the realm of EM black holes of higher dimensions. This makes construction of black hole types from pure YM charge (with negligible mass, for example) possible and enriches our list of black holes with new properties. What follows for N ≥ 7, for technical reasons, we assume a relation between α 2 and α 3 which are completely free otherwise. Not only black holes but the asymptotical behaviors and properties of our space times are determined by these parameters as well. It is not difficult to anticipate that by the same token the same problem can further be generalized to cover the forth (α 4 ), fifth (α 5 ), etc. order terms in the action to be superimposed to the Einstein-Hillbert (EH) Lagrangian. By studying the relative weight of contribution from higher Lovelock terms it is not difficult to anticipate that the EH and GB terms dominate over the higher order, much more tedious terms. For this reason we restrict ourselves in this paper to maximum third order (α 3 ) terms in the Lagrangian.

II. ACTION AND FIELD EQUATIONS
The action which describes the third order Lovelock gravity coupled with Yang-Mills field without a cosmological constant in N dimension reads [1] where tr(.) = is the third order Lovelock Lagrangian. Here R, R µνγδ and R µν are the Ricci Scalar, Riemann and Ricci tensors respectively, while the gauge fields F where C µ are the gauge potentials, and α 2 and α 3 are GB and third order Lovelock coefficients. Variation of the action with respect to the space-time metric g µν yields the Einstein-Yang-Mills-Gauss-Bonnet-Lovelock (EYMGBL) equations where the stress-energy tensor is G E µν is the Einstein tensor, while G GB µν and G µν are given explicitly as [2] Variation of the action with respect to the gauge potentials A while the integrability conditions are * F (a)µν ;µ in which * means duality [6].
The N−dimensional line element is chosen as in which the S N −2 line element will be expressed in the standard spherical form We use the Wu-Yang Ansatz [7] in N−dimensional case as where we imply (to have a systematic process) that the super indices a is chosen according to the values of i and j in order.
The YM field 2-forms are defined as follow We note that our notation follows the standard exterior differential forms, namely d stands for the exterior derivative while ∧ stands for the wedge product [7]. The integrability are easily satisfied by using (12). The YM equations are also satisfied. The energy-momentum tensor (5), becomes after with the non-zero components The EYMGBL equations (4) reduce to the general equation in which a prime denotes derivative with respect to r, n = N − 2,α 2 = (n − 1) (n − 2) α 2 andα 3 = (n − 1) (n − 2) (n − 3) (n − 4) α 3 . This equation is valid for N ≥ 4 (i.e. n ≥ 2), but for N = 4 (i.e. n = 2) we get which clearly is α 2,3 independent and therefore it will be the Einstein-Yang-Mills equation admitting the well-known Reissner-Nordstrom form For N = 5 (i.e. n = 3) the Eq. (18) has already been considered in [5]. We note that in Eq. (18), r,α 2 andα 3 can all be scaled by Q (i.e., r → |Q| r,α 2 → Q 2α 2 andα 3 → Q 4α 3 , as long as Q = 0) so that we set in the sequel, without loss of generality, Q = 1.
IV. THE EYMGB CASE, α 2 = 0, α 3 = 0 Eq. (18) with α 3 = 0 takes the form which may be called as Einstein-Gauss-Bonnet-Yang-Mills (EGBYM) equation. This equation admits a general solution in any arbitrary dimensions N as follows where m is the usual integration constant to be identified as mass.

A. The EYMGB solution in 6−dimensions
In this section we shall explore some physical aspects of the solution (22) in 6-dimensions. This is interesting for the reason that, N = 5 and N = 6 are the only dimensions which will not be effected by the non-zero third order Lovelock gravity. For N = 6 (n = 4), the metric function f (r) in Eq. (22) takes the form in whichα 2 (= 6α 2 ) and ± refer to the two different branches of the solution. Asymptotic behaviors of f ± (r) can be shown to be as lim r→∞ f + (r) → 1 + r 2 α 2 , and lim r→∞ f − (r) → 1, which imply that, the positive branch is Asymptotically-de Sitter (A-dS) with positive α 2 and Asymptotically-Anti de Sitter (A-AdS) with negative α 2 . It is seen obviously that the negative branch is an Asymptotically Flat (A-F) space. One can also show that lim r→0 + f + (r) → +∞, and lim r→0 + f − (r) → −∞, which clearly, shows that, f + (r) is an A-dS solution while f − (r) represents an A-F black hole solution. In the sequel we shall consider α 2 > 0 with the negative branch of the solution (i.e., the A-F black hole solution). One can easily show that, this solution admits a single horizon (i.e., event horizon) given by which is real and positive for any values of m andα 2 . In Fig. (1), (i.e., the dashed curves) we plot the radius of event horizon r + , in terms of α 2 (i.e., Gauss-Bonnet parameter), for some fixed values for m. This figure displays the contribution of the Gauss-Bonnet parameter to the possible radius of the event horizon of the black hole. By looking at the Fig. (1), one may comment that for any value of m, lim α 2 →∞ r + → 0. We notice further that, with α 2 = 0, we get the radius of event horizon for the six dimensional EYM black hole solution which was given in the Ref. [5]. That is, for very large α 2 , the event horizon coincides with the central singularity. As a particular choice, we consider α 2 = 1 2 which implies that The surface gravity, κ defined by [8] takes the value The associated Hawking temperature depending on mass m and α 2 = 1 2 becomes in the choice of units c = G = = k = 1.

B. The EYMGB solution in 7−dimensions
In this section we represent some physical aspects of the solution (22) in 7-dimensions. In 7-dimensions, both second and third order Lovelock terms contribute but still we set α 3 = 0 in order to identity the contribution of α 3 . For N = 7 (n = 5), the metric function f (r) in Eq. (22) takes the form in whichα 2 (= 12α 2 ) and ± refers to the two individual branches of the solutions. In Fig. (2) we plot f − (r) which goes to asymptotically flat value for r → ∞. Asymptotic behaviors of f ± (r) can by written as which imply that, the positive branch is Asymptotically-de Sitter (A-dS) for positive α 2 and Asymptotically-Anti de Sitter (A-AdS) for negative α 2 , while the negative branch leads to an Asymptotic Flat (A-F) space. Also one can show that lim r→0 + f + (r) → +∞, and lim r→0 + f − (r) → −∞, which clearly, manifests that, f + (r) is an A-dS solution while f − (r) represents an A-F black hole solution. Hence in the sequel we just consider α 2 > 0 and the negative branch of the solution (i.e., the A-F black hole solution). One can easily show that, this solution admits only an (event horizon) which can be written as which implies that r + is real and positive for any values of m andα 2 . In the Fig. (1) we plot the radius of event horizon r + , in terms of α 2 (i.e., the solid curves), with some fixed values of m. This figure displays the contribution of the Gauss-Bonnet parameter in place of possible radius of the event horizon of the EYM black hole. We notice that, with α 2 = 0, we recover the radius of event horizon for the 7−dimensional EYM black hole solution which is given in the Ref. [7].
As a particular choice, we consider α 2 = 1/6 which implies that and the surface gravity, (26) in this case has the value With the associated Hawking temperature given by which is comparable with the 6-dimensional case (28).
V. THE EYML CASE WITH, α 2 = 0, α 3 = 0 In this section we just consider the effect of α 3 on the solution of the field equation. Eq.

therefore). The proper general solution of this equation is
given by in which One can easily show that in the limitα 3 → 0, we obtain which is the EYM solution.
VI. THE GENERAL CASE, α 3 = 0 = α 2 For N ≥ 7 (n ≥ 5), with α 3 = 0 = α 2 we shall see the role of the third order Lovelock parameters as well as the second order. This leads us to a tedious set of differential equations which fortunately reduces to Eq. (18) and can be integrated exactly. Indeed, the general solution of the Eq. (18) with arbitrary values of α 2 and α 3 , in any arbitrary dimension N ≥ 7 can be expressed by the following expression f (r) = 1 + (4Ω 2 /∆) 1 3 6 (n + 1) (n − 3)α 3 r n +α in which we use the abbreviations To proceed further with these expressions doesn't seem feasible from technical points, therefore in the sequel we shall adopt a special simplifying relation between α 2 and α 3 .
in which .
The associated Hawking temperature T H = κ 2π can be found by using the above result, on which one may expect that, T H is stronglyα 2 (andα 3 ) dependent.

B. Asymptotically dS (AdS) property
The general solution (39) as r → ∞ reads where It is observed that the metric function (39) It is seen that in this effective cosmological constant both α 2 , and α 3 play role. One can easily show that, depending onα 2 andα 3 ,Λ can take zero, positive or negative values, and consequently the general solution becomes asymptotically flat, dS or AdS, respectively. For instance, from Eq. (43), one can show that the choiceα 3 =α 2 2 /3 is asymptotically flat. These results verify that, the Lovelock parametersα 2 andα 3 significantly modify the properties of EYM black holes as well as their asymptotic behaviors.

VII. CONCLUSION
We introduced YM fields through the Wu-Yang Ansatz into the third order Lovelock gravity with spherical symmetry. The Ansatz and symmetry aided in overcoming the technical difficulty and obtaining exact solutions in higher dimensions. In this sense our work is partly an extension of our previous work which included only the GB parameter and for N = 5 [5]. Our solutions include black hole possessing parameters of mass, magnetic charge (which