Lovelock black holes with a power-Yang-Mills source

We consider the standard Yang-Mills (YM) invariant raised to the power q, i.e., $(F_{\mu \nu}^{(a)}F^{(a) \mu \nu})^{q}$ as the source of our geometry and investigate the possible black hole solutions. How does this parameter q modify the black holes in Einstein-Yang-Mills (EYM) and its extensions such as Gauss-Bonnet (GB) and the third order Lovelock theories? The advantage of such a power q (or a set of superposed members of the YM hierarchies) if any, may be tested even in a free YM theory in flat spacetime. Our choice of the YM field is purely magnetic in any higher dimensions so that duality makes no sense. In analogy with the Einstein-power-Maxwell theory, the conformal invariance provides further reduction, albeit in a spacetime for dimensions of multiples of 4.


I. INTRODUCTION
N−dimensional static, spherically symmetric Einstein-Yang-Mills (EYM) black hole solutions in general relativity are well-known by now for which we refer to [1], and references cited therein. YM theory's non-linearity naturally adds further complexity to the already non-linear gravity, thus expectedly the theory and its accompanied solutions become rather complicated. Extension of the Einstein-Hilbert (EH) action with further non-linearities, such as Gauss-Bonnet (GB) or Lovelock have also been considered. These latter theories involve higher order invariants in such combinations that the field equations remain second order.
More recently there has been aroused interest about black hole solutions whose source is a power of the Maxwell scalar i.e., (F µν F µν ) q , where q is an arbitrary positive real number [2].
Subsequently this will be developed easily into a hierarchies of YM terms. In the standard Maxwell theory we have q = 1, whereas now the choice q = 1 is also taken into account which adds to the theory a new dimension of non-linearity from the electromagnetism. Nonlinear electrodynamics, such as Born-Infeld (BI) involves a kind of non-linearity that is more familiar for a long time [3]. From the outset we express that the non-linearity involved in the power-Maxwell formalism is radically different from that of BI. An infinite series expansion of the square root term in the latter reveals this fact. For the special choice q = N 4 , where N =dimension of the spacetime is a multiple of 4, it yields a traceless Maxwell's energymomentum tensor which leads to conformal invariance. That is, in the absence of different fields such as self-interacting massless scalar field and /or a cosmological constant we have a vanishing scalar curvature. This implies a relatively simpler geometry under the invariance g µν → Ω 2 g µν and naturally attracts interest. The absence of black hole solutions in higher dimensions for a self-interacting scalar field was proved long time ago [4]. Self-interacting Maxwell field with a power of invariant, however, which conformally interacts with gravity admits black hole solutions [2].
Being motivated by the black holes sourced by the power of Maxwell's invariant we investigate in this work the existence of black holes with a power of YM source. That is we shall choose our source as F µν is the YM field with its internal index 1 ≤ a ≤ 1 2 (N − 1) (N − 2) and q is a real number such that q = 1 recovers the EYM black holes. Similar to the power-Maxwell case we obtain the conformally invariant YM black holes with a zero trace for the energy-momentum tensor. It turns out in analogy that the dimensions of spacetimes are multiples of 4. The power q can be chosen arbitrary provided the conformal property is lost. It will also be shown that q < 0, will lead to the violation of the energy and causality conditions. This will restrict us only to the choice q > 0. As before, our magnetically charged YM field consists of the Wu-Yang ansatz in any higher dimensions [1]. The EYM metric function admits an integral proportional to ∼ ln r r 2 for N = 5 and ∼ 1 r 2 for all N > 5. The fixed r−dependence for N > 5, was considered to be unusual i.e., a drawback or advantage, depending on the region of interest. Now with the choice of the power q on the YM invariant we obtain dependence on q as well, which brings extra r−dependence in the metric. The possible set of integer q values in each N > 5 is determined by the validity of the energy conditions. For N = 4 and 5 we show that q = 1, necessarily, but for N > 5 we can't accommodate q = 1 unless we violate some energy conditions. We consider next the GB (i.e., second order Lovelock) and successively Lovelock's third order term added to the first order EH Lagrangian. Our source term throughout the paper is the YM invariant raised to the power q (and its hierarchies). In each case, separately or together, we seek solutions to what we call, the Einstein-power-YM (EPYM) field with GB and Lovelock terms. It is remarkable that such a highly nonlinear theory with nonlinearities in various forms admits black hole solutions and in the appropriate limits, with q = 1, it yields all the previously known solutions. In the presence of both the second and third order Lovelock terms, however, we impose for technical reasons an algebraic condition between their parameters. This we do for the simple reason that the most general solution involving both the second and third order terms is technically far from being tractable.
Useful thermodynamic quantities such as the Hawking temperature, specific heat and free energy are determined and briefly discussed.
Organization of the Letter is as follows. Sec. II contains the action, field equations, energy-momentum for EPYM gravity and solutions to the field equations. Sec.s III and IV follow a similar pattern for the GB and third order Lovelock theories, respectively. Yang-Mills hierarchies are discussed in Sec. V. We complete the Letter with Conclusion which appears in Sec. VI.

II. FIELD EQUATIONS AND THE METRIC ANSATZ FOR EPYM GRAVITY
The N (= n + 2) −dimensional action for Einstein-power-Yang-Mills (EPYM) gravity with a cosmological constant Λ is given by (8πG = 1) in which F is the YM invariant R is the Ricci Scalar and q is a positive real parameter. Here the YM field is defined as in which C (b)(c) has been described elsewhere [5]. We note that the internal indices {a, b, c, ...} do not differ whether in covariant or contravariant form. Variation of the action with respect to the spacetime metric g µν yields the field equations where G µν is the Einstein tensor. Variation with respect to the gauge potentials A (a) yields the YM equations where ⋆ means duality. It is readily observed that for q = 1 our formalism reduces to the standard EYM theory. Our objective in this work therefore is to study the role of the parameter q in the black holes. Our metric ansatz for N (= n + 2) dimensions, is chosen as in which f (r) is our metric function and where The choice of these metrics can be traced back to the form of the stress-energy tensor (5), (12) below) and consequently G 0 0 − G 1 1 = 0, whose explicit form, on integration, gives |g 00 g 11 | = C =constant. We need only to choose the time scale at infinity to make this constant equal to unity.

A. Energy momentum tensor
Recently we have introduced and used the higher dimensional version of the Wu-Yang ansatz in EYM theory of gravity [1,5]. In this ansatz we express the Yang-Mills magnetic gauge potential one-forms as 2 ≤ j + 1 ≤ i ≤ n + 1, and 1 ≤ a ≤ n(n + 1)/2, ..
One can easily show that these ansaetze satisfy the YM equations [1,5]. In consequence, the energy momentum tensor (5), with Tr F (a) We observe that the trace of T a b is T = − 1 2 F q (N − 4q) which vanishes for the particular case q = N 4 . It is also remarkable to give the intervals of q in which the Weak Energy Condition (WEC), Strong Energy Condition (SEC), Dominant Energy Condition (DEC) and Causality Condition (CC) are satisfied [6]. It is observed from Tab. (1) that the physically meaningful range for q is n+1 4 ≤ q < n+1 2 , which satisfies all the energy and causality conditions. The choice q < 0, violates all these conditions so it must be discarded. In the sequel we shall use this energy momentum tensor to find black hole solutions for the EPYM, EPYMGB and EPYMGBL field equations with the cosmological constant Λ.
in which f (r) = 1 + g (r) . Direct integration leads to the following solutions where m is the ADM mass of the black hole. It is observed that physical properties of such a black hole depends on the parameter q. The location of horizons, f (r h ) = 0, involves an algebraic equation whose roots can be found numerically. The entropy, Hawking temperature and other thermodynamics properties all can be calculated accordingly and they are dependent on q. Tab. (1) shows that the minimum possible value for q which provides all the energy conditions to be satisfied is given by q min = n+1 4 , that is, the case of solution with logaritmic term. In 5−dimensions q min = 1, which recovers the usual EYM solution found in [1,5]. With the exception of N = 5 where q = 1 is part of possible q ′ s (which satisfy all the energy conditions), in higher dimensions q must be greater than one. For instance, in 6−dimensions 5 4 ≤ q < 5 2 and in 7−dimensions 3 2 ≤ q < 3. If one constrains q to be an integer, Tab. (2) gives the possible q values in some dimensions. From this table we can identify the dimensions in which the logarithmic term appears naturally. These are N = 5, 9, 13, ..., for which q min = N −1 4 is an integer. Let us remark that since for N = 4 our YM field gauge transforms to an Abelian form [7], our results become automatically valid also for N = 4.
We observe that although the metric function f (r) at infinity goes to − Λ 3 r 2 its behavior about the origin is quite different and strongly depends on q i.e., This is important because for the case of q ≥ n+1 4 one may adjust the mass and charge to have a metric function in contradiction with the Cosmic Censorship Conjecture (CCC). One statement of this conjecture is that all singularities (here r = 0) are hidden behind event horizons. Of course, nature may restrict Q and m in order not to violate this conjecture.
Note that r = 0 is a singularity for the metric whose Ricci scalar is given by

C. Extremal Black Holes
Closely related with a usual black hole is an extremal black hole whose horizons coincide.
As it is well known to get extremal solution one should solve f (r) = 0, and f ′ (r) = 0 simultaneously. This set of equations for the solution (14), without cosmological constant, leads to r e = (n (n − 1)) where r e is the radius of degenerate horizon and m e and Q are the extremal mass and charge of the black hole, respectively. One may check the case of q = 1, resulting in which clearly in 4-dimensions gives r e = Q = m e , as it should.

D. Thermodynamics of the EPYM black hole
In this section we present some thermodynamical properties of EPYM black hole solution with cosmological constant. Here it is convenient to rescale our quantities in terms of some different powers of radius of the horizon r h , i.e., we introducě where is the heat capacity for constant Q and F = M ADM − T H S is the free energy of the black hole as a thermodynamical system. Therein is the Bekenstein-Hawking entropy where Γ (.) stands for the gamma function. As one may notice in (14) m represent the ADM mass of the black hole. This helps us to writě which imposes some restrictions onQ i andΛ in order to have a positive and physically acceptableM ADM .
In terms of the event horizon r h Hawking temperature becomeš For the case of q = n+1 4 clearly by imposingM ADM ,Ť H > 0 one findsΛ 3 < 1 − 1 2q and for the case of q = n+1 4 and choosing r h = 1, one getsΛ 3 < 1 −Q 2 +2 n+1 . The heat capacityČ is given byČ which reveals the thermodynamic instability of the black hole. In fact the possible roots of denominator ofČ present a phase transition which can be interpreted as thermodynamical instability.
For completeness we give also the free energy F of our black hole as a thermodynamical system, which iš By letting q = 1 and n = 2 for the 4−dimensional Reissner-Nordström metric, the foregoing expressions become

ITY
The EPYMGB action in N(= n + 2)−dimensions is given by (8πG = 1) where α is the GB parameter and L GB is given by Variation of the new action with respect to the space-time metric g µν yields the field equa- where and T µν is given by (12).

A. EPYMGB Black hole solution for N ≥ 5 dimensions
As before, the rr component of Einstein equation (33) can be written as in which α 2 = (n − 1) (n − 2) α 2 . This equation admits a solution as The asymptotic behavior of the metric reveals that which depending on Λ it is de Sitter, Anti de Sitter or flat. Abiding by the (anti) de Sitter limit forα 2 → 0, we must choose the (−) sign.

B. Thermodynamics of the EPYMGB black hole
By using the above rescaling plusα 2 =α 2 /r 2 h , one can find the Hawking temperature of the EPYMGB black hole solutions (36) aš here (±) state the correspondence branches. Here we observe thatŤ H (−) in the limit of α 2 → 0 correctly reduces to the Hawking temperature of EPYM black hole (23) as expected.
It is remarkable to observe thatα 2 = − 1 2 is a point of infinite temperature, or instability of the black hole. This means that ifα 2 /r 2 h = − 1 2 , the black hole will be unstable. For the positive branch one should be careful aboutα 2 → 0 which is not applicable.
In the sequel we give the other thermodynamical properties of the BH solution (36) in separate cases.

Negative branch
The ADM mass:M The heat capacity: The free energy: The ADM mass:M The heat capacity: The free energy: 3. Positive branch q = n+1 4 The ADM mass:M The heat capacity: The free energy: The ADM mass:M The heat capacity: The free energy: .
Finally in this section we look atČ which clearly, in general, vanishes atα 2 = − 1 2 . Also any possible root for the denominator ofČ gives instability point or a phase transition.

ITY
In this section we consider a more general action which involves, beside the GB term, the third order Lovelock term [8,9]. The EPYMGBL action in N(= n + 2)−dimensions is given by (8πG = 1) where α 2 and α 3 are the second and third order Lovelock parameters respectively, and [8] is the third order Lovelock Lagrangian. Variation of the new action with respect to the space-time metric g µν yields the field equations where .
1. The particular case ofα 3 =α 2 2 /3 In the third order Lovelock theory we first prefer to impose a condition on Lovelock's parameters such asα 3 =α 2 2 /3. This helps us to work with less complicity and in the sequel for the sake of completeness we shall present the general solution without this restriction as well. The metric function after this condition is given by where as usual m is the mass of the black hole. One may find which gives the asymptotical behavior of the metric such as de Sitter, Anti de Sitter or flat (Λ = 0). We note that in the limitα 2 → 0, we have f (r) → 1 − Λ 3 r 2 , as it should.
not be possible therefore we only stress on the specific case ofα 3 =α 2 2 /3. Given this particular choice, we start with the ADM mass of the BH which readš whose Hawking temperature is given by We notice here that the Hawking temperature diverges asα 2 approaches to −1.
The heat capacity: The free energy: . (67) The heat capacity: The free energy: In the foregoing expressions it is observed that forα 2 = −1, the free energy diverges, signalling the occurrence of a critical point. Further, the sign of the heat capacity can be investigated to see whether thermodynamically the system is stable (Č > 0) or unstable (Č < 0), which will be ignored in this Letter.

V. YANG-MILLS HIERARCHIES
In this section we investigate the possible black hole solutions for the case of a superposition of the different power of the YM invariant F and any further investigation in this line is going to be part of our future study. It is our belief that a detailed analysis of the energy conditions for the YM hierarchy exceeds the limitations of the present Letter, we shall therefore ignore it. The YM hierarchies in d−dimensions has been studied by D. H.
Tchrakian, et. al. [10] in a different sense. Here we start with an action in the form of in which F is the YM invariant, b 0 = n(n+1)

3
Λ and b k≥1 is a coupling constant. Variation of the action with respect to the spacetime metric g µν yields the field equations and variation with respect to the gauge potentials A (a) yields the YM equations Our metric ansatz for N (= n + 2) dimensions, is given by (7) and the YM field ansatz is as before such that the new energy momentum tensor reads as [1, 1, γ, γ, .., γ] , and γ = 1 − 4k n .
The solution of Einstein equation for α 2 = α 3 = 0 reveals that where m is the ADM mass of the black hole and Ψ = r n q k=0 b k F k dr =         