Non-Abelian Lovelock-Born-Infeld Topological Black Holes

TL;DR

This paper explores new asymptotically AdS black hole solutions in higher-dimensional Einstein gravity with non-Abelian Born-Infeld fields, revealing unique properties related to mass, charge, and singularity types compared to Einstein-Yang-Mills solutions.

Contribution

It introduces novel non-Abelian Lovelock-Born-Infeld black hole solutions, analyzing their properties and asymptotic behaviors across various dimensions, extending previous Einstein-Yang-Mills models.

Findings

Solutions differ in mass, charge, and singularity nature from Einstein-Yang-Mills black holes.

Existence of extremal black holes with non-negative mass in Einstein-Yang-Mills-Born-Infeld theory.

Singularity types can be timelike or spacelike depending on parameters and dimensions.

Abstract

We present the asymptotically AdS solutions of the Einstein gravity with hyperbolic horizons in the presence of $So(n(n-1)/2-1, 1)$ Yang-Mills fields governed by the non-Abelian Born-Infeld lagrangian. We investigate the properties of these solutions as well as their asymptotic behavior in varies dimensions. The properties of these kind of solutions are like the Einstein-Yang-Mills solutions. But the differences seems to appear in the role of the mass, charge and born-Infeld parameter $\beta$, in the solutions. For example in Einstein-Yang-Mills theory the solutions with non-negative mass cannot present an extreme black hole while that of in Einstein-Yang-Mills-Born Infeld theory can. Also the singularities in higher dimensional Einstein-Yang-Mills theory for non-negative mass are always spacelike, while depending on choosing the parameters, we can find timelike singularities in the similar case of Einstein-Yang-Mills-Born-Infeld theory. We also extend the solutions of Einstein to the case of Gauss-Bonnet and Lovelock gravity. It is shown that, these solutions in the limits of $\beta\to0$, and $\beta\to\infty$, represent pure gravity and gravity coupled with Yang-Mills fields, respectively.

Equations

$${\mathcal{G}}_{\mu \nu }=\sum _{k=0}^{[n/2]}{\alpha }_k^{\prime }{\mathcal{G}}_{\mu \nu }^{(k)}=8\pi T_{\mu \nu }$$

$${\mathcal{G}}^{\mu }{}_{\nu }{}^{(k)}=-\frac{1}{2^{k+1}}{\delta }_{\nu j_1{j}_2....j_{2k}}^{\mu i_1{i}_2....i_{2k}}{R}_{i_1{i}_2}^{j_1{j}_2}....R_{i_{2k-1}{i}_{2k}}^{j_{2k-1}{j}_{2k}}$$

$${\gamma }_{ab}=-C_{ad}^c{C}_{bc}^d/{\left|\det C_{ad}^c{C}_{bc}^d\right|}^{1/M},$$

$$T_{\mu \nu }=\frac{1}{4\pi }\left(\frac{{\gamma }_{ab}}{\Gamma }{F}_{\mu \lambda }^{(a)}{F}_{\nu }^{(b)\lambda }+\frac{1}{4}{g}_{\mu \nu }{\mathcal{L}}_m\right)$$

$$\Gamma =\sqrt{1+\frac{{\gamma }_{ab}}{2{\beta }^2}{F}_{\lambda \sigma }^{(a)}{F}^{(b)\lambda \sigma }}$$

$$F_{\mu \nu }^{(a)}={\nabla }_{\mu }{A}_{\nu }^{(a)}-{\nabla }_{\nu }{A}_{\mu }^{(a)}+\frac{1}{2e}{C}_{bc}^a{A}_{\mu }^{(b)}{A}_{\nu }^{(c)}$$

$${\mathcal{L}}_m=4{\beta }^2\left(1-\Gamma \right).$$

$${\nabla }_{\mu }(\frac{1}{\Gamma }{F}^{\mu \nu (a)})+\frac{1}{\Gamma e}{C}_{bc}^a{A}_{\mu }^{(b)}{F}^{(c)\mu \nu }=0,$$

$$d{s}^2=-f(r)d{t}^2+\frac{d{r}^2}{f(r)}+r^{2}d{\Omega }_{n-1}^2$$

$$A^{(a)}$$

$$A^{(b)}$$

$$x_1$$

$$x_i$$

$$x_n$$

$$\epsilon_{a}=\left\{\begin{array}[]{ll}-1&\ \ \ 1\leq a\leq n-1\\ 1&\ \ \ n\leq a\leq\frac{n(n-1)}{2}\end{array}\right.$$

$$\Gamma (r)=\sqrt{1+\frac{(n-1)(n-2)e^2}{2{\beta }^2{r}^4}}$$

$${\mathcal{G}}_{\mu \nu }^{(1)}-\frac{n(n-1)}{2l^2}{g}_{\mu \nu }=8\pi T_{\mu \nu }.$$

$${\left[r^{n-2}(1+f)\right]}^{\prime }-\frac{n}{l^2}{r}^{n-1}-\frac{4{\beta }^2}{n-1}{r}^{n-1}\left(1-\Gamma \right)=0$$

$$f(r)=-1-\frac{3m}{r^2}+\frac{r^2}{l^2}-\frac{{\beta }^2{r}^2}{3}\left(1-{\Gamma }_4\right)-\frac{2e^2\ln (r)}{r^2}-\frac{e^2\ln \left(1+{\Gamma }_4\right)}{r^2},$$

$$f(r)=-1-\frac{(n-1)m}{r^{n-2}}+\frac{r^2}{l^2}+\frac{4{\beta }^2{r}^2}{n(n-1)}\left[1-ϝ(r)\right],$$

$$ϝ(r)=\frac{n}{r^n}\int r^{n-1}\Gamma (r)dr={}_2{F}_1\left(\left[-\frac{1}{2},-\frac{n}{4}\right],\left[1-\frac{n}{4}\right],{\Gamma }^2-1\right)$$

$$f(r)=-1-\frac{(n-1)m}{r^2}+\frac{r^2}{l^2}$$

$$f(r)\to -1-\frac{(n-1)m}{r^{n-2}}+\frac{r^2}{l^2}-\frac{(n-2)e^2}{(n-4)r^2}$$

$$f=-1-\frac{7m}{r^6}+\frac{r^2}{l^2}+\frac{{\beta }^2{r}^2}{14}\left(1-{\Gamma }_8\right)-\frac{3e^2}{4r^2}{\Gamma }_8\left[1-\ln {\left(r^2+r^2{\Gamma }_8\right)}^{\frac{{\Gamma }_8^2-1}{{\Gamma }_8}}\right]$$

$$f\to -\frac{(8a-1)}{r^{8a-2}}\left[m-\lambda e^{4a}\ln (\varsigma )\right],$$

$$\lambda$$

$$\varsigma$$

$$f\to -\frac{(8a+3)}{r^{8a+2}}\left[m+\lambda e^{4a+2}\ln (\varsigma )\right],$$

$$\lambda$$

$$\varsigma$$

$$f\to -2a\frac{m}{r^{2a-1}}$$

$$f_{n=6}=-1-\frac{5m}{r^4}+\frac{r^2}{l^2}+\frac{2{\beta }^2{r}^2}{15}\left(1-{\Gamma }_6\right)-\frac{4e^2}{3r^2}{\Gamma }_6,$$

$$f_{n=10}=-1-\frac{9m}{r^8}+\frac{r^2}{l^2}+\frac{2{\beta }^2{r}^2}{45}\left(1-{\Gamma }_{10}\right)-\frac{8e^2}{15r^2}{\Gamma }_{10}(3-2{\Gamma }_{10}^2),$$

Full text

I Introduction

Non-linear electrodynamics was proposed in the thirties to remove singularities associated with charged pointlike particles. Among the non-linear theories of electrodynamics Born-Infeld (BI) theory BI is distinguished, since BI type actions arise in many different contexts in superstring theory Fradkin ; Leigh . Also In the case of a D-brane, it has been known that the effective action on the brane is Born-Infeld action if all derivative terms are neglected Abo . The effective action of several D-branes is very important in the development of the understanding of the non-perturbative superstring theory, such as Matrix theory Banks . Tseytlin Tse proposed that if all derivative terms are neglected, the effective action on the branes is a non-Abelian generalization of the Born-Infeld action. There are a lot of outstanding classical solutions to both the Abelian and non-Abelian Born-Infeld theories. Born-Infeld black holes in (A)dS spaces has been discussed in Cai1 . A type of a particle-like solution in the non-Abelian Born-Infeld model was obtained by Gal’tsov and Kerner (GK) Gal . More close relationship between these two particle-like configurations becomes clear in Kunz ; Gal2 ; HalilBI . Among theories of non-Abelian gauge fields, the Yang-Mills theory may be regarded as the most fundamental one in elementary particle physics. The role of the Yang-Mills field in gravity has become an interesting topic of studies. Physical significance of the particle-like solutions of Einstein-Yang-Mills (EYM) field equations found by Bartnik and McKinnon (BK) BK as well as their possible role in the string-inspired models remains rather obscure. The attentions towards the Einstein-Yang-Mills system even became more, after the discovery of the first known example of hairy black holes Vol1 ; Bizon which are not uniquely characterized by their conserved charges and so violate manifestly the no-hair conjecture theory. In particular, particle-like, soliton-like and black hole solutions in the combined Einstein-Yang-Mills (EYM) models in different dimensions, shed new light on the complex features of compact object in these models HalilE . (See Vol for an overview).

With the advent of string theory, the possibility to have extra-dimensions became one of the most promising possibility to extend the standard model of particles physics. Higher dimensional theories of gravity may present some new features which are absent in four dimensions. Indeed in four dimensions the only gravitational action that can be built from curvature invariants leading to second order equations in the metric is the Einstein-Hilbert action. The situation changes in higher dimensions. In five dimensions one can add for example a Gauss-Bonnet term to the action . In higher dimensions one can add higher curvature powers to the action. Such higher curvature power theories leading to second order equations for the metric are known as Lovelock theories Lov . There are lots of considerable works in Lovelock gravity (for example see Cai2 ).

In this present Letter, we will study the topological black hole of lovelock gravity in the present of non-abelian version of Born-Infeld theory in (n+1)-dimensions with the gauge group $SO(n(n-1)/2-1,1)$. The coupling of the two non-linear , Yang-Mills and Born-Infeld fields, is a candidate for effective action for superstrings and D-branes, besides the fundamental importance in the context of gravitation theory. To obtain an exact solutions, we use a modified version of Wu-Yang ansatz Wu which was originally introduced in $N=4$ field theory and for the first time was applied by Yasskin Yasskin . Recently, this ansatz has been used to find the EYM and GBYM solutions with negative cosmological constant in higher dimensions BoE ; BoGB .

The out line of this paper is as follows. In Sec. II we introduce the field equations of Lovelock gravity in the presence of the energy-momentum tensor of the coupling of Yang-Mills and Born-Infeld fields. In section III, we obtain the (n+1)-dimensional solutions of the Einstein gravity and investigate their properties in varies dimensions. Section IV devotes to the general solutions to the Gauss-Bonnet and Lovelock gravity, and their properties respectively. We finish our paper with some concluding remarks.

II Field Equations

As was shown by Lovelock in the early seventies Lov , the possible corrections to Einstein gravity are quite limited, since the only symmetric, divergence free tensor that can be constructed out of the metric and its first two derivatives in a (n+1)- dimensional space-time is

[TABLE]

[TABLE]

Where $[d/2]$ denotes the integer part of $d/2$, $T_{\mu\nu}$ is the energy-momentum tensor and $\alpha^{\prime}_{k}$ are Lovelock constants which represent the coupling of the terms in the whole Lagrangian and give the proper dimensions. Usually in order the Einstein gravity to be recovered in the low energy limit, the constant $\alpha^{\prime}_{0}$ should be identified as the cosmological constant up to a constant, $\alpha^{\prime}_{0}=-2\Lambda$, and $\alpha^{\prime}_{1}$ should be positive (for simplicity one may take $\alpha^{\prime}_{1}=1$). As In this context, we are interested in asymptotically anti-de Sitter (AdS) solutions, we set $\Lambda=-n(n-1)/2l^{2}$.

We want to examine the Lovelock-Yang-Mills-Born-Infeld system for a compact, semi-simple gauge group ${\mathcal{G}}$ with structure constants $C_{bc}^{a}$. The basic elements of the model are $({\cal M},g_{\mu\nu},A_{\mu}^{\left(a\right)})$, where ${\cal M}$ is the spacetime manifold with metric $g_{\mu\nu}$ and $A_{\mu}^{\left(a\right)}$ are the gauge potentials. The metric tensor of the gauge group is BoE

[TABLE]

where $M$ is the dimension of the gauge group, the Latin indices $a$, $b$…. go from $1$ to $M$, and the repeated indices is understood to be summed over.

In Eq.(1), $T_{\mu\nu}$ is the energy momentum tensor given as

[TABLE]

where therein

[TABLE]

and

[TABLE]

In the equations (2),(3) and (4), $\beta$ is the Born-Infeld parameter which has the dimension of $length^{-2}$, $F_{\mu\nu}^{(a)}$’s are the Yang-Mills fields, $e$ is the coupling constant and ${\cal L}_{m}$, The matter term, is the F(2) nonabelian and nonlinear action density

[TABLE]

In the limit $\beta\rightarrow 0$, $\mathcal{L}_{m}$ becomes equal to zero and in the limit $\beta\rightarrow\infty$ it reduces to the standard Yang-Mills form, $\mathcal{L}_{m}=-\gamma_{ab}F_{\mu\nu}^{(a)}F^{(b)\mu\nu}$, so we expect to obtain the corresponding solutions in these both limits.

The YMBI gauge fields $F_{\mu\nu}^{(a)}$’s satisfy the nonlinear-nonabelian equation

[TABLE]

We consider the metric of the following form:

[TABLE]

where $d\Omega_{n-1}^{2}=d\theta_{1}^{2}+\sinh\theta_{1}^{2}d\theta_{2}^{2}+\sinh\theta_{1}^{2}\sum_{i=3}^{n-1}\prod_{j=2}^{i-1}\sin\theta_{j}^{2}d\theta_{i}^{2}$, is the line element of $(n-1)$- dimensional hypersurface with constant negative curvature and volume $V_{n-1}$.

In order to obtain The gauge potential in higher dimensional spacetimes we use the (n+1)- dimensional Wu-Yang Ansatz Wu as follows

[TABLE]

where $a$ and $b$ run from $1$ to $n-1$ and $n$ to $n(n-1)/2$, respectively and

[TABLE]

The Lie algebra of this gauge group is $SO(n(n-1)/2-1,1)$ with the metric tensor of $\gamma_{ab}=\epsilon_{a}\delta_{ab}$, which there is no sum on $a$ and

[TABLE]

Substituting Eqs. (7) and (4) in (3), we have

[TABLE]

III Static Solutions in

Einstein Gravity

In this section the goal is to find the exact solutions of Einstein gravity in the presence of YMBI fields. In this case, the field equation (1) becomes

[TABLE]

Where ${\cal G}_{\mu\nu}^{(1)}$ is just the Einstein tensor. To find the function $f(r)$, one may use any components of Eq. (9). The tt-component of the above equation using (2) and (8) is

[TABLE]

For $N=n+1=5$ the solution is obtained to be:

[TABLE]

Where $\Gamma_{4}=\Gamma\>(n=4)$. For $N=n+1>5$ the solution of the field equation (10) is :

[TABLE]

$\digamma(r)$ is defined as:

[TABLE]

Where ${}_{2}F_{1}([a,b],[c],z)$ is hypergeometric function. It is apparent that in the limit $\beta\rightarrow 0$ Eq. (11) will be

[TABLE]

which is the AdS solution to the Einstein equation. Using the fact that ${}_{2}F_{1}([a,b],[c],z)$ has a convergent series expansion for $|z|<1$, we can find the limit of the $f(r)$ for large $r$ and $\beta$ as

[TABLE]

As expected this is the solution to the Einstein-Yang-Mills equations introduced in BoE .

The global structure of the spacetime is characterized by properties of the singularities and horizons. It is easy  to show that the Kretschmann scalar $R_{\mu\nu\lambda\kappa}R^{\mu\nu\lambda\kappa}$ diverges at $r=0$ and is finite everywhere else, so $r=0$ is an essential singularity. The behavior of the solution $f(r)$  at infinity is dominated by the term $r^{2}\left(1/l^{2}+4\beta^{2}/n(n-1)\right)$ and so one can see that $f(r)\rightarrow+\infty$  for large r’s. Also one can see that the function $f(r)$ has a negative or positive value near $r=0$ depending on the value of $-\frac{(n-1)m}{r^{n-2}}-\frac{4\beta^{2}r^{2}}{n(n-1)}\digamma(r)$ as this is the dominating term in $f(r)$ near $r=0.$ The hypergeometric function ${{}_{2}F_{1}}\left(\left[-{\frac{1}{2},-\frac{{n}}{{4}}}\right],\left[1-{\frac{{n}}{{4}}}\right],\Gamma^{2}-1\right)$, has different forms in different dimensions.

By considering an integer number $a$, we obtain the explicit form of the hypergeometric function as follows.

III.1 $\mathbf{n=8a,a=1,2,..}$

For these values of $n,$ as in ${{}_{2}F_{1}}\left(\left[a,b\right],\left[{c}\right],z\right),$ $\ c$ is always an integer, the integral (12), is expected to include a logarithmic term. As a case for $n=8$ the metric function $f(r)$ is

[TABLE]

Where $\Gamma_{8}=\Gamma(n=8)$. Of course one should note that this solution in $\beta\rightarrow+\infty,$ reduces to give the relation (13) as expected, and the logarithmic term vanishes in this case. But other than this case, this logarithmic case plays a significant term in the properties of the Yang-Mills-Born-Infeld solutions.

To see these properties we obtain the limit of $f(r)$ near $r=0$ for these values of  $n=8a$. In this case we obtain:

[TABLE]

Where

[TABLE]

When $m>\lambda e^{4a}\ln(\varsigma)$, $f(r)$ is negative and as $f(r)\rightarrow+\infty$, when r$\rightarrow+\infty$, so $f(r)$ has certainly one real root and a black hole with one horizon exists. For non-negative mass if $\varsigma>1$ and $m<\lambda e^{4a}\ln(\varsigma)$, $f(r)$ is positive near $r=0$ and the solution may present an extreme black hole, a black hole with two horizons or a spacetime without a horizon. This is a property that does not happen in the Einstein-Yang-Mills theory as the spacetime always presents naked singularity for non-negative mass.

Also for non-negative mass of this case the singularity at $r=0$ is timelike, but in the Einstein-Yang-Mills theory the singularity for the dimensions higher than five, is always spacelike.

for negative mass, the possibility to have spacelike singularity exists if $\varsigma<1$ and $\mid$ $m\mid<\mid\lambda e^{4a}\ln(\varsigma)\mid$, while in EYM gravity the singularity for negative mass is timelike.

III.2 $\mathbf{n=4\>(2a+1)}$,  a= 1,2, …

In this case, the hypergeometric function again includes a logarithmic term. For the behavior of the metric function near $r=0$ we have

[TABLE]

Where in this case

[TABLE]

By choosing proper parameters, this term can be negative or positive. For negative mass this term can be positive if $\varsigma<1$, or if $\varsigma>1$ and also $\mid$ $m\mid>\mid\lambda e^{4a+2}\ln(\varsigma)\mid.$ In this case extreme black hole exists for the proper choose of the parameters.   For non-negative mass also if $\beta$ be sufficiently small and $\varsigma<1$, $f(r)$ will be positive and the solution may present a black hole with one or two horizons or a spacetime without a horizon. This is also the case that does not happen in Yang-Mills theory without Born-Infeld or in Maxwell-Born-Infeld theory.

III.3 $\mathbf{n=2a+1,a=1,2,\>...}$

In this case

[TABLE]

This is similar to the case that happens in Maxwell-Born-Infeld theory and just when $m<0,$ extreme black hole may exists and for non-negative mass spacetime always presents naked singularity.

III.4 $\mathbf{n=2\>(4a\pm 1),a=1,2,\>...}$

In these two cases, the integral in the relation (12) can be solved easily. For example for $n=6$ and $n=10$ respectively, solutions are obtained as

[TABLE]

[TABLE]

Where $\Gamma_{6}$ and $\Gamma_{10}$ are the amount of $\Gamma$ for $n=6$ and $n=10$ respectively. The behavior of $f(r)$ near $r=0$, for the case $n=2(4a-1)$ is

[TABLE]

[TABLE]

While that of For $n=2(4a+1)$ is as follows

[TABLE]

[TABLE]

So depending on proper choosing of the parameters $m$ and $e,$ black hole with one or two horizons, an extreme black hole and a naked singularity may exist. Also in these two cases a spacelike or timelike singularities may exist depending on the values of $m$ and $e$ and $\beta$.

We know a horizon is a null hypersurface defined by $r=r_{h}$ such that $f(r_{h})=0$ with finite curvatures, where $r_{h}$ is a constant horizon radius. For all the cases that $f(r)$ is positive near $r=0$, the extreme black hole may exists which therein, both $f(r)$ and $f^{\prime}(r)$ are zero on the horizon radius $r=r_{\mathrm{ext}}$ and can be calculated from (10) to be

[TABLE]

For these cases the spacetime of Eqs. (7) and (11) presents a naked singularity if $m<m_{\mathrm{ext}}$, an extreme black hole for $m=m_{\mathrm{ext}}$ and a black hole with two horizons provided $m>m_{\mathrm{ext}}$, where $m_{\mathrm{ext}}$ is

[TABLE]

The Hawking temperature is given by

[TABLE]

Where $r_{+}$ is the largest real root of $f(r)$ and

[TABLE]

$\digamma^{\prime}(r_{+})$ is the value of the first derivative of $\digamma(r_{+})$ at $r=r_{+}$. It’s notable that $T$ vanishes for $m=m_{\mathrm{ext}}$.

IV Static Solutions in Lovelock Gravity

The Gauss-Bonnet-Yang-mills field equation in the presence of YMBI fields may be written as

[TABLE]

Where ${\cal G}_{\mu\nu}^{(2)}$ is the second order Lovelock tensor introduced in Hoi . The tt-component of the above field equation for the metric (7) is:

[TABLE]

Where $\alpha_{2}=(n-2)(n-3)\alpha^{\prime}_{2}$. It is a matter of calculation to show that the solution of the field equation (17), for $N=n+1=5$ may be written as

[TABLE]

Where $\Gamma_{4}=\Gamma\>(n=4)$. Solving the equation (17) for $N\geq 5$, we get

[TABLE]

Where $\digamma(r)$ was introduced in (12). There are two families of solutions which correspond to the sign in front of the square root in Eq. (18). We call the family with minus (plus) sign the minus (plus)-branch solution. The minus-branch solution reduces to the solution in the Einstein-Yang-Mills-Born-Infeld solution in the limit of $\alpha\rightarrow 0.$ On the other hand, $f(r)$ diverges for the plus-branch solution in this limit, and there is no counterpart in the Einstein theory.

We notice that $f(r)$ takes the form

[TABLE]

as $\beta\rightarrow 0$.  Also in the limit $\beta\rightarrow\infty$ it takes the form

[TABLE]

This is the solution to the Gauss-Bonnet-Yang-Mills equation obtained in BoGB  as expected.

Of course one may note that $f(r)$ is imaginary for $r<r_{0}$ and real for $r>r_{0}$ where $r_{0}$ is the largest real root of the following equation:

[TABLE]

Thus one cannot extend the spacetime to the region $r<r_{0}.$ To get rid of this incorrect extension we introduce the new radial coordinate $\rho$ as

[TABLE]

With this new coordinate the metric (7) becomes

[TABLE]

Where now one should substitute $r=\sqrt{\rho^{2}+r_{0}^{2}}$ in Eq. (18). The new metric function has a singularity at $\rho=0$ $(r=r_{0})$.

Like in Einstein gravity, In Gauss-Bonnet gravity, black hole with one or two horizons, an extreme black hole and a naked singularity may exist. here, we just consider the condition of having extreme black holes, for which the temperature vanishes. The Hawking temperature can be obtained as

[TABLE]

Where $\Gamma(r_{+})$ was given in Eq. (15) and $r_{+}$ is the radius of outer horizon. It is a matter of calculation to show that $m=m_{\mathrm{ext}}$,

[TABLE]

is the solution of $T=0$.

To find explicitly the solutions to the Yang-Mills-Born-Infeld equation in third order Lovelock gravity in the presence of cosmological constant, we have to consider the third order Lovelock-Yang-mills field equation as

[TABLE]

Where now ${\cal G}_{\mu\nu}^{(3)}$ is the third order Lovelock tensor introduced in Hoi . The tt-component of the field equation (21) for the metric ansatz (7) is derived to be:

[TABLE]

where prime denotes the derivative with respect to $r$ and we define $\alpha_{2}^{{}^{\prime}}=\alpha_{2}/(n-2)(n-3)$ and $\alpha_{3}^{{}^{\prime}}=\alpha_{3}/3(n-2)(n-3)(n-4)(n-5)$ for simplicity.

First we solve the equation (22) for the case $N=n+1=7.$ The equation in this case admits the solution:

[TABLE]

where

[TABLE]

Where $\Gamma_{6}$, as was mentioned before, is the amount of $\Gamma$, for $n=6$. Also for $N=n+1>7$ the generalized solution is the same as the relation (23) with $j(r)$ being:

[TABLE]

Now for this solution as $\beta\rightarrow 0,$ $k(r)$ reduces to

[TABLE]

which gives the solution to the Third order Lovelock gravity as expected and for $\beta\rightarrow\infty,$ $j(r)$ will be

[TABLE]

The solution given by Eqs. (23) and (25) represents the AdS solution to the Third order Lovelock-Yang-Mills equation.

To see the asymptotic behavior of the solution to the Third order Lovelock gravity, we write it for a special case that $\alpha_{3}=\alpha_{2}^{2}.$ The solution then will be

[TABLE]

This solution has a singularity at $r=0$ as the kretschmann scalar diverges at $r=0$.

The Hawking temperature for this solution is given by:

[TABLE]

where $r_{+}$ is the radius of event horizon. Also in this case, we see that the black hole solutions may present an extreme black hole with horizon radius $r_{\mathrm{ext}}$, where $r_{\mathrm{ext}}$ is one of the real roots of $T=0$.

Now by solving the equations in second and third order Lovelock theories, we deduce that the tt-component of the field equation in Lovelock gravity is

[TABLE]

We can solve this equation to obtain the general solution in Lovelock gravity as

[TABLE]

V Concluding Remarks

We obtained the solutions to Einstein and Lovelock theories considering the coupling of two non-linear fields, Yang Mills and Born-Infeld fields and investigated the properties of the solutions . The properties of these kind of solutions are like the Yang-Mills solutions. But the difference seems to appear in the role of the mass in the solutions, as for small r’s in Yang-Mills gravity the dominant term is the term containing $m$, but in the the Yang-Mills-Born-Infeld, the dominant term indicates both $m$, and $\digamma(r),$ which includes $\beta,$ the Born-Infeld parameter and $e$. As the function $\digamma(r),$ takes different forms in different dimensions, we saw that this term modifies the properties of the solutions. For example in Einstein-Yang-Mills theory the solutions with nonnegative mass cannot present an extreme black hole but we found conditions for some of the solutions in Einstein-Yang-Mills-Born Infeld theory that, an extreme black hole or naked singularity can exist for non-negative mass. The singularities in Einstein-Yang-Mills theory are always spacelike and therefore unavoidable, but the singularities in Einstein-Yang-Mills-Born-Infeld theory can be spacelike or timelike depending on the choosing of the parameters. We also obtained the solutions for the second and third order Lovelock theories and from that we introduced the solution for the n-order Lovelock-Yang-mills-Born-Infeld theory.

Acknowledgements

N. Bostani wishes to thank H. Hendi for many helps in this work. This work has been supported in part by Research Institute for Astrophysics and Astronomy of Maragha.