A way of decoupling gravitational sources in pure Lovelock gravity

TL;DR

This paper introduces an algorithm to decouple gravitational sources in pure Lovelock gravity, enabling the derivation of new and existing solutions, including regular black holes in anti-de Sitter spacetime with higher curvature effects.

Contribution

The paper presents a novel algorithm for decoupling sources in pure Lovelock gravity, facilitating the construction of analytic solutions with extra dimensions and higher curvature terms.

Findings

Derived new analytic solutions in pure Lovelock gravity.

Showed how to obtain regular black holes from anti-de Sitter space.

Demonstrated the application of minimal geometric deformation in this context.

Abstract

We provide an algorithm that shows how to decouple gravitational sources in Pure Lovelock gravity. This method allows to obtain several new and known analytic solutions of physical interest in scenarios with extra dimensions and with presence of higher curvature terms. Furthermore, using our method, it is shown that applying the minimal geometric deformation to the Anti de Sitter space time it is possible to obtain regular black hole solutions.

Equations

$$T_{AB}={\bar{T}}_{AB}+{\theta }_{AB},$$

$${\nabla }_A{T}^{AB}=0.$$

$$L=\sqrt{-g}\sum _{n=0}^N{\gamma }_n{L}_n,$$

$$L_n=\frac{1}{2^n}{\delta }_{{\alpha }_1{\beta }_1...{\alpha }_n{\beta }_n}^{{\mu }_1{\nu }_1...{\mu }_n{\nu }_n}{\Pi }_{r=1}^n{R}_{{\mu }_r{\nu }_r}^{{\alpha }_r{\beta }_r},$$

$${\delta }_{{\alpha }_1{\beta }_1...{\alpha }_n{\beta }_n}^{{\mu }_1{\nu }_1...{\mu }_n{\nu }_n}=\frac{1}{n!}{\delta }_{[{\alpha }_1}^{{\mu }_1}{\delta }_{{\beta }_1}^{{\nu }_1}...{\delta }_{{\alpha }_n}^{{\mu }_n}{\delta }_{{\beta }_n]}^{{\nu }_n}$$

$$\sum _{n=0}^N{\gamma }_n{\mathcal{G}}_{AB}^{(n)}=T_{AB},$$

$$G_B^A+{\gamma }_2{H}_B^A=T_B^A$$

$$H_{AB}=$$

$$\displaystyle+R^{CDE}_{A}R_{BCDE}\Big{)}-\frac{1}{2}g_{AB}L_{2}.$$

$$L=\sqrt{-g}{\gamma }_n{L}_n=\sqrt{-g}{\gamma }_n\frac{1}{2^n}{\delta }_{{\alpha }_1{\beta }_1...{\alpha }_n{\beta }_n}^{{\mu }_1{\nu }_1...{\mu }_n{\nu }_n}{\Pi }_{r=1}^n{R}_{{\mu }_r{\nu }_r}^{{\alpha }_r{\beta }_r},$$

$${\mathcal{G}}_{AB}^{(n)}=T_{AB},$$

$$({\mathcal{G}}^{(n)}{)}_B^A=-\frac{1}{2^{n+1}}{\delta }_{B{A}_1...A_{2n}}^{A{B}_1...B_{2n}}{R}_{B_1{B}_2}^{A_1{A}_2}\cdots R_{B_{2n-1}{B}_{2n}}^{A_{2n-1}{A}_{2n}}.$$

$$d{s}^2=-e^{\nu (r)}+e^{\lambda (r)}d{r}^2+r^{2}d{\Omega }_{d-2}^2,$$

$$T_B^A=\text{mbox}diag(-\rho ,p_r,p_{\theta },p_{\theta },...),$$

$$\frac{1}{2}(p_r+\rho ){\nu }^{\prime }+p_r^{\prime }+\frac{d-2}{r}(p_r-p_{\theta })=0.$$

$$T_B^A={\bar{T}}_B^A+\alpha ({\theta }_1{)}_B^A$$

$$\rho$$

$$p_r$$

$$p_{\theta }$$

$$T_B^A=$$

$$+{\alpha }^{n-1}({\theta }_{n-1}{)}_B^A+{\alpha }^n({\theta }_n{)}_B^A$$

$$\rho =$$

$$-{\alpha }^{n-1}({\theta }_{n-1}{)}_0^0-{\alpha }^n({\theta }_n{)}_0^0$$

$$p_r=$$

$$+{\alpha }^{n-1}({\theta }_{n-1}{)}_1^1+{\alpha }^n({\theta }_n{)}_1^1$$

$$p_{\theta }=$$

$$+{\alpha }^{n-1}({\theta }_{n-1}{)}_2^2+{\alpha }^n({\theta }_n{)}_2^2$$

$$\Pi =$$

$$\displaystyle+...+\alpha^{n-1}\Big{(}(\theta_{n-1})^{2}_{2}-(\theta_{n-1})^{1}_{1}\Big{)}$$

$$\displaystyle+\alpha^{n}\Big{(}(\theta_{n})^{2}_{2}-(\theta_{n})^{1}_{1}\Big{)}.$$

$$\displaystyle\frac{2}{d-2}r^{d-2}\Big{(}\bar{\rho}-\alpha(\theta_{1})^{0}_{0}-\alpha^{2}(\theta_{2})^{0}_{0}-...-\alpha^{n-1}(\theta_{n-1})^{0}_{0}$$

$$\displaystyle-\alpha^{n}(\theta_{n})^{0}_{0}\Big{)}=\frac{d}{dr}\Big{(}r^{d-2n-1}\left(1-e^{-\lambda}\right)^{n}\Big{)},$$

$$\displaystyle\frac{2}{d-2}r^{2n}\Big{(}\bar{p}_{r}+\alpha(\theta_{1})^{1}_{1}+\alpha^{2}(\theta_{2})^{1}_{1}+...+\alpha^{n-1}(\theta_{n-1})^{1}_{1}$$

$$\displaystyle+\alpha^{n}(\theta_{n})^{1}_{1}\Big{)}=nr\nu^{\prime}e^{-\lambda}\left(1-e^{-\lambda}\right)^{n-1}$$

$$-(d-2n-1){\left(1-e^{-\lambda }\right)}^n$$

$$\frac{1}{2}({\bar{p}}_r+\bar{\rho }){\nu }^{\prime }+{\bar{p}}_r^{\prime }+\frac{d-2}{r}({\bar{p}}_r-{\bar{p}}_{\theta })$$

Full text

1 Introduction

In the last years, several branches of theoretical physics have predicted the presence of extra dimensions. Thus, it makes sense to think of the existence of geometrical objects in space times with a number of dimensions greater than four as for example branes, strings, or higher dimensional black holes. In regarding this, theories of gravity emerged which present higher curvature correction terms when the space time has a number of dimensions greater than four. One interesting theory among of them is Lovelock gravity Lovelock . One achievement of Lovelock theory is that it shares the following features with the General Relativity :

• •

Its equation of motion are symmetric.

• •

Its equations of motion are of second order on the derivative of the metric tensor.

• •

Free divergence.

Into the Lovelock gravities, we can find the Pure Lovelock theory. It is well known that the General Relativity has a no non-trivial vacuum solution (without cosmological constant) when $d=3$ (i.e $d=2n+1$, where $n=1$), one interesting feature is that Pure Lovelock keeps this property for $d=2n+1$ with $n>1$, see reference indio6 . On the other hand, including the cosmological constant, General Relativity has a unique (Anti) de Sitter ground state for $\Lambda(<0)>0$, in regarding this, other interesting feature of Pure Lovelock theory is that, it keeps this feature for $n$ odd, however, for $n$ even, this theory has a double Anti de Sitter or de Sitter ground state for $\Lambda>0$ and does not have ground state for $\Lambda<0$ milko1 ; Cai:2006pq .

On the other hand, in Einstein Hilbert theory, finding new solutions of physical interest is not a simple task due to the highly nonlinear behavior of its equations. In regarding this, Ovalle in (2017) Ovalle1 proposed a method called Gravitational Decoupling of Sources, which corresponds to the first algorithm that shows how to decouple gravitational sources in General Relativity. This method applies a Minimal Geometric Deformation (MGD) to the temporal and radial metric components together with a decoupling of sources. The method is explained in reference Ovalle2 : “given two gravitational sources: a source A and an extra source B, standard Einstein’s equations are first solved for A, and then a simpler set of quasi-Einstein equations are solved for B. Finally, the two solutions can be combined in order to derive the complete solution for the total system.”.

By applying this method to a solution of Einstein equations, named seed solution, it is possible to obtain new analytic solutions of physical interest. Related to this, by deforming some isotropic well known and well behaved solutions, new anisotropic and well behaved solutions have been obtained that represent stellar distributions in references Ovalle2 ; Camilo ; Tello1 ; Graterol ; Tello2 ; Tello3 ; Luciano ; Milko . By deforming the Schwarzschild space time new black hole solutions have been obtained in reference Ovalle3 ( other black hole solutions obtained with this method in references Contreras1 ; Contreras2 ; Contreras3 ; Contreras4 ). Other examples of applications of the method are: solutions in Einstein Klein Gordon system Ovalle4 ; solutions in $f(G)$ gravity Sharif1 ; solutions in $f(R)$ gravity Sharif5 ,cloud of strings solutions Angel . See other applications in references Ovalle5 ; Contreras6 ; Sharif2 ; Sharif3 ; Luciano1 ; Sharif4 ; Regginaldo ; Ovalle:2019lbs ; Ovalle:2019Iso ; Gabbanelli:2019txr .

Thus, motivated by the fact that in Pure Lovelock theory there are several kind of solutions in literature as for example: black hole solutions in references Cai:2006pq ; indio3 ; indioBH ; PureLovelock1 ; PureLovelock2 ; Aranguiz:2015voa and stellar distributions in references indio2 ; indio4 ; indio5 , it seems of physical interest to provide an algorithm to decouple gravitational sources in Pure Lovelock gravity, and thus, to apply the method to known ( or unknown) solutions and to test if it is possible to obtain new solutions of physical interest. See also milko1 ; milko2 ; indio1 .

We start by deforming the seed energy momentum tensor $\bar{T}_{AB}$ by an additional source $\theta_{AB}$, which causes anisotropic effects on the self-gravitating system. This additional source can contain new fields, like scalar, vector and tensor fields Ovalle2 . Therefore the energy momentum tensor is:

[TABLE]

since Pure Lovelock theory has free divergence, then, the energy momentum tensor satisfies the conservation equation:

[TABLE]

In this work we provide a Gravitational Decoupling method in Pure Lovelock gravity, and therefore we show a simple approach to decoupling gravitational sources in this theory. We will show that the Pure Lovelock equations of motion can be solved for each component $\{\bar{T}_{AB},\theta_{AB}\}$ separately, at least for the spherically symmetric and static case. For each component will be obtained a particular metric tensor $\{\bar{g}_{AB},g_{AB}^{\,\,\,\theta}\}$, and the final metric $g_{AB}$ is a simple combination of these metrics. As a simple test, we will apply our method to an Anti de Sitter space time and, we will test if it is possible to obtain solutions that represent regular black holes.

2 Lovelock Gravity and the Pure Lovelock case

The Lovelock Lagrangian is :

[TABLE]

where $N=\frac{d}{2}-1$ for $d$ even and $N=\frac{d-1}{2}$ for $d$ odd and, $\gamma_{n}$ are arbitrary coupling constants. $L_{n}$ is a topological density defined as:

[TABLE]

where $R^{\alpha\beta}_{\mu\nu}$ is a $n$ order generalization of the Riemann tensor for the Lovelock theory, and:

[TABLE]

is the generalized Kronecker delta.

It is worth to stress that, the terms $L_{0},L_{1}$ and $L_{2}$ are proportional to the cosmological constant, Ricci Scalar and the Gauss Bonnet Lagrangian, respectively. The corresponding equation of motion is given by:

[TABLE]

where $\mathcal{G}^{(n)}_{AB}$ is a $n$ order generalization of the Einstein tensor due to the topological density $L_{n}$. As example $\mathcal{G}^{(1)}_{AB}$ is just the Einstein tensor associated with the Ricci scalar (Einstein Hilbert theory is a particular case of Lovelock theory), and $\mathcal{G}^{(2)}_{AB}$ is the Lanczos tensor $H_{AB}$ associated with the Gauss Bonnet Lagrangian.

For example, the Einstein Gauss Bonnet equations of motion up to $n=2$, without cosmological constant are:

[TABLE]

where the Lanczos tensor is:

[TABLE]

2.1 Pure Lovelock case

Pure Lovelock is a theory that involving only a single fixed value of $n$ (with $n\geq 1$), without sum over the lower order. In some cases it is considered one single value of $n\geq 1$ plus the $n=0$ term, i.e. $L=L_{0}+L_{n}$, as for example in references milko1 ; PureLovelock1 ; PureLovelock2 . For simplicity, in this work we take a single value of $L_{n}$ without the $L_{0}$ term, as for example in references indio1 ; indio2 . Thus, the Lagrangian is:

[TABLE]

The equations of motion are given by:

[TABLE]

where

[TABLE]

and where the coupling constants were set to unity as in references indio1 ; indio2 .

In this work we study the static $d$ dimensional spherically symmetric metric, wich in Schwarzschild-like coordinates reads:

[TABLE]

where $d\Omega^{2}_{d-2}$ corresponds to the metric of a $(d-2)$ unitary sphere. The energy momentum tensor corresponds to a neutral perfect fluid:

[TABLE]

where, from the spherical symmetry, we have for all the $(d-2)$ angular coordinates that $p_{\theta}=p_{\phi}=...$ . The conservation law $T^{AB}_{;B}=0$ gives:

[TABLE]

Note that the analogue Einstein tensor has free divergence $(\mathcal{G}^{(n)})^{AB}_{;B}=0$ Dadhich:2008df , and the Bianchi Identities are satisfied Camanho:2015hea .

It is worth stressing that the $(d-2)$ angular components of the equations of motion are similar $(\theta,\theta)=(\phi,\phi)=...$ and the conservation equation can be written as a combination of the $(t,t)$,$(r,r)$ and $(\theta,\theta)$ components indio2 . In this way, there are three field equations $(t,t),(r,r),(\theta,\theta)=(\phi,\phi)=...$ and one conservation equation 14. But only 3 equations are independent. Thus, any one equation could be ignored and the system will be satisfied if the other three are solved.

3 Pure Lovelock equations of motion for multiples sources

In the equations of motion 10, $T^{A}_{B}=\mbox{diag}(-\rho,p_{r},p_{\theta},p_{\theta},...)$ is given by equation 1, and the seed energy momentum tensor is given by $\bar{T}^{A}_{B}=\mbox{diag}(-\bar{\rho},\bar{p}_{r},\bar{p}_{\theta},\bar{p}_{\theta},...)$.

In the Gravitational Decoupling method, reference Ovalle1 , developed for Einstein Hilbert theory, is introduced an additional source $(\theta_{1})^{A}_{B}$ coupled with the seed energy momentum by the constant $\alpha$. It is worth to notice that the power of $\alpha$ coincides with the value $n=1$ corresponding to the Einstein Hilbert theory. Thus, the energy momentum 1 is:

[TABLE]

where the source $(\theta_{1})^{A}_{B}$ is arbitrary. Thus, it is easily see that:

[TABLE]

In this work, inspired by the above mentioned method, for a generic value of $n$, it is proposed the following energy momentum tensor:

[TABLE]

therefore the number of sources is determined by the value of $n$. It is worth to stress that the energy momentum 15 is a particular case of 3 for $n=1$. As example, for the Pure Gauss Bonnet case with $n=2$ the energy momentum 3 has the form $T^{A}_{B}=\bar{T}^{A}_{B}+\alpha(\theta_{1})^{A}_{B}+\alpha^{2}(\theta_{2})^{A}_{B}$. Now:

[TABLE]

[TABLE]

[TABLE]

where $p_{\theta}=p_{\phi}=...$ . In equation 22 $\bar{p}_{\theta}=\bar{p}_{\phi}=...$ and we impose that $(\theta_{i})^{2}_{2}=(\theta_{i})^{3}_{3}=...$

So, for $(\theta_{i})^{1}_{1}\neq(\theta_{i})^{2}_{2}=(\theta_{i})^{3}_{3}=...$ and $\bar{p}_{r}\neq\bar{p}_{\theta}=\bar{p}_{\phi}=...$ these sources induce an anisotropy:

[TABLE]

For the isotropic case where $\bar{p}_{\theta}=\bar{p}_{r}$, the addition of our source $\theta_{AB}$ is a simple way to generate an anisotropy.

So, the $(t,t)$ and $(r,r)$ components of the equations of motion are given by indio1 ; indio2 :

[TABLE]

and

[TABLE]

We solve the $(t,t)$ and $(r,r)$ components of the Pure Lovelock equations together with the conservation equation. Using the Bianchi identities, we ignore the remaining $(\theta,\theta)=(\phi,\phi)=...$ components (the suspense points indicate that all the tangential components of the Pure Lovelock equations are similar).

By inserting equations 20,21 and 22 into equation 14:

[TABLE]

Thus, the system to solve corresponds to equations 3,3 and 3. At this stage we would deal with five unknown functions, namely, the metric components $\lambda(r)$ and $\nu(r)$, and the effective thermodynamics functions $\rho$, $p_{r}$ and $p_{\theta}=p_{\phi}=...$. However, below, we implement the gravitational decoupling method where this scenario is modified.

The Pure Lovelock equations of motion for the seed energy momentum tensor are recovered for the limit $\alpha\to 0$ in the system 3, 3 and 3. So, it is fulfilled that

[TABLE]

and next, the first line of equation 3 is conserved. For $n=1$ both components of energy momentum tensor are directly conserved, ı.e $\nabla_{A}(\theta_{1})^{A}_{B}=0$, however, for $n>1$ one can notice that:

[TABLE]

where the covariant derivative is computed by using the line element 12. In this work we impose in arbitrarily way that:

[TABLE]

Thus, we will solve the system 3, 3, 27 and 29. Under this assumption each source is separately conserved, and thus, there is no exchange of energy momentum between them. Therefore, our energy momentum tensor 3 is a way of decoupling the system inspired by the approach of reference Ovalle1 .

4 Gravitational decoupling by MGD in Pure Lovelock gravity

We start with a solution to equations 3, 3, 27 and 29 with $\alpha=0$, namely seed solution $\{\eta,\mu,\bar{\rho},\bar{p}_{r},\bar{p}_{t}\}$, where $\eta$ and $\mu$ are the corresponding metric functions:

[TABLE]

Turning on the parameter $\alpha$, the effects of the sources $(\theta_{i})_{AB}$ appear on the seed solution $\{\eta,\mu,\bar{\rho},\bar{p}_{r},\bar{p}_{t}\}$. These effects can be encoded in the geometric deformation undergone by the seed fluid geometry $\{\eta,\mu\}$ in equation 30 as follows:

[TABLE]

[TABLE]

It means that only the radial component of the line element 30 is deformed, where $g(r)$ is the corresponding deformation of the radial part. This is known as Minimal Geometric Deformation Ovalle1 .Thus, replacing equations 31 and 32 into of equations 3 and 3:

[TABLE]

and

[TABLE]

Thus we must solve the system of equations 4, 4, 27 and 29. We use the Binomial theorem :

[TABLE]

thus:

[TABLE]

and

[TABLE]

Thus, replacing equations 41 and 4 into of equations 4 and 4, the system splits into the following sets of equations:

• •

The standard Pure Lovelock equations for a seed solution (with $\alpha=0$) :

[TABLE]

and

[TABLE]

and the respective conservation equation:

[TABLE]

• •

The terms of order $\alpha$ give rise to the following quasi-Pure Lovelock equations of order $\alpha^{1}$, which include the source $\theta_{AB}$:

[TABLE]

[TABLE]

and the respective conservation equation:

[TABLE]

Thus, following the iteration, it is possible to obtain the quasi pure Lovelock equations of order $\alpha^{2}$, $\alpha^{3}$…$\alpha^{n-3}$, $\alpha^{n-2}$.

• •

The terms of order $\alpha^{n-1}$ give rise to the following quasi-Pure Lovelock equations of order $\alpha^{n-1}$:

[TABLE]

and

[TABLE]

and the respective conservation equation

[TABLE]

• •

The terms of order $\alpha^{n}$ give rise to the following quasi-Pure Lovelock equations of order $\alpha^{n}$:

[TABLE]

and

[TABLE]

and the respective conservation equation

[TABLE]

It is worth stressing that each quasi Pure Lovelock equation cannot be formally identified as the spherically symmetric Pure Lovelock equations for $n>1$, because the right sides of each quasi Pure Lovelock equation do not have the standard expressions for the Generalized Einstein tensor components $\mathcal{G}^{(n)}_{00}$ and $\mathcal{G}^{(n)}_{11}$. Furthermore, the Bianchi identities are not satisfied for each quasi Pure Lovelock equation. For $n=1$ the quasi Einstein equations can be transformed in the standard Einstein equations after a convenient redefinition of the energy momentum tensor Ovalle1 , however, the method of the reference Ovalle1 has been widely used to find new solutions without using this mentioned redefinition in several works.

Despite the above mentioned, our imposed way for solving the system 4, 4, 27 and 29, based in the decoupling of sources by means of the standard and quasi Pure Lovelock equations, ensures us to solve successfully the original system 4, 4 and 3. Furthermore, under our assumptions, each conservation equation 49, 52,$...$ • ‣ 4, 58 is separately conserved, and thus, there is no exchange of energy momentum between the seed fluid and each sector $(\theta_{i})_{AB}$. So, in our gravitational decoupling method there is only purely gravitational interaction.

It is worth stressing that as a consequence of the application of the MGD:

• •

We start with the indefinite system 3,3 and 3. After the application of MGD, we have a set of equations for the seed fluid $(\nu,\mu,\bar{\rho},\bar{p}_{r},\bar{p}_{\theta})$ given by the standard Pure Lovelock equations.

Next, we suppose that we have already found a seed fluid solution $(\nu,\mu)$ and the sources $(\bar{\rho},\bar{p}_{r},\bar{p}_{\theta})$, thus we have :

• •

A much simpler system of four unknown functions $(g,(\theta_{n})^{0}_{0}$ ,$(\theta_{n})^{1}_{1}$,$(\theta_{n})^{2}_{2})$ given by the quasi Pure Lovelock equations of order $\alpha^{n}$.

• •

Supposing that we have found the values of $g$ and $(\theta_{n})_{AB}$, we have $n-1$ systems, given by the quasi Pure Lovelock equations of order $\alpha^{i}$, where each of them has three unknown functions $((\theta_{i})^{0}_{0},(\theta_{i})^{1}_{1},(\theta_{i})^{2}_{2})$

For the study of well behaved solutions that represent stellar distributions it is necessary to analyse the matching conditions Milko . This is outside of the scope of this work and could be studied in elsewhere.

5 A special case

We impose the condition $\nu=-\lambda$ in equation 12. So, the $(t,t)$ component of Pure Lovelock equation keeps its form as in equation 3. However, it is direct to check that the $(r,r)$ component, equation 3, now is:

[TABLE]

see reference indio3 . Now, the system to solve is : equation 3 that corresponds to $(t,t)$ component, equation 5 that corresponds to $(r,r)$ component, and equations 27 and 29.

Furthermore, we impose the condition $\rho=-p_{r}$, as in references milko1 ; Dymnikova:2010zz , where $\rho$ corresponds to equation 20 and $p_{r}$ to equation 21. Additionally, we will impose arbitrarily that $\bar{\rho}=-\bar{p}_{r}$ and $(\theta_{i})^{0}_{0}=(\theta_{i})^{1}_{1}$, with $i=1,2...n$. Thus, the condition $\rho=-p_{r}$ is fulfilled.

So, the $(t,t)$ and $(r,r)$ components are similar to equation 3, whereas, now the conservation equation 3 takes the following form:

[TABLE]

In this way, the system to solve corresponds to the equations 3, 27 with $\bar{\rho}=-\bar{p}_{r}$ and 29 with $(\theta_{i})^{0}_{0}=(\theta_{i})^{1}_{1}$. Our seed solution, which is solution of this system with $\alpha=0$, is:

[TABLE]

Again, turning on $\alpha$, the effects of the source $\theta_{AB}$ appear on the seed solution. These effects are encoded in the geometric deformation undergone by the seed fluid geometry in equation 61 as follows:

[TABLE]

[TABLE]

Thus, taking into account the geometric deformation of equations 62 and 63, the $(t,t)$ and $(r,r)$ components are similar to the equation 4 and, the system to solve is given by equations 4, 27 with $\bar{\rho}=-\bar{p}_{r}$ and 29 with $(\theta_{i})^{0}_{0}=(\theta_{i})^{1}_{1}$.

Finally, using the binomial development 41, the system splits into the following sets of equations:

• •

The standard Pure Lovelock equations for a seed solution (with $\alpha=0$), that correspond to the equations 47 and the conservation equation given by:

[TABLE]

• •

The terms of order $\alpha$ give rise to the quasi-Pure Lovelock equations of order $\alpha^{1}$, that correspond to the equations 50 and the conservation equation given by:

[TABLE]

Again, following the iteration, it is possible to obtain the quasi pure Lovelock equations of order $\alpha^{2}$, $\alpha^{3}$…$\alpha^{n-3}$, $\alpha^{n-2}$.

• •

The terms of order $\alpha^{n-1}$ give rise to the quasi-Pure Lovelock equations of order $\alpha^{n-1}$, that correspond to the equations 53 and the conservation equation given by:

[TABLE]

• •

The terms of order $\alpha^{n}$ give rise to the quasi-Pure Lovelock equations of order $\alpha^{n}$, that correspond to the equations 56 and the conservation equation given by:

[TABLE]

5.1 A very simple test: Applying the method to the AdS space time.

As a simple test, we apply our method to a seed Anti de Sitter space time and, we test if it is possible to obtain a solution of physical interest that represents regular black holes.

The Schwarzschild black hole has a singularity where the laws of physics cease to operate. From the classical point of view, Bardeen in reference Bardeen proposed the first model of regular black holes, where the singularity is avoided, due to the formation of a dense core near the origin, whose effective cosmological constant $\Lambda_{eff}$ causes repulsive effects and, whose internal geometry has de Sitter form. This is achieved, by the change of the constant mass parameter by a mass function $M\to m(r)$ , such that near the origin $m(r)\approx\frac{\Lambda_{eff}}{6}r^{3}$, and $\displaystyle\lim_{r\to\infty}m(r)=M$. Thus, the factor $f=1-\frac{2m(r)}{r}$ near the origin behaves as de Sitter space time, and far from origin behaves as Scharzschild space time. After this, the formation of the Sitter core is associated to quantum fluctuations, where the energy density is of order of Planck units near the origin. This model is called *Planck star *, see references Planckstar1 ; Planckstar2 . Other more recent studies about regular black holes have been developed in references RBH1 ; RBH2 ; RBH3 ; RBH4 ; milko1 .

Due that regular black holes with de Sitter ground state have a cosmological horizon, we apply the method to a seed solution with Anti de Sitter structure. This is because, the presence of the cosmological horizon prevents a correct definition of the mass. Conversely, Anti de Sitter space time has a well defined asymptotically region, and thus, it is possible to define the mass.

As was said above, Pure Lovelock is a theory that, for $\Lambda>0$, has a unique de Sitter ground state for $n$ odd, and has a double Anti de Sitter or de Sitter ground state for $n$ even. Conversely, for $\Lambda<0$, has a unique Anti de Sitter ground state for $n$ odd and does not have physical solution for $n$ even.

So, to find a seed solution with Anti de Sitter structure, we study the case where the seed fluid has the following energy density:

[TABLE]

and

[TABLE]

thus, the energy density represents a $d$ dimensional cosmological constant, and $l^{n}$ is the Anti de Sitter radius.

Then, we will apply the Gravitational Decoupling method as follows:

• •

We solve the standard Pure Lovelock equations for our seed fluid:

First, we test the case with $n$ even. Inserting the seed energy density 68 into equation 47, we get to

[TABLE]

where $d-2n-1>0$ and,whose solution is $\pm\frac{r^{2}}{l^{2}}=(1-\mu)$. Taking the minus sign:

[TABLE]

For the case with $n$ odd, with $d-2n-1>0$, inserting the seed energy density 69 into equation 47, it is direct that the unique solution is equation 71.

Thus, both for $n$ odd or $n$ even, our seed solutions represent an Anti de Sitter space time. The tangential seed pressure is determined by equation 64.

• •

We solve the quasi-Pure Lovelock of order $\alpha^{n}$:

We choose a source $-(\theta_{n})^{0}_{0}=F(r)$ as in reference milko1 : $F(r)$ has a maximum value at $r=0$, such that the $m(r)$ function near the origin behaves as :

[TABLE]

where $k$ is a constant, and the $m(r)$ function is computed as:

[TABLE]

By solving equation 56:

[TABLE]

So, our line element is given by

[TABLE]

where:

[TABLE]

thus, near the origin the function behaves as:

[TABLE]

thus, for

[TABLE]

the solution behaves as de Sitter near the origin, and represents to a Pure Lovelock regular black hole by gravitational decoupling.

One example of $F(r)$ function is found in reference milko1 , wich is a $d$ dimensional generalization of Hayward density and, in our case is:

[TABLE]

where, one can notice that:

[TABLE]

on the other hand:

[TABLE]

where $Q$ is defined in this reference as a regulator and $M$ is the total mass. There is a deep analysis of unities in reference milko1 . So, $k^{2n}=Q^{d-2}$ and if the condition 78 is satisfied, then the function $F(r)$ is suitable for represent a Pure Lovelock Regular black hole. The fact that $\displaystyle\lim_{r\to\infty}m(r)=M$ allows to define correctly the mass in a space time with AdS ground state.

The source $(\theta_{n})^{2}_{2}$ is determined with equation 67

• •

Known $\bar{\rho},\bar{p}_{r},\bar{p}_{t},(\theta_{n})^{0}_{0},(\theta_{n})^{1}_{1},(\theta_{n})^{2}_{2}$, $\mu$, $\nu$ and $g$, the remaining sources $(\theta_{i})^{0}_{0},(\theta_{i})^{1}_{1},(\theta_{i})^{2}_{2}$ are determined by the quasi-Pure Lovelock equations of order $\alpha^{i}$.

6 Conclusion and discussion

We have shown an approach that represents a simple method for decoupling gravitational sources in Pure Lovelock gravity. Thus, it is possible to decouple gravitational sources under the effects of higher curvature correction terms for space times with a number of dimensions greater than four.

Applying our method, the final solution obtained is the result of the decoupling of the Pure Lovelock equations in a seed sector described by the seed energy momentum tensor $\bar{T}^{A}_{B}$ and the quasi Pure Lovelock equations of order $\alpha^{i}$ described by the sources $(\theta_{i})^{A}_{B}$. Thus, the equations of motion are solved for each sector separately and, by the superposition of these solutions, the complete solution is obtained.

The seed and the extra sources are separately conserved under the assumptions imposed in this work. The quasi Pure Lovelock equations are of order $\alpha^{1},\alpha^{2},...\alpha^{n-1},\alpha^{n}$. Therefore, the combination of these $n+1$ sectors only has gravitational interaction and does not have exchange of energy momentum. The order $\alpha^{n}$ corresponds to the order of the Generalized Einstein Tensor $\mathcal{G}^{(n)}_{AB}$. Thus, for $n=1$, where $\mathcal{G}^{(1)}_{AB}$ represents to the Einstein Tensor in Einstein Hilbert theory, the quasi Pure Lovelock are of order $\alpha^{1}$, and then, the quasi Einstein equations described in reference Ovalle1 are a particular case of our method. As indicates reference Ovalle1 , the quasi Einstein equations are useful to study the interaction between ordinary matter and the conjectured dark matter, so, our quasi Pure Lovelock equations perhaps could serve to investigate this problem in presence of higher curvature terms and in space times with a number of dimensions greater than four.

Furthermore, we have presented our method for the case where the line element has the form of equation 12 in section 4, and for the case where $\nu=-\lambda$ and $\rho=-p_{r}$ in section 5. As a simple test, we have applied our method to an Anti de Sitter seed solution. Choosing a source $F(r)$ that fulfills the condition 72, we have found the Pure Lovelock Regular black hole solution by gravitational decoupling 76. This solution differs in its structure with the Pure Lovelock regular black hole found in reference milko1 , whose structure is:

[TABLE]

wich takes the sign $+(-)$ for $\Lambda>(<)0$. Thus, equation 76 is a new solution in Pure Lovelock gravity, and thus a new Pure Lovelock regular black hole. In our solution, the ground state is obtained with the seed fluid, and when the extra sources $(\theta_{i})_{AB}$ are turn off. Although both solutions 76 and 82 differ in its structure, both share the same structure of ground state : for $\Lambda>0$ both solutions have a double Anti de Sitter or de Sitter ground state for $n$ even (where we have chosen the AdS branch) , whereas for $\Lambda<0$ both solutions have a single Anti de Sitter ground state for $n$ odd.

Although our solution 76 is new by using Pure Lovelock gravity, one solution with similar mathematical structure was found by using the $n$-fold degenerated ground state theory in reference milko1 . Therefore both solutions share the horizons structure and thermodynamics features described in reference milko1 . However, both solutions have different ground state structure. The ground state in the $n$-fold degenerated ground state theory is obtained by choosing a suited election of the coupled constants $\gamma$ such that the solution has one unique Anti de Sitter ground state (or $n$ fold degenerated Anti de Sitter ground state) with $\Lambda<0$, whereas in our case the ground state structure is different and was described in the last paragraph.

So, we have showed that our Gravitational Decoupling method is a direct way to obtain regular black holes. In regarding this, the Minimal Geometric Deformation of the Anti de Sitter space time give rise to regular black holes, under the assumptions used in this work.

A simple recipe to apply our method could be:

• •

Pick up a seed Pure Lovelock solution $\{\mu,\nu,\bar{\rho},\bar{p}_{r},\bar{p}_{t}\}$ and solve the standard Pure Lovelock equations.

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Solve the quasi Pure Lovelock equations of order $\alpha^{n}$. In regarding this, we impose a form of the source $(\theta_{n})^{0}_{0}$ and the function $g(r)$ is directly determined by the $(t,t)$ component. Furthermore, in direct way, the sources $(\theta_{n})^{1}_{1}$ and $(\theta_{n})^{2}_{2}$ are obtained by solving the $(r,r)$ and $(\theta,\theta)$ components, respectively.

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Well known $\bar{\rho},\bar{p}_{r},\bar{p}_{t},(\theta_{n})^{0}_{0},(\theta_{n})^{1}_{1},(\theta_{n})^{2}_{2}$, $\mu$, $\nu$ and $g$, the remaining functions $(\theta_{i})^{0}_{0},(\theta_{i})^{1}_{1},(\theta_{i})^{2}_{2}$ are directly determined by the quasi-Pure Lovelock equations of order $\alpha^{i}$.

Thus, our method is an easy algorithm to search new analytical solutions of physical interest in Pure Lovelock gravity. In Einstein Hilbert theory, the Gravitational Decoupling method of reference Ovalle1 have been used to find new $4D$ black hole solutions in references Ovalle3 ; Contreras1 ; Contreras2 ; Contreras3 ; Contreras4 and new $4D$ well behaved solutions that represent stellar distributions in references Ovalle2 ; Camilo ; Tello1 ; Graterol ; Tello2 ; Tello3 ; Luciano ; Milko ; Luciano1 . So, inspired by this method, we have presented a useful tool that could serve to find new black hole solutions or stellar distributions in space times with a number of dimensions greater than four, and in presence of higher curvature correction terms. This applications could be studied in elsewhere.