# Minimal geometric deformation in asymptotically (A-)dS space-times and the isotropic sector for a polytropic black hole

## Abstract

In the context of the minimal geometric deformation method, in this paper we implement the inverse problem in a black hole scenario. In order to deal with an anisotropic polytropic black hole solution of the Einstein field equations with cosmological constant, the deformation method is slightly extended. After obtaining the isotropic sector and the decoupler for an anisotropic (A-)dS polytropic black hole solution, we emphasize a possible relation between anisotropization/isotropization and the violation of the energy conditions.

## 1 Introduction

In recent years, the use of the minimal geometric deformation (MGD) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33] as a systematic and powerful method to obtain new and relevant solutions of the Einstein field equations, has considerable increased [18, 22, 23, 24, 25, 26, 27, 30, 32]. For example, the method has allowed to induce local anisotropies in spherically symmetric systems leading to both more realistic interior solutions of compact objects [25, 26] and hairy black holes [24]. More recently, the method has been extended to solve the inverse problem [32], namely, given any anisotropic solution of the Einstein field equations it is possible to recover the isotropic source and the decoupler matter content which, after gravitational interaction, led to the anisotropic configuration. In that work, it was found that, for a simple anisotropic solution violating all the energy conditions, the free parameters involved in the MGD can be fitted in such a manner that both the isotropic source and the matter decoupler content satisfy all the energy conditions. The importance of this result lies in the fact that the inverse problem allowed to interpret the MGD as some kind of mechanism which leads to the apparition of exotic matter after gravitational interaction of well behaved matter content.

In the same spirit of Ref. [32], it could be interesting to explore if the same duality exotic/non-exotic matter content occurs in other scenarios after the application of the inverse MGD problem. To be more precise, we could study if such a duality persists in situations where the starting point is a solution sourced by a matter content satisfying all the energy conditions. In order to do so, in this work we implement the inverse problem program in a polytropic black hole (BH) originally studied in reference [34] and extended to the scale-dependent scenario in Ref. [35]. As it will be shown in the rest of the manuscript, the choice of such a system is twofold: first, to extend the MGD in order to deal with Einstein field equations with cosmological constant and, second, to implement the inverse problem in an anisotropic system which satisfies all the energy conditions.

This work is organized as follows. In the next section we briefly review the MGD-decoupling method. In Sect. 3 we develop the method to obtain the generator of any anisotropic solution of the Einstein Field Equations and then we implement the method for a polytropic BH solution in Sect. 4. The last section is devoted to final comments and conclusion.

## 2 Einstein equations with cosmological constant and extended MGD-decoupling

*r*only. Considering Eq. (4) as a solution of the Einstein field equations, we obtain

*f*is the geometric deformation undergone by the radial metric component \(\mu \), “controlled” by the free parameter \(\alpha \). By doing so, we obtain two sets of differential equations: one describing an isotropic system sourced by the conserved energy–momentum tensor of a perfect fluid \(T^{\mu (m)}_{\nu }\) an the other set corresponding to quasi-Einstein field equations sourced by \(\theta _{\mu \nu }\). After taking into account that the cosmological constant can be interpreted as some kind of isotropic fluid, we include the \(\varLambda \)-term in the isotropic sector and we obtain

^{1}We would like to emphasize that that the addition of the cosmological constant only affects the isotropic sector because Eqs. (15), (16) and (17) remain unchanged. At this point, we are ready to implement the inverse problem program.

## 3 MGD-decoupling: the inverse problem

*f*in terms of well known quantities of anisotropic solution. To be more precise, with this constraint we do not need any artificial equation of state for the \(\theta \)’s components. It is worth mentioning that, in the case of Einstein equations with cosmological constant, the previous constraint leads to the same result obtained in Ref. [32] because there is no contribution of \(\varLambda \). More precisely, after subtracting Eqs. (9) and (10), the cosmological constant disappears and the solution remains the same. In this sense, the combination of Eqs. (16) and (17) with the constraint (18) leads to a differential equation for the decoupling function

*f*given by

*f*, is given by

In the next section, we shall briefly review the mains aspects of the polytropic BH reported in Ref. [34] and then we shall implement the method to obtain its isotropic generator and decoupler matter content.

## 4 Isotropic sector of a polytropic BH solution

*f*, reads

The results obtained so far could be interpreted as follows. The isotropic sector of the polytropic BH reported in Ref. [34] corresponds to a BH solution hiding exotic matter in its interior. What is more, the new isotropic BH solution obtained in this work could be thought as the remnant of a gravitational collapse involving exotic matter which is the responsible of sustain traversable wormholes [36, 37, 38, 39].

## 5 Conclusions

In this work we have extended the Minimal Geometric Deformation approach when a cosmological constant is present showing that, in this case, only the isotropic sector is modified. In particular, the inverse problem in the context of polytropic black holes has been explored, obtaining the isotropic sector from which an anisotropic (A-)dS polytropic black hole is obtained. Moreover, the isotropic sector contains a singularity which does not depends on any of the free parameters of the deformation approach. This singularity is hidden inside an event horizon, which remarkably coincides with that of the anisotropic black hole. Finally, we have noted that the isotropic sector is deeply linked with the appearance of exotic matter, although it can be located inside the horizon. In this sense, this work shows a nice example of how one could, in principle, control the energy conditions by tuning the isotropy/anisotropy of a black hole solution.

## Footnotes

- 1.
In what follows we shall assume \(\kappa ^{2}=8\pi \).

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