# Anisotropic neutron stars by gravitational decoupling

## Abstract

In this work we obtain an anisotropic neutron star solution by gravitational decoupling starting from a perfect fluid configuration which has been used to model the compact object PSR J0348+0432. Additionally, we consider the same solution to model the Binary Pulsar SAX J1808.4-3658 and X-ray Binaries Her X-1 and Cen X-3 ones. We study the acceptability conditions and obtain that the MGD-deformed solution obey the same physical requirements as its isotropic counterpart. Finally, we conclude that the most stable solutions, according to the adiabatic index and gravitational cracking criterion, are those with the smallest compactness parameters, namely SAX J1808.4-3658 and Her X-1.

## 1 Introduction

The first exact interior solution of the Einstein’s equations which describe a self-gravitating perfect fluid with constant density embedded in a static and spherically symmetric vacuum was obtained by Karl Schwarzschild [1] and for a long time isotropic solutions have been broadly considered as suitable interior models of compact objects (see, for example, [2] and references therein). However, very few of these solutions can be considered as physically relevant because they violate some of the elementary conditions that a realistic solution has to satisfy (for a list of physical conditions of interior solutions see, for example, [3]). Even more, in the cases where acceptable solutions can be found, the perfect fluid model can not be used to deal with situation where it is assumed local anisotropy of pressure, that seems to be very reasonable for describing the matter distribution under a variety of circumstances [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

In this sense, we may wonder if given an acceptable perfect fluid interior solution, in the sense of [3], one can be able to extend it to anisotropic domains holding its physical acceptability to some extent. Fortunately, this question has an affirmative answer. Indeed, the so-called Minimal Geometric Deformation (MGD) method [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] have been extensively used to obtain new black hole solutions and to extend interior isotropic models, given the number of ingredients which convert it in a versatile and powerful tool to solve the Einstein’s equations. To be more specific, the method has been used to obtain anisotropic like-Tolman IV solutions [34, 38], anisotropic Tolman VII solutions [66] and black holes in \(3+1\) and \(2+1\) dimensional space–times [41, 52, 55]. Moreover, in the context of modified theories of gravitation, the method has been used to obtain solutions in \(f({\mathcal {G}})\) gravity [45], Lovelock [63], *f*(*R*, *T*) [61], Rastall [69] and more recently black holes and interior solutions in the context of braneworld [68].

Recently, a perfect fluid interior solution has been reported in Ref. [70] and has been used to model the neutron star PSR J0348+0432. It is our main goal here extend this solution by MGD to model not only the compact object PSR J0348+0432 but the Binary Pulsar SAX J1808.4-3658 and X-ray Binaries Her X-1 and Cen X-3 ones. Additionally, we want to explore the physical acceptability of the anisotropic solution obtained by MGD.

This work is organized as follows. In the next section we review the main aspects related to the MGD-decoupling method. Section 3 is devoted to introduce the new static and spherically symmetric solution and the anisotropic solution obtained by MGD is studied in Sect. 4. In the last section, we present our final comments and conclusions.

## 2 Einstein equations and MGD-decoupling

This section is devoted to review the main aspects of MGD. To this end, we shall follow closely the original paper on this topic in the context of gravitational decoupling in General Relativity reported in [34].

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

As explained in previous sections, the MGD has been successfully used to extend isotropic solutions to anisotropic domains. More precisely, given a metric functions \(\{\nu ,\mu \}\) sourced by a perfect fluid \(\{\rho ,p\}\) that solve Eqs. (13), (14) and (15), the deformation function *f* can be found from Eqs. (17), (18) and (19) after choosing suitable conditions on the anisotropic source \(\theta _{\mu \nu }\). It is worth mentioning that the case we are dealing with demands for an exterior Schwarzschild solution. In this case, the matching condition leads to the extra information required to completely solve the system.

*f*after replacing (27) in (18). In this sense, the only remaining step to construct an anisotropic interior solution following MGD is to choose a suitable perfect fluid as a seed to solve the system (17), (18) and (19) via the mimic constrain to finally obtain the interior solution with local anisotropies described by \(\{\nu ,\lambda ,\tilde{\rho },\tilde{p}_{r},\tilde{p}_{\perp }\}\).

Compactness parameter for some compact stars

Compact Star Model | Mass in \(M_{\odot }\) | Radius | Compactness factor |
---|---|---|---|

SAX J1808.4-3658 | 0.85 | 9.5 | 0.13 |

Her X-1 | 0.9 | 8.1 | 0.15478 |

Cen X-3 | 1.49 | 10.8 | 0.2035 |

PSR J0348+0432. | 1.97 | 12.957 | 0.229365 |

## 3 Perfect fluid neutron star

In this section we briefly review some aspects about the isotropic spherically symmetric solution reported in Ref. [70].

*C*,

*a*and

*A*are constants. The metric (28) is a solution of the Einstein Field equation \(R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }R=-\kappa ^{2}T^{(m)}_{\mu \nu }\) where the isotropic source reads

*R*is the radius of the star. Furthermore, the normalized radius, \(x\in (0,1)\), and the parameter \(\omega \) are dimensionless quantities. It is worth noticing that, on one hand, the parametrization allows to write the quantities \(\kappa ^{2}R^{2}\rho \) and \(\kappa ^{2}R^{2}p\) as functions of

*x*and \(\omega \) only. On the other hand, the matching conditions lead to (see [70] for details)

*u*given by

*u*is fixed. For example, in Ref. [70] the authors fixed the compactness parameter using data associated to PSR J0348+0432.

## 4 Anisotropic neutron star by MGD

Note that, for fixed \((u,\omega )\) the decoupling function, *f*, is a function of *x* only. However, it is worth noticing that after implementing MGD the metric function \(\lambda \) given by Eq. (12) and the matter sector of the total solution, namely, \(\tilde{\rho }\), \(\tilde{p}_{r}\) and \(\tilde{p}_{\perp }\) which are obtained from Eqs. (12), (17), (18) and (19) respectively, will depend on the free parameter \(\alpha \).

The next step in the program consist in to calculate \(\{\lambda ,\tilde{\rho },\tilde{p}_{r},\tilde{p}_{\perp }\}\). However, as can be checked by the reader, the resulting functions are too long that their explicit form is not illuminating at all. In this sense, we shall dedicate the rest of the manuscript in to illustrate their properties by graphical analysis. Furthermore, we shall fix the compactness parameter taking values from Table 1 (see [59] and references therein).

## 5 Conditions for physical viability of interior solutions

The study of acceptability conditions of interior solutions is important because, as it is well known, the goal is not only to solve Einstein’s equations but to demonstrate that they are suitable to describe a physical system. In this section we explore some of these conditions.

### 5.1 Matter sector

Note that all the profiles satisfy the first requirement about positivity and monotonously decreasing for all the values of \(\alpha \). Interestingly, as \(\alpha \) increases, the density at the center becomes greater in contrast to the behaviour on the surface. On the contrary, the radial and the traverse pressure decrease when \(\alpha \) grows.

### 5.2 Energy conditions

### 5.3 Causality

### 5.4 Adiabatic index

### 5.5 Convection stability

### 5.6 Stability against gravitational cracking

*r*between the center and some value beyond which the force reverses its direction [72]. In Ref. [73] it is stated that a simple requirement to avoid gravitational cracking is

## 6 Final remarks

In this work we have implemented the Minimal Geometric Deformation decoupling method to extend an isotropic model of Neutron Stars to anisotropic domains. More precisely, we studied the model used in [70] to describes the PSR J0348+0432 compact object but, additionally, we have considered the models SAX J1808.4-3658, Her X-1 and Cen X-3, also. Furthermore, we have studied the physical acceptability of all the models for different values of the MGD-parameter, \(\alpha \), starting from the isotropic case (\(\alpha =0\)) and exploring how their behaviour are modified when the anisotropy is induced (\(\alpha \ne 0\)). In particular we studied the profiles of density and pressures, energy condition, causality, and the stability of the solution studying the adiabatic index, the convection and cracking conditions. We found that, for all the values of compactness parameters considered, the density and the pressures satisfy the basic requirements of an interior solution, namely, they reach their maximum value at the center and are monotonously decreasing toward the surface of the star. Additionally, the solution satisfies the dominant and the strong energy conditions and their sound velocities (radial and tangential) are less than the speed of light as expected. We found that, for compactness parameters corresponding to SAX J1808.4-3658 and Her X-1 compact objects, the solution is stable according to the adiabatic index criteria for all the values of the decoupling parameter \(\alpha \) considered. However, for Cen X-3 and PRS J0348+0432 the adiabatic index is less than 4 / 3 for the highest values of \(\alpha \) considered and as a consequence, the configuration is unstable in this situations. This situation is similar to the obtained in the study of gravitational cracking, more precisely, the configuration is stable for the most compact object considered and unstable for the other two models. Interestingly, in this case, the instability appears for any value of \(\alpha \) we considered. Regarding the convection stability, we found that the solution is unstable for all the models under study for any value of \(\alpha \) including the isotropic case which corresponds to \(\alpha =0\). In conclusion, we have obtained that according to the adiabatic index and gravitational cracking criterion, the most stable anisotropic models are those with the smallest compactness parameters, namely SAX J1808.4-3658 and Her X-1.

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