# Gravitational decoupled anisotropies in compact stars

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## Abstract

Simple generic extensions of isotropic Durgapal–Fuloria stars to the anisotropic domain are presented. These anisotropic solutions are obtained by guided minimal deformations over the isotropic system. When the anisotropic sector interacts in a purely gravitational manner, the conditions to decouple both sectors by means of the minimal geometric deformation approach are satisfied. Hence the anisotropic field equations are isolated resulting a more treatable set. The simplicity of the equations allows one to manipulate the anisotropies that can be implemented in a systematic way to obtain different realistic models for anisotropic configurations. Later on, observational effects of such anisotropies when measuring the surface redshift are discussed. To conclude, the consistency of the application of the method over the obtained anisotropic configurations is shown. In this manner, different anisotropic sectors can be isolated of each other and modeled in a simple and systematic way.

## 1 Introduction

The study of analytical solutions of Einstein field equations plays a crucial role in the discovery and understanding of relativistic phenomena. Theoretical grounds provide many exact and not exact isotropic solutions in general relativity; however most of them have no physical relevance and do not pass elementary tests of astrophysical observations [1, 2, 3, 4]. Theoretical grounds gives very few isotropic solutions under static and spherically symmetric assumptions. Worse yet, even fewer of these solutions have physical relevance passing elementary tests of astrophysical observations [1, 2, 3, 4]. Furthermore, no astronomical object is constituted of perfect fluids only; hence isotropic approximation is likely to fail.

Anisotropic configurations have continuously attracted interest and are still an active field of research. Strong evidence suggests that a variety of very interesting physical phenomena gives rise to a quite large number of local anisotropies, either for low or high density regimes (see [5] and references therein). For instance, among high density regimes, there are highly compact astrophysical objects with core densities ever higher than nuclear density \((\sim 3\times 10^{17}\,\text {kg/m}^3)\) that may exhibit an anisotropic behaviour [6]. Certainly, the nuclear interactions of these objects must be treated relativistically. The anisotropic behaviour is produced when the standard pressure is split in two different contributions: the radial pressure \(p_r\) and the transverse pressure \(p_t\), which are not likely to coincide.

Anisotropies in fluid pressure usually arise due to the presence of a mixture of fluids of different types, rotation, viscosity, the existence of a solid core, the presence of a superfluid or a magnetic field [7]. Even are produced by some kind of phase transitions or pion condensation among others [8, 9]. The sources of anisotropies have been widely studied in the literature, particularly for different highly compact astrophysical objects such as compact stars or black holes, either in four dimensions [10, 11] as well as in the context of braneworld solution in higher dimensions [12, 13, 14].

*minimal geometric deformation*approach [15, 16, 17] (MGD hereinafter). This method was originally proposed in the context of the Randall–Sundrum braneworld [18, 19] and was designed to deform the standard Schwarzschild solution [20, 21]. It describes the effective 4D geometry of a spherically symmetric stellar distribution with a physically admissible anisotropic behaviour produced by bulk corrections over the braneworld. The details of this method will be shown later, however the main lines goes as follows: let us start with a well known spherically symmetric gravitational source \(T_{\mu \nu }^{\scriptscriptstyle (0)}\). This source can be as simple as one would desire; one can start with any known perfect fluid or even with vacuum itself. Any classical solution works as a seed for this method. After this, one switch on a new source of anisotropy

This method for decoupling non-linear differential equations can be applied in a systematic way and has a vast unexplored territory where it could give different novel perspectives. MGD does not only give consistent interior solutions for different perfect fluids in GR; it could also be conveniently exploited in relevant theories such as *f*(*R*)–gravity [25, 26, 27], intrinsically anisotropic theories as Hořava–aether gravity [28] or to study the stability of novel proposals of Black Holes, described by Bose Einstein gravitational condensate systems of gravitons [29, 30, 31]. This is a robust method to extend physical solutions into an anisotropic domain preserving the physical acceptability.

The paper is organized as follows: after this introduction, we present the Einstein field equations for an anisotropic fluid. In Sect. 3 we explain how the MGD approach is implemented to generate arbitrary anisotropic solutions. Section 4 is dedicated to apply this method to a particular seed, the Durgapal–Fuloria model for compact stars. We present some physical anisotropic solutions and discus possible observational effects. In Sect. 5 we extend the method to seeds which are already anisotropic. The last two sections are dedicated to discuss the main results and summarize our conclusions.

## 2 Anisotropic effective field equations

*r*. There is another equation consequence of the Bianchi identities: the covariant conservation of the stress–energy tensor

*a priori*restriction for the components of \(\theta _{\mu \nu }\); however, if \(\theta _r{}^r\ne \theta _\varphi {}^\varphi \) when solving the equation system (4)–(6), we will be in the presence of the pressure anisotropy

## 3 Minimal geometric deformation approach

*minimal geometric deformation*over the metric without breaking the spherical symmetry of the initial solution; this is

*minimal distortion*of the metric, the system of Eqs. (4)–(6) results quasi-decoupled: we obtain the Einstein equations for the chosen perfect fluid; and an effective ‘pseudo-Einstein’ system of equations governing the \(\theta \)-sector. The only parameter that connects the two sectors is the temporal geometric function \(\nu (r)\). At the same order as before, the temporal, radial and angular equations of motion relating the geometry of the spacetime to the thermodynamic characteristic of the perfect fluid sector reduce to

To conclude this section, let us summarize. First we started with an indeterminate system of Eqs. (4)–(6). Then, we performed a linear mapping of the radial geometric function of the metric (16) that results in a ‘decoupling’ of the Einstein field equations. We ended with two sets of equations: a perfect fluid sector \(\{\rho ;\,p;\,\nu ;\,\mu \}\), given by Eqs. (17)–(21) where everything is known once a perfect solution of GR is chosen; and a simpler sector of three linearly independent equations that can be chosen from (22) to (26), for determining four unknown functions \(\{f^*;\,\theta _t{}^t;\,\theta _r{}^r;\,\theta _\varphi {}^\varphi \}\). Once the second sector is solved, we can identify directly the effective physical quantities introduced in (10), (11) and (12). At this point, is mandatory to recall that the underlying anisotropic effect which appears as a consequence of breaking the isotropic condition over the effective pressures, \(\widetilde{p}_t \ne \widetilde{p}_r\), causes the appearance of the anisotropy \(\varPi (\alpha ;r)\) defined in Eq. (14).

## 4 Anisotropic Durgapal–Fuloria compact star

*C*an integration constant. The gravitational mass of a sphere of radius

*r*is obtained integrating the density inside the corresponding volume; in spherical coordinates is

*m*vanishes faster than

*r*as one can easily check from Eq. (29) inside Eq. (31).

*A*is the second (and last) integration constant to be determined, both

*A*and

*C*, using boundary conditions over the surface \(r=R\). In the present article the outer metric will be chosen to satisfy the Schwarzschild form—for simplicity, an uncharged compact star. Extensions to more complex outer spacetimes is straightforward. We can choose a Kerr metric for obtaining a more realistic rotating object, or even switch on a charge. Both effects are more complicated and interesting sources of anisotropies that give rise to more realistic scenarios. Nonetheless we will maintain the distribution in vacuum to get the treatment as simple as possible. Both constants

*A*and

*C*are positive; however they are expected to change as far as anisotropies begin to be considered.

The remaining equations after the decoupling, Eqs. (22)–(24), have to be solved if a generic anisotropic self-gravitating system is desired. The system of equation is as explain before underdetermined. A reasonable constrain is needed to close the system, but it is mandatory not to lose the physical acceptability of the solution. These issues will be discussed in what follows when three different anisotropic solutions (of many) are presented.

### 4.1 Pressure-like constraint for the anisotropy

As we have closed the system of equations with the constrain (33), we can compute all the effective magnitudes that characterized the fluid; but first, the values of the integration constants *A* and *C* are needed to be fixed. This will be done by means of consistent matching conditions.

#### 4.1.1 Matching conditions

*C*not to vary from the perfect fluid solution once the anisotropies are considered. The value is

*r*/

*R*for different values of \(\alpha \). At first sight one can observe that the higher \(\alpha \) is, the smaller the radial pressure becomes.

This issue is not surprising at all. The anisotropy mimics the radial pressure, hence the radial and tangential pressures start to drift apart in the region close to the solid surface. For this anisotropic behaviour to happen, both pressures must decrease in magnitude at the inner region. Of course this pressure discrepancy with respect to the isotropic solution makes the density to be disturbed. The equilibrium between gravitational collapse and pressure repulsion is modified; hence the mass function is redistributed to the center of the star. Despite this, the total mass of the anisotropic object remains unmodified and so as the compactness parameter \(\xi \).

*A*from Eq. (39). The temporal component of the MGD metric (30) should match smoothly with the outer Schwarzschild region

*A*remains unchanged. This constant close one branch (\(\alpha \)-dependent) of anisotropic solutions analogous to Durgapal–Fuloria; namely \(\{\nu ;\,\lambda ;\,\widetilde{\rho };\,\widetilde{p}_r;\,\widetilde{p}_t\}\). Of course, this solution is not unique. Different anisotropic solutions can be obtained starting from the Durgapal–Fuloria solution by means of requiring different constrains when closing the indeterminate system of equations. In next section we will consider a different constrain and we will see that a different anisotropic solution is obtained.

### 4.2 Density-like constrain for the anisotropy

*R*, \(R_{\mu \nu }R^{\mu \nu }\) and \(R_{\mu \nu \gamma \sigma }R^{\mu \nu \gamma \sigma }\) to remain smooth and finite all over the inner region. Note that with this constraint the radial deformation is again totally determined by the solution of the perfect fluid. Eventually, one computes the relevant component of the metric; the minimally deformed component is written as in Eq. (16) (naming \(\beta \) to the coupling between sectors) as

*m*(

*r*) presented in Eq. (29)

Once the system is closed and the minimal deformation obtained, the remaining magnitudes are easily derived. As before this will be done by means of the smooth matching between the inner and outer region of the star.

#### 4.2.1 Matching conditions

*A*and

*C*; this time for the density ansatz (50). It is already known that the constant

*C*is determined by the second fundamental form (42). Its value is

*minimal geometric deformation*over the metric in only one ‘direction’; the ‘direction’ to where the density and the mass is increase. The anisotropies restricted to the present constrain change the integration constants; for instance

*C*is \(\beta \) dependent. An analysis over Eq. (54) shows that

*C*increases when \(\beta \) becomes more negative. This behaviour makes Eqs. (53) and (55) to increase when \(\beta \) increase in modulus. Of course, theses both parameters can not increase without a limit; as in the previous case, anisotropies develop instabilities. In the first curve of Fig. 3 it is seen how the density function increases in the inner region, while it slightly decreases its value over the surface’s surroundings softening the crust. The mass function rises throughout the interior and the total effective mass is also increased.

*A*is found in an analogous manner than in the previous section, i.e. by means of Eq. (49). The value of this constant changes with \(\beta \). The usual constant of Durgapal–Fuloria is recovered in the limit of \(\beta \rightarrow 0\) as it should be.

### 4.3 Detectability and observational differences in anisotropic distributions

*t*. This relation is given by the standard formula \(\mathrm {d}\tau ^2=g_{tt}\,\mathrm {d}t^2\) that yields the following for the surface redshift

Let us start with the first solution derived in Sect. 4.1. The interest should focus in the mass function (48) evaluated over the surface. As we have explained after this equation the Schwarzschild mass remains unmodified with respect to the Durgapal–Fuloria mass, this is \(M_{\scriptscriptstyle Schw}\equiv M_{\scriptscriptstyle DF}(R)=\widetilde{m}(R)\). Hence, there is no observational evidence to differentiate an isotropic star to these anisotropic counterpart.

## 5 Anisotropizing an anisotropic Durgapal–Fuloria star

In Sect. 3, we present a method to generate different anisotropic solutions of Einstein field equations using any well known perfect fluid as a seed. After this, we apply this prescription to the Durgapal–Fuloria perfect sphere. In Sect. 4, with some reasonable constrains we found two novel physical anisotropic solutions analogous to the Durgapal–Fuloria compact star.

*minimally deform*the anisotropic solution along the radial component of the metric. While the temporal geometric function in Eq. (30) remains unchanged, the minimal distortion takes place only over the radial component

*minimal geometric deformation*(62) decouples the two anisotropic sectors. On the one hand, the seed sector characterized by the density given by Eq. (47), the radial pressure \(\widetilde{p}_r\) (44) and the tangential anisotropic pressure \(\widetilde{p}_t\) obtained in Eq. (45). This parameters solve the already known equation system

*C*is obtain by means of the continuity of the second fundamental form, analogous condition to Eq. (42). Imposing the annulment of the latter effective radial pressure at the surface \(\varSigma \), we get

*C*as well as the effective pressure \(\bar{p}_r\) recover the corresponding values: Eqs. (43) and (44) in the limit of \(\beta \rightarrow 0\) or Eqs. (54) and (56) when \(\alpha \rightarrow 0\). Besides, this constant is required to plot the thermodynamic parameters.

In Fig. 4 we present the corresponding evolution of the parameters of the theory: for a comparison, we include also the Durgapal–Fuloria isotropic solution (\(\alpha =\beta =0\) in solid line). If for instance, one of the couplings move away from zero but the other remains null, the thermodynamic quantities behave as in Figs. 2 (if \(\alpha \ne 0\)) or 3 (if \(\beta \ne 0\)), as it is expected. After this, we plot the anisotropic seed to be *minimally deformed* by fixing the coupling \(\alpha \) (red dashed-line). Finally, \(\beta \) drifts away the parameters again. We choose one smaller order or magnitude for the second deformation to make notorious the effect over the seed solution. An important statement is that as the seed is anisotropic and the corresponding tangential pressure is nonnull over the surface, then there is no restriction for \(\beta \) to be negative. \(\beta \) is allowed to be positive until either it decrease the tangential pressure until it becomes null, or the anisotropy becomes unstable.

*minimal geometric deformations*computed in Sect. 4

*minimal deformation*induced by an anisotropy subjected to a pressure structure, makes the stress energy tensor to become \(\widetilde{T}_{\mu \nu }=T^{\scriptscriptstyle (PF)}_{\mu \nu }+\alpha \,\theta _{\mu \nu }^{\scriptscriptstyle (pressure)}\). After this, a second

*minimal deformation*acts over the already anisotropic solution, but now subjected to a density constrain. The new contribution is given by Eq. (82), therefore the effective energy–momentum tensor (63) is decomposed as

From a perturbation theory point of view, one can think that the deformations over the metric (*zero* order) is due to the existence of the anisotropic term which acts at \(\mathcal{O}(\alpha )\); being \(\alpha \) the coupling strength to the anisotropies. We must emphasize that, although the MGD approach seems like a perturbation technique, the method, in fact, is not, and this is easily visualized by noticing that the couplings do not necessarily have to be small, which is a crucial ingredient in perturbation theories. The deformation being a perturbation is just a well behaved limit of the theory, and means that we can softly deform the seed configuration. Being the theory noncommutative, successive and mixed perturbative deformations give different configurations depending on the order in which each of them are implemented. This provides infinite manners of deforming realistic configurations controlling rigorously the physical acceptability of the resulting anisotropic distribution.

## 6 Conclusions

In this paper we have presented different branches of solutions that models non-rotating and uncharged anisotropic superdense stars. Each anisotropic branch opens a possibility for new physically acceptable configuration obtained by guided deformations over the isotropic Durgapal–Fuloria stars, and exemplify some possible anisotropic distributions among the many that the MGD method generates. This prescription has been design to decouple the field equations of static and spherically symmetric self-gravitating systems. It associate the anisotropic sector with a deformation over the geometric potentials. In this work we have reported radial deformations only, but extensions to the temporal deformation may bring intriguing results. After the decoupling, one obtains a sector which solution is already known (seed sector) and the anisotropic sector which obeys a set of simpler ‘pseudo-Einstein’ equations associated to the metric deformation. It is worth to note that the *minimal geometric deformations* stem in an exclusively gravitational interaction between sectors; i.e. there is no exchange of energy–momentum among them.

When the equations are decoupled, no new information is introduced. Then we have an underdetermined system of equations; consistent constrains are needed. We have shown how intuitive constrain leads to new physical anisotropic solutions. Variations in the couplings between the seed and the anisotropic sector reveals consistent evolution of the thermodynamical parameter giving to the MGD method a new prove of validity. We also have discussed the observational features of the anisotropic sectors. When the anisotropy changes the compactness of the star, the observed redshift increases as it is expected. Not all anisotropic contributions have observational effects because some anisotropies only readjust the thermodynamical parameters in the interior. However, when the anisotropy tweaks the compactness parameter, the star suffer a redshift. Therefore, observational data would bound the parameters of the model.

After presenting two branches of solutions that provides an infinite number of physical compact stars, we have proceed to generalize the method to deform anisotro-pic solutions. Any solution of Einstein equation admits a minimal deformation. Different anisotropic sources have additive effect, however these effects are noncommutative. The path to the final configuration matters, and normally deformations in reversed order produces different resulting configurations. Hence, the method provides a ‘fine tunning structure’ that generates an enormous amount of different physically acceptable anisotropic stars.

## Notes

### Acknowledgements

The author AR was supported by the CONICYT-PCHA /Doctorado Nacional/2015-21151658. LG acknowledges the FPI Grant BES-2014-067939 from MINECO (Spain). The author CR was supported by Conicyt PhD Fellowship No. 21150314.

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