An anisotropic interior solution of Einstein equations

1. Introduction

The first investigations made about stellar solutions in the frame of the general theory of relativity consider that their interior is formed by a perfect fluid in a static and spherically symmetric spacetime.¹ The most recent investigations show that this type of proposals are adequate in some cases, since they are consistent with observational data of mass and radius for some stars,^{2,3,4,5,6,7,8,9} the concordance of these models with the observational data can occur due to the models being a good approximation for these stars or for the range of the values for the masses and radii of the respective stars. However, for the cases in which the density is greater than the nuclear density, some theoretical studies show that locally anisotropic pressures can be generated, that is to say, that the radial and tangential pressures are no longer equal as it is in the case of a perfect fluid. The difference in the pressures can be present when the density of the matter is greater than the nuclear density^10,11 originated by different causes¹² as it occurs in the presence of nucleonic superfluids,¹³ pion condensations,¹⁴ some phase transitions¹⁵ or due to a little rotation.¹⁶ One of the first solutions proposed in which the anisotropy in the pressure is supposed to be present locally and that allows to understand some differences with the solutions given for a perfect fluid was developed for the case of an incompressible fluid.¹⁷ Starting from this solution and from the requirement that the pressure is finite in the center of the star, a bound was obtained for the compactness $u=\frac{GM}{c^{2}R}<\frac{1}{2}[1-3^{\frac{2}{\xi -1}}]$ with $\xi <1$, relation between the mass $M$ and the radius of the star. As such in this case, for $\xi >0$, there can be a compactness value greater than in the case of a perfect fluid ($\xi =0$, $u<\frac{4}{9}$). Although we arrived to this conclusion for the case of an anisotropic fluid with constant density, in the most general anisotropic case, with physically acceptable conditions, the compactness value¹⁸ can be greater than one from a perfect fluid.¹⁹ Considering anisotropic fluids people have been able to construct a wide variety of solutions, some of these have the property that they can be reduced to the case of a perfect fluid in absence of the anisotropy,^{20,21,22,23,24,25,26} others have been developed from imposing a state equation for the radial pressure as function of the density,^{27,28,29,30,31} also its consistency, between the theoretical values and the observational data, has been verified for some compact objects, as are the stars 4U 1538-52, LMCX-4, PSRJ1614-2230, EXO1785-248, SAXJ1808.4-3658, HER X-1 and Vela X-1.^{32,33,34,35,36} The analysis of stellar models is still an active theme both for the case of Einstein’s general theory of relativity^{37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56} as well as other alternative theories of gravitation, among these the $f(R)$ gravity^57,58,59 the $f(R,G)$ gravity,^60,61 the $f(R,T)$ theory,^62,63,64,65 the $f(T,\tau )$⁶⁶ theory and the $f(Q)$ theory,⁶⁷ in which $R$ is the curvature scalar, $G$ denotes the Gauss–Bonnet invariant, $T$ represents the trace of the energy–momentum tensor, $\tau$ is the torsion scalar and $Q$ is the so-called superpotential in modified symmetric teleparallel gravitation.

In the construction of exact solutions, some techniques or mechanisms have been developed. In the case of a perfect fluid, one of these mechanisms is based on assuming a relation between the metric functions and their derivatives⁶⁸; another technique originates from a specific form of one of the metric potentials, regularly the choice has been $g_{tt}=-{(1+a{r}^2)}^n$¹ although recently there have been explorations on a form of $g_{tt}$ as the quotient of two functions^2,3,4,5,6; also there have been theorems shown that support the structure of the differential equation that originates from the isotropy condition $P_t=P_r$ (radial pressure equal to tangential pressure) and these allow to construct new solutions with perfect fluid starting from a previously known condition^69,70,71; although there is also the possibility of imposing a state equation $P=P(\rho )$, constructing solutions in this case has only been developed by means of numerical methods. In the case of solutions with anisotropic fluid (with a perfect charged fluid or with an anisotropic charged fluid), the presence of an additional function (or two in the anisotropic charged case), with the same number of differential equations as in the perfect fluid case, but with more function to determine, has made it easier to obtain a greater number of solutions than the case with perfect fluid. All the solutions with perfect fluid, that are physically acceptable, can be used as seeds to generate anisotropic solutions, the procedure is to chose in an appropriate manner, anisotropy functions^{20,21,22,23,24,25,26} (or the intensity of the electric field).^{72,73,74,75,76,77,78,79} Inside of this case we can find the so-called gravitational decoupling or Minimal Geometric Deformation (MGD) procedure, which employs a decomposing mechanism from the Einstein equations in which the sources of anisotropy create a new equations system and are associated to the deformation.^{80,81,82,83,84} In this case, it is possible to obtain solutions with a state equation, regularly associated with the radial pressure $P_r=P_r(\rho )$ in this case, one of the metric functions is chosen a priori and through the state equation the other metric function is determined and once the geometry is known the anisotropy function is determined.^{27,28,29,30,31} Another technique starts from a geometric condition, called Karmarkar condition,⁸⁵ which guarantees that the solution in the four dimension spacetime can be embedded in a flat five dimension space, the Karmarkar equation related to the derivatives of the metric functions, and for adequate choices from one of the metric factors this equation can be integrated, which allows to obtain a solution.^41,86,87 Another mechanism, for the charged case, is to assume a metric function and a state equation starting from those that we can determine the intensity of the electric field and the other metric function. Recently, the mechanism of Embedding and Gravitational Decoupling has been conjoined^88,89,90,91 and from the most recent techniques we can find the usage of the complexity factor concept as a supplementary condition to construct anisotropic solutions, which guarantees a relation that determines the anisotropy factor and facilitates finding new solutions.^{92,93,94,95,96,97} One of the mechanisms less utilized consists in giving the explicit form of the metric functions, in a way that they satisfy regularity conditions and absence of event horizons, so that we can determine the hydrostatic, electric field and anisotropy functions from these,^98,99,100 this same technique has been employed for obtaining solutions with sources that consider the existence of quintessence matter in the interior of the star.⁴⁹ The diversity of investigations on anisotropic stellar solutions that continues to be approached both in the frame of the general relativity theory where recently the Complexity Factor Formalism has been employed to obtain solutions,¹⁰¹ and some cases have been adopting the extended Chaplygin gas equation-of-state and master polytrope for anisotropic matter¹⁰² as well as solutions to represent compact objects through the Karmarkar condition with dark energy sources.¹⁰³ Besides considering anisotropic models in alternative theories as is the squared gravity,¹⁰⁴ f(Q)-gravity theory,¹⁰⁵ f(R) gravity¹⁰⁶ and $f(R,G)$ gravity.¹⁰⁷

In the current report, we will focus on presenting a generalization from a solution with perfect fluid⁹ obtained in the frame of Einstein’s general theory of relativity. To obtain the solution, we give a particular form of the anisotropy factor $\mathrm{\Delta }=\frac{2}{r}(P_t-P_r)$ consistent with the requirement that it is only nullified in the center of the star and that it allows to obtain a solution which satisfies the requirements necessary for it to be physically acceptable. This paper is arranged as follows. In Sec. 2, we formulate the equations system for an anisotropic fluid in a static and spherically symmetrical spacetime and we obtain the solution. Section 3 focuses in the determination of the system constants in terms of the physical quantities from the union conditions between the interior solution and the exterior solution given by Schwarzschild’s geometry. Section 4 centers in the graphic description of the solution to show that it is physically acceptable and stable. Section 5 is dedicated to the discussion and conclusions of the new solution as well as describing future works.

2. The System

In the interior description of compact objects, assuming that the fluid is anisotropic in the pressures, Einstein’s equations, $G_{\alpha \beta }=k{T}_{\alpha \beta }$, $k=8\pi ∕G{c}^4$, have as source of the energy–momentum tensor :

where ${\chi }^{\mu }$ and $u^{\mu }$ denote the unit vector in the radial direction and the four-velocity, respectively, $P_r$, $P_t$ and $\rho$ represent the radial pressure, tangential pressure and density of the matter. Meanwhile, $g_{\alpha \beta }$ denotes the components of the metric. In our case, the geometry is static and spherically symmetric and an adequate form for the line element is

For this form of the metric and with the energy–momentum tensor of a fluid with anisotropic pressures, Einstein’s equations and the Bianchi identity (or the equation for the conservation of the energy–momentum tensor) generate the system of differential equations :

where $\prime$ represents the derivative in respect to the radial coordinate. From this set of equations, only three are independent. In particular, the conservation equation (6) that can be obtained from the conservation equation ${\nabla }_{\mu }{T}_{\nu }^{\mu }=0$, can also be obtained by deriving Eq. (5) and replacing the derivative of $B$ given by Eq. (3) and the second derivative of $y$ given by (5). The form of (6) which generalizes the Tolman–Oppenheimer–Volkoff (TOV) equation^68,108 is useful since its interpretation gives us a relation for the forces present in the interior and represents the hydrostatic equilibrium.

3. The Solution

The manner in which we will determine the solution with fluid that presents anisotropic pressures is starting from the form of a metric potential $g_{tt}=-y{(r)}^2$, with

which was previously employed in the construction of an interior solution with perfect fluid and applied for the case of the star LMC X-4.⁹ It is convenient to highlight that not just any choice of a metric potential $g_{tt}=-y{(r)}^2$ generates an analytical solution that is physically acceptable, more than the 80% of the solutions presented in different investigation reports are not physically acceptable,¹ as such the construction of exact solutions is still a challenge and these are the ones that, even for idealized cases, these solutions have guided some of the general properties of the compact objects. Such is the case of the bound for the rate between the mass and the radius of the stars $u=GM∕c^{2}R<4∕9$, better known as the Buchdahl limit,¹⁹ characteristic that originated from the Schwarzschild interior solution, deducted for a perfect incompressible fluid. Some of the conditions that must be imposed on the metric potentials $g_{tt}=-y{(r)}^2$ or $g_{rr}=1∕B(r)$ arise from the analysis of the regularity of the curvature in its interior through the Kretschmann scalar²²

As such for $r\in [0,a]$ with $a>0$ it must be satisfied that

in addition to this the absence of event horizon must be imposed, that is to say, $g_{tt}=-y{(r)}^2<0$$\forall r\in [0,R]$, where $R$ is the radius of the star. The choice of the function (7) satisfies the conditions given in Eq. (9), meanwhile that the requirement (10) on the function $B$ will be analysed further ahead. From Eqs. (4) and (5), we have which allows to notice that for the requirement to be met (10) it is necessary to make an adequate choice of the anisotropy factor $\mathrm{\Delta }(r)=P_t(r)-P_r(r)$. To show the relevance that the choice of the anisotropy factor has, for simplicity sake, let’s consider the potential of the Tolman IV solution $g_{tt}=-(1+\nu r^2)$, By integrating Eq. (11), we obtain

The presence of $\delta$ generates the function $lnr$, which is singular in $r=0$. If we take $\delta =0$, the function $B(r)$ is regular in $r=0$. The presence of the term $\sigma r$ implies that the derivative of the density $\rho {(0)}^{\prime }=-8\sigma ∕(k{c}^2)$, as such the density would not have its maximum value in the center of the star, which is contrary to the expectations. As such, to begin with the anisotropy factor in the vicinity of $r=0$ must be of the form $\mathrm{\Delta }\approx s{r}^{2}F(r)$, with $F(0)\ne 0$; satisfying that the radial and tangential pressures have the same value in the center of the star, although this does not guarantee that if this behavior is satisfied, that the resulting solution will be physically acceptable. For this work, we choose

which guarantees that $P_t(0)=P_r(0)$, satisfies the condition of being a monotonic increasing function and facilitates the integration of the system of equations. Also, we will ask that $N\ne 0$, for $N=0$ we will recover the case of a perfect fluid, meanwhile $N>0$ guarantees us a model that could be physically acceptable, since in the contrary case it could occur that the tangential pressure $P_t(R)$ on the surface is negative. Taking these two functions (7) and (13) and subtracting (4) and (5), we obtain the differential equation

$$\begin{array}{l}{B}^{\prime }-\frac{4[\sqrt{2}+3+4(2\sqrt{2}-1)a{r}^2+7a^2{r}^4]B}{(\sqrt{2}+2a{r}^2)(3\sqrt{2}+2+7a{r}^2)r}+\frac{2(1+a{r}^2)(3\sqrt{2}+2+7a{r}^2)}{(\sqrt{2}+2a{r}^2)(\sqrt{2}+3+7a{r}^2)r}\\ -\frac{8(3\sqrt{2}+2+7a{r}^2)N{a}^2{r}^3}{7{(\sqrt{2}+2a{r}^2)}^3(31\sqrt{2}+44)}=0.\end{array}$$(14)

After the integration of the equation, the results are

$$\begin{array}{ccc}B(x)\hfill & =\hfill & 1-\frac{(-22+17\sqrt{2})(732\sqrt{2}+1832+7(555\sqrt{2}+251)x+4606x^2)x}{2303{(\sqrt{2}+2x)}^3}\hfill \\ \hfill & \hfill & -\frac{(22\sqrt{2}-31){(3\sqrt{2}+2+7x)}^{3}x}{343{(1+\sqrt{2}x)}^3}\left[C+\frac{N}{3\sqrt{2}+2+7x}+ln\frac{\sqrt{2}+3+7x}{3\sqrt{2}+2+7x}\right],\hfill \end{array}$$(15)

where, for the reduction of typographic space, we define $x=a{r}^2$. Once the metric functions are determined, and given the anisotropy function, we obtain the form of the hydrostatic functions expressed as

$$\begin{array}{ccc}\rho (x)\hfill & =\hfill & \frac{3[\sqrt{8}+6+(15\sqrt{2}-4)x+14x^2][1-B]a}{k{c}^2(\sqrt{2}+2x)(3\sqrt{2}+2+7x)x}-\frac{8(\sqrt{18}+2+7x)Nax}{7k{c}^2(31\sqrt{2}+44){(\sqrt{2}+2x)}^3}\hfill \\ \hfill & \hfill & -\frac{14(12\sqrt{2}-13+7x)xa}{(\sqrt{2}+2x)(\sqrt{2}+3+7x)(3\sqrt{2}+2+7x)k{c}^2},\hfill \end{array}$$(16)

3.1. Physical conditions

Obtaining a solution for the system of Eqs. (3)–(5) does not guarantee that it is physically acceptable. There are additional conditions that must be verified, some of these correspond to that of the union between the interior solution and the exterior solution (given by Schwarzschild’s solution), help us determine the constants or parameters that appear in the solution or to express the physical quantities in terms of these, in our case the parameters are $(a,C,N)$. In general terms, the requirements are the following^1,109:

• The solution must be regular, which implies that the metric functions satisfy that $y^2(r)>0$ and $B(r)>0$ in the range $0\le r\le R$ (where $R$ is the radius of the star).

• The density and radial pressure must be positive, finite and monotonically decreasing, that is to say $\rho >0$, ${\rho }^{\prime }<0$, $P_r>0$ and $P_r^{\prime }<0$, with their maximum value on the center of the star.

  The radial pressure must be zero on the border, i.e. $P_r(R)=0$; meanwhile the tangential pressure must be positive $P_t>0$ and match the value of the radial pressure in the center $P_t(0)=P_r(0)$.

• The causality condition must be met, this is, the speed of sound must be positive and lower or equal to the speed of light in a vacuum, $0\le \frac{dP}{d\rho }\le 1$.

• The solution must be stable, two applicable criteria in the anisotropic case are: (a) the stability under radial adiabatic oscillations, which is met if^{110,111,112,113} :

  $$\gamma =\frac{P+c^2\rho }{P_r}\frac{d{P}_r}{d\rho }>{\gamma }_{\mathrm{\text{crit}}}=\frac{4}{3}+\frac{19}{21}u.$$
  (b) Stability in regards to the craking concept, which indicates that, if: $v_t^2-v_r^2\le 0$ the solution is potentially stable^16,114

• The energy conditions must be satisfied: null energy condition (NEC), weak energy condition (WEC), strong energy condition (SEC), dominant energy condition (DEC):

  $$\begin{array}{c}\mathrm{\text{NEC}}:c^2\rho +P_r\ge 0,c^2\rho +P_t\ge 0,\hfill \\ \mathrm{\text{WEC}}:c^2\rho +P_r\ge 0,c^2\rho +P_t\ge 0,c^2\rho \ge 0,\hfill \\ \mathrm{\text{SEC}}:c^2\rho +P_r\ge 0,c^2\rho +P_t\ge 0,c^2\rho \ge 0,c^2\rho +P_r+2P_t\ge 0,\hfill \\ \mathrm{\text{DEC}}:c^2\rho \pm P_r\ge 0,c^2\rho \pm P_t\ge 0,c^2\rho \ge 0.\hfill \end{array}$$

• The union between the interior and exterior geometries must be continual, with the exterior geometry given by the Schwarzschild metric

  $$d{s}^2=\left(1-\frac{2GM}{c^{2}r}\right)d{t}^2-{\left(1-\frac{2GM}{c^{2}r}\right)}^{-1}d{r}^2+r^{2}d{\mathrm{\Omega }}^2+r^2(d{\theta }^2+{sin}^2\theta d{\varphi }^2),r\ge R.$$

From this set of requirements, the continuity between the exterior and interior geometries and that the radial pressure is zero on the surface of the star allows us to determine the following relations:

The rest of the restrictions for the solution to be physically acceptable imply inequalities that involve the constants and restrict their values. A detailed analysis of the solution allows us to distinguish that it is the adiabatic index that determines the maximum compactness value $u_{max}=0.23577$ which occurs when $s_{max}=0.90378$, to guarantee the stability of the solution when faced with infinitesimal radial adiabatic perturbation. Meanwhile, the interval for the anisotropy parameter $N$ is determined by the positivity of the tangential speed of sound on the surface of the star.

4. Graphic Analysis

Although from a detailed analysis, it is found that the solution constructed is useful for describing stars with compactness $u\le 0.23577$, in this section, we will focus on the case of the star Her X-1, although similar behaviors occur for other compactness values. The star Her X-1 was detected by the Uhuru satellite in the Hercules constellation, located at a distance of approximately $6\mathrm{\text{kpc}}$ from Earth and reported in 1972,¹¹⁵ with its characteristics analyzed in different investigative reports.^{116,117,118,119,120,121,122} In one of these,¹²¹ by means of observational data its mass was determined $(0.85\pm 0.15)M_{\odot }$ and the prediction of the radius $R=8.1\pm 0.41\mathrm{\text{km}}$¹²² was done. The values that we will consider in the graphic analysis will be those of the shortest radius and highest mass $(7.69\mathrm{\text{km}},1M_{\odot })$ that generate the maximum compactness $u_{max}=0.1919$ and those of the greatest radius and lowest mass $(8.51\mathrm{\text{km}},0.7M_{\odot })$ that imply the minimum compactness $u_{min}=0.1214$. And we define the dimensionless functions $k{c}^2{R}^2\rho (x)$, $k{R}^2{P}_r(x)$, $k{R}^2{P}_t(x)$, $v_r{(x)}^2∕c^2$, $v_r{(x)}^2∕c^2$ and $\gamma (x)$, from the variable $x=r∕R$.

4.1. Causality condition

From the set of required restrictions for the solution to be physically acceptable, two are the ones that determine the range of the anisotropy parameter $N$: the stability in regards to the craking concept, that implies $N\ge 0$, the case $N=0$ corresponds to the one with the perfect fluid; and the positivity of the tangential speed of sound, which determines the maximum value of $N$ independently of the compactness value.

In the left graph of Fig. 1, we represent the tangential speed of sound for the maximum compactness, being positive for $N\in [0,60.247]$, if $0\le N<49.996$ the function is monotonically increasing and if $49.996\le N\le 60.247$ the function is monotonically decreasing. In the case of the minimum compactness, right graph of Fig. 1, $N\in [0,66.481]$, for $0\le N<50.723$ the function is monotonically increasing and for $50.723\le N\le 66.481$ the function is monotonically decreasing.

The radial speed of sound, Fig. 2 presents a similar behavior to that of the tangential speed of sound. In both cases, the anisotropy parameter $N$ influences the monotonically increasing and decreasing behavior. However, for both the greatest and lowest compactness, the value of the radial speed of sound ends up being greater than the tangential speed of sound which is reflected in Fig. 3, implying that the model is stable in regards to the cracking concept.

4.2. The stability of the solution: The adiabatic index and the Harrison–Zeldovich–Novikov criteria

Although it has already been shown that the solution is stable in regards to the cracking concept, it is convenient to verify the stability by means of other criteria. During the breakout works, there was the introduction of the analysis starting from the adiabatic index perturbation versus the radial coordinate.^110,111,112 The study of the stability analysis for the case with anisotropy is centered only on the analysis of the adiabatic index in the radial direction, since this is the only direction that is affected against an eventual gravitational collapse. Recently, a correction has been presented in relation to the critical value that defines the stability, since this critical value depends on the amplitude of the Lagrangian displacement from the equilibrium and the compactness factor $u$, giving as a result that the stability occurs if $\gamma >{\gamma }_{\mathrm{\text{crit}}}=4∕3+19u∕21$.¹¹³ In Fig. 4, it is shown that the solution is also stable in relation to the radial oscillations, in the case of greater compactness ${\gamma }_{max,\mathrm{\text{cri}}}=1.507$ and the adiabatic index of the solution has its minimum value in the center of the star $\gamma (0)>1.6>1.507$; meanwhile for the minimum compactness ${\gamma }_{min,\mathrm{\text{cri}}}=1.443$ and $\gamma (0)>2.5>1.443$, which guarantees the stability.

In the case of static and spherically symmetric stellar models, the Harrison–Zeldovich–Novikov criteria guarantee the stability in relation to radial pulsations, if it is satisfied that the mass $M=M({\rho }_c,P_{rc},R)$ as function of the central density ${\rho }_c$ satisfies that $\frac{\partial M}{\partial {\rho }_c}>0$.^123,124 For the case that is being analysed, we have that

where $S_1=270\sqrt{2}-324$. After obtaining the derivative in relation to the central density with $S_2=6(5-3\sqrt{2})$ and $S_3=810-486\sqrt{2}$, from where we can conclude that the criteria mentioned is satisfied, since this last expression is positive given that ${\rho }_c+3P_{rc}>0$.

4.3. The hydrostatic function and the energy conditions

In Fig. 5, we show the positive, finite and monotonically decreasing behavior of the density, its values for the maximum compactness (graph on the left) are greater than the values for the minimum compactness (graph on the right). For a fixed value of the compactness, as the anisotropy parameter $N$ increases the central density is greater and the density on the surface is lower for higher values of $N$. In Fig. 6, it can be seen that the radial pressure is regular, positive, monotonically decreasing and it is zero on the surface, identified by $x=1$. In this case, the presence of the anisotropy generates that the radial pressure is lower as the values of the parameter $N$ become lower. Also, just like for the density, the values with higher compactness have greater pressure, which is consistent given the presence of a greater density.

Although it is not a requirement that the tangential pressure is a monotonically decreasing function, for the solution constructed if it has this behavior it is positive and nonzero on the surface, except for $N=0$ which corresponds to the case of the perfect fluid in which $P_t(x)=P_r(x)$. Also the tangential pressure is greater than the radial pressure which guarantees that the anisotropy factor $\mathrm{\Delta }\ge 0$ and as a consequence we have the existence of a repulsive force associated to the anisotropy, allowing for the representation of objects that are more compact than in the case of a perfect fluid.¹²⁵ In regards to the analysis of the energy conditions, from the graphs of Figs. 5–7, we have that the positivity of the functions for the density, radial pressure, tangential pressure and we can also observe that the density is greater than the pressures, this allows us to assert that the energy conditions are satisfied.

4.4. The hydrostatic equilibrium and the continuity of the geometry

The hydrostatic equilibrium is described through Eq. (6) which generalizes the TOV equation for the case with anisotropic pressures. The decomposing of it gives us the forces present in the interior of the star, which are the gravitational force $F_g$, the hydrostatic force $F_h$ and the anisotropic force $F_a$ :

In the graphs of Fig. 8, each one of these forces are represented. The attractive gravitational force is counteracted by the forces of hydrostatic and anisotropic repulsion, the value of these last ones increases as the anisotropy coefficient $N$ increases. Also, the forces have greater values for the case of greater compactness.

The metric functions for each one of the compactness values and from the value of the anisotropy parameter $N$ are shown in Fig. 9, in it we observe the continuity of the exterior and interior geometry. The temporal metric component does not change with the value of the parameter $N$, meanwhile, the radial metric function $g_{rr}$ does change.

5. Discussion and Conclusions

In this work, we formulate the proposal of a new solution to Einstein’s equations for the static and spherically symmetrical case assuming an anisotropic fluid. Satisfying different criteria as are causality, energy conditions, and the non-existence of event horizons criteria. It represents a generalization to a previously presented model for the case of a perfect fluid. Given the compactness value obtained, the solution was applied to represent the star Her X1 from observational data. Showing that the solution is physically acceptable, that is to say, the density and pressures are positive and monotonically decreasing functions, the condition of causality is met, as well as the energy conditions, in addition we have that the solution is stable in relation to the criteria of the adiabatic index and the cracking concept. Also, as it is shown in Tables 1 and 2, the orders of magnitude from the density are those expected for stars with these compactness values. In particular, we have that the density and the pressure in the interior of the star are one order of magnitude greater for the value of maximum compactness compared with the one with the minimum compactness.

On the other hand, the solution constructed gives more evidences about the relevance that the metric potential $g_{tt}=-y{(r)}^2$ has in the construction of stellar solutions. One question that arises from this is if it’s possible to obtain physically acceptable solutions taking as a starting point the same function $y(r)$ but choosing a more general form of the anisotropy factor $\mathrm{\Delta }$ or imposing a state equation. These questions could be approached in future works.

Acknowledgments

We thank the support from our institutions for the facilities provided during the realization of this work as well as the CONACYT for the support given. NCM thanks the TECNM–Instituto Tecnólogico del Valle de Morelia and the PRODEP for the support to the “Cuerpo Académico ITVAMO-CA-7” during the realization of this investigation work. We thank the reviewers for the suggestions done to improve this work.