We obtain a new nonsingular exact model for compact stellar objects by using the Einstein field equations. The model is consistent with stellar star with anisotropic quark matter in the absence of electric field. Our treatment considers spacetime geometry which is static and spherically symmetric. Ansatz of a rational form of one of the gravitational potentials is made to generate physically admissible results. The balance of gravitational, hydrostatic, and anisotropic forces within the stellar star is tested by analysing the Tolman-Oppenheimer-Volkoff (TOV) equation. Several stellar objects with masses and radii comparable with observations found in the past are generated. Our model obeys different stability tests and energy conditions. The profiles for the potentials, matter variables, stability, and energy conditions are well behaved.

1. Introduction

The studies on the structure and behaviour of compact relativistic stellar spheres have been possible through investigation of solutions for field equations. Neutral and charged spheres have been investigated under specified spacetime geometries. Charged stellar models under static and spherically symmetric spacetime include the performance by [110]. Uncharged stellar models include recent works performed by [1115]. The studies by Ruderman [16] and Canuto [17] indicate that radial and transverse pressures within the stellar bodies may not be equal. The quantity defining this difference is called the pressure anisotropy. Different phenomena explain the indicators of this imbalance of pressures within the stellar object. Some of these phenomena include existence of condensate states such as pion and kaon condensations as indicated by Kippenhahn and Weigert [18] which exist when the stellar core has solid matter as observed by Sawyer [19]. This can also happen when the stellar matter passes through several phase transitions as indicated by Sokolov [20]. Bowers and Liang [21] asserted that no compact stellar sphere is composed wholly of perfect fluid. Ultrahigh density and gravitational pull in these objects result to enormous pressure anisotropy. Several studies have recognized the significance of pressure anisotropy in stellar models including recent performance by [6, 7, 12, 2225]. Imposition of equation of state for stellar matter in modelling relativistic stellar objects is very significant. Various studies have adopted different equations defining the state of stellar fluid to obtain well-behaved stellar models. We observe that [23, 2630] obtained well-behaved charged stellar models by applying linear equation of state. The treatment by [11, 3135] applied quadratic equation of state to generate stellar models with astrophysical significance. The works by [13, 3639] generated physically significant stellar models with viable results in astrophysics and astronomy using Chaplygin equation of state. The recent relativistic models by [4, 4042] were generated by imposing polytropic equation of state. On the other hand, [43, 44] applied van der Waals equation of state to obtain astrophysically significant stellar models. The use of linear equation of state in modelling self-gravitating stellar objects is evident in various studies. Models consistent with strange stars composed of quark matter have been found with this equation of state. The simplified form of linear equation of state has the form as indicated by Witten [45], where and represent energy density and bag constant, respectively. We are interested to investigate the geometry and physical characteristics of strange stars with quark matter by introducing linear form of equation of state along with a new form of one of the gravitational potentials missing in the previous studies to obtain a well-behaved anisotropic strange star model. Additionally, we undertake several physical tests which are missing in most of the models generated in the past. This article is organised in seven sections. The basic stellar equations are presented in Section 2 followed by the transformations in Section 3. The model and its physical properties are introduced in Sections 4 and 5, respectively. Discussion of findings is presented in Section 6, and the closing remarks are provided in Section 7.

2. Basic Stellar Equations

We generate a new exact solution describing the interior of the strange star in general relativity. The geometry of spacetime is considered to be static and spherically symmetric. The line element of the star interior is considered to have the form with and being the gravitational functions. The exterior of the strange star is considered to conform to Schwarzschild exterior spacetime given by the line element where is the mass of the star. The energy momentum tensor for the stellar star in the absence of electric field is given by where , , and are the energy density, radial pressure, and tangential pressure, respectively. These matter variables are determined relative to a comoving unit timelike fluid four-velocity . The value representing the coupling constant and the speed of light is considered to be united.

The Einstein field equations which govern neutral stellar stars can be represented as

The primes () in the field equations represent the derivatives of the gravitational functions with respect to the radial distance . The function representing the mass of the neutral stellar star is given by

The stellar star is considered to compose quark matter which admit the linear equation of state of the form where is the bag constant.

3. Transformations

To ease the integrability of the field equations in systems (4a), (4b), and (4c), we embrace Durgapal and Bannerji’s [46] transformations given by where and are arbitrary real constants. When we subject system (2) to these transformations, we obtain

and the line element becomes

Applying the transformations in Equation (7) to Equation (5), the mass function becomes

Incorporating the equation of state (6) in systems (4a), (4b), and (4c), the field equations becomes

We notice that (20) can also be written as where and is the constant of integration. The anisotropic factor is defined by , and the force exerted due to anisotropy is given by . The study by [47] indicated that when , the anisotropic force acts outward and it is repulsive in nature and when , the force acts inward with attractive behaviour. When , it implies that and there is no anisotropic force and the model becomes isotropic.

4. The Model

The matter variables involved in our model include , and . Since the systems (11a), (11b), (11c), (11d), and (11e) have lesser number of equations as compared to unknown variables, we specify one of the variables in the system to simplify the integration process. We followed the method recently applied by Mathias and Sunzu [30] with a different choice of metric function. The choice is carefully made to yield physically admissible model. We specify the gravitational potential in a rational form given by where and are real constants such that and . This function is continuous and regular throughout the interior of the stellar star. This choice takes reciprocal form of the potential in [48, 49] to obtain a well-behaved strange star model. The choice of potential made by [49] had a rational form of with quadratic expression in the numerator and linear expression in the denominator. The choice in the current paper has linear expression in the numerator and quadratic expression in the denominator. Our choice of potential is a new form, and it guarantees a new strange star model that obeys various physical tests on stability, state of hydrostatic equilibrium, and energy conditions. The choice made in this study enlightens more the investigations of strange stars.

When we apply Equation (14) into Equation (12), we obtain the metric function in the form where is a constant of integration and the function is given by where is given by

The exact model obtained by solving systems (4a), (4b), and (4c) can be written in the form

5. Physical Properties

We notice that the solution functions presented in systems (18a), (18b), (18c), (18d), and (18e) which are obtained by solving the system of field equations in systems (11a), (11b), (11c), (11d), and (11e) are in elementary form. The mass function in Equation (10) then becomes and the line element has the form

5.1. Compactness and Redshift

The compactness factor for the stellar stars proposed by [50] is given by where is the total mass of the stellar star and is the stellar radius. The critical value of compactification factor for isotropic fluid spheres , however for anisotropic spheres , may exceed this limit [51]. By applying Equation (19) to Equation (21), we obtain

The gravitational redshift of a stellar star is given by where is the quantity defining compactness of the stellar star. In the present model, is found by substituting (22) into (23), and the resulting function has the form

5.2. The TOV Equation

The Tolman-Oppenheimer-Volkoff (TOV) equation is an important tool to examine the balance of forces acting within the stellar object. When the forces are well balanced, then the stellar object is said to be in the state of hydrostatic equilibrium. The balance of forces implies that the energy within the stellar star is conserved. The TOV equation is given by

The balancing condition for a neutral anisotropic stellar star is that all forces (gravitational force , hydrostatic force , and anisotropic force ) act within the stellar fluid sum to zero. That is, where

The expressions for the forces in (27a), (27b), and (27c) are in the form

5.3. Stability Conditions

The stability of a stellar model can be tested under different stability criteria. The nature of radial and tangential variations of equation of state parameter prompts information on model stability. The equation of state parameter in radial and transverse orientations is given by and , respectively. In the present model,

The stability of the stellar model can also be examined based on the value of adiabatic index given by

The stability condition for Newtonian spheres can be examined based on the value of adiabatic index satisfying the inequality . However, the work by [52] indicates that the stability of stellar sphere with anisotropic fluid satisfies the inequality where , , and represent initial radial pressure, tangential pressure, and energy density, respectively. The works by [5355] show that the stability of anisotropic stellar spheres satisfies the inequality , where is the critical adiabatic index defined by where is the compactification factor.

In the present model, the expression for is given by

The proposition by [56] provides another stability criterion for stellar spheres termed as cracking of a star. For stable stellar configurations,

This condition is essential to prevent stellar star from overturning. The variables and are given by and , respectively. In the current model,

5.4. Energy Conditions

The physically admissible models for stellar objects with anisotropic matter distribution should satisfy different energy conditions. The works by [5759] indicate different kinds of energy conditions including null energy condition (), weak energy condition (), dominant energy condition (), and strong energy conditions (). These conditions satisfy the following inequalities:

We examine the viability of current stellar model graphically by testing these conditions.

5.5. Boundary Conditions

We match the interior solution to exterior Schwarzschild solution at the boundary of the stellar star. The conditions for continuity of metric coefficients from the line elements (1) and (2) at the boundary are given by

The radial pressure vanishes at . This implies that

Substituting the expressions for , , , and from Equations (18a), (18b), and (19), respectively, into Equations (37), (38), and (39), we obtain where

We observe that systems (40a), (40b), and (40c) have seven parameters (i.e., , , , , , , and ) in the three equations. The system has sufficient free parameters for matching conditions.

6. Discussion

In this section, we describe the structure and behaviour of the model generated in Section 4, relative to acceptable behaviour of stellar models in general relativity. Mathematical computations have been performed with Mathematica software while plots have been generated by using Python programming language. All the plots are generated against the radial distance , and the values for the parameters used are , , , , and .

We observe that the profiles for the gravitational potentials presented in Figure 1 show finite, regular, and continuous behaviour throughout the stellar interior. The plots coincide at the surface of the stellar star. We also observe that these potentials are positive increasing functions at each interior point of the stellar star. Figures 2 and 3 show that the energy density , radial pressure , and tangential pressure are monotonically decreasing functions as radius increases with negative gradient, i.e., , , and . Figure 3 also indicates that the radial and tangential pressures have the same maximum value at the centre. Similar decreasing profiles are evident in recent studies by [5, 14, 60, 61]. The anisotropic factor in Figure 4 has zero value at the centre and then increases to the maximum value away from the centre and then slightly decreases as it approaches the stellar surface. This feature can also be observed in the works by [28, 29, 34]. We notice that at each point within the stellar interior. This implies that the force due to anisotropy acts outward and it is repulsive in nature based on the observation by [47]. In Figure 5, we clearly observe that the mass function increases monotonically with radial distance to the surface. This is a necessary condition for well-behaved stellar models. In Figure 6, the compactification factor () of the stellar star is in acceptable range for anisotropic spheres. This profile also shows that the compactness factor increases with the increase in radial distance. The upper bound for the surface redshift indicated in Figure 7 for the current model is found to be . This value conforms to [50] limit, for realistic compact stars. The variations of forces in TOV equation are indicated in Figure 8. We observe that the gravitational , hydrostatic , and anisotropic forces within the star are balanced such that . The gravitational force countervails the sum of hydrostatic and anisotropic forces. This condition confirms that our model satisfies the equilibrium test. Figure 9 clearly indicates that which is less than unity showing that the sound speed within the stellar star is less than the speed of light. This condition is necessary for stable stellar configurations. Figure 10 shows that the equation of state parameter for the current model is well behaved. We notice that decreases monotonically with radial distance. This feature is also evident in the performance by [3, 62]. This is a necessary stability condition as per [63] observation. We also observe that show slightly increasing behaviour to a local maximum and then slightly decreases shortly as it approaches the stellar surface but remaining positive. Figure 11 clearly shows that the adiabatic index . This indicates that our model is stable [64]. In Figure 12, we can easily note that . This agrees the Herrera [56] cracking condition for stability of the model. This model also admits several energy conditions. In Figure 13, we note that the inequalities , , , , , and are all satisfied. This indicates that our model satisfies the null, weak, dominant, and strong energy conditions.

We applied the mass function (19) to obtain stellar masses compatible with the observations. We have obtained the radii and masses compatible with the compact stellar stars such as SAXJ1808.4 with  km and , 4 U1538-52 with  km and , LMCX-4 with  km and , and HerX-1 with  km and as indicated in [6567], respectively. Values of the parameters for these stellar masses and radii are indicated in Table 1.

7. Conclusion

In the current paper, we generated a new nonsingular exact solution of the Einstein field equations consistent with relativistic static sphere with anisotropic matter distribution. New form of one of the gravitational potentials along with a linear equation of state has been applied to obtain a strange star model consistent with stellar spheres with quark matter. All the gravitational potentials and matter variables are well behaved and are free from central singularities. The state of hydrostatic equilibrium for our model has been tested by analysing Tolman-Oppenheimer-Volkoff (TOV) equation. The profiles for gravitational potentials, matter variables, the speed of sound, compactness, redshift, hydrostatic equilibrium, stability, and energy conditions are acceptable on physical grounds. Stellar masses and radii comparable with the previous experimental observation have been generated with our model. These are compatible with relativistic compact stellar objects like SAXJ1808.4, 4U1538-52, LMCX-4, and HerX-1 observed by [6567], respectively.

Data Availability

We have no associated data in this paper.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.


We would like to acknowledge the University of Dodoma as our employer for providing conducive environment for research and by ensuring the availability of facilities and resources.