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H. Nazar, G. Abbas, "Model of Charged Anisotropic Strange Stars in Minimally Coupled Gravity", Advances in Astronomy, vol. 2021, Article ID 6698208, 25 pages, 2021. https://doi.org/10.1155/2021/6698208
Model of Charged Anisotropic Strange Stars in Minimally Coupled Gravity
In the present article, we have investigated a new family of nonsingular solutions of static relativistic compact sphere which incorporates the characteristics of anisotropic fluid and electromagnetic field in the context of minimally coupled theory of gravity. The strange matter bag model equation of state (EoS) has been considered along with the usual forms of the Karori–Barua metric potentials. For this purpose, we derived the Einstein–Maxwell field equations in the assistance of strange matter EoS and type ansatz by employing the two viable and cosmologically well-consistent models of and . Thereafter, we have checked the physical acceptability of the proposed results such as pressure, energy density, energy conditions, equation, stability conditions, mass function, compactness, and surface redshift by using graphical representation. Moreover, we have investigated that the energy density and radial pressure are nonsingular at the core or free from central singularity and always regular at every interior point of the compact sphere. The numerical values of such parameters along with the surface density, charge to radius ratio, and bag constant are computed for three well-known compact stars such as (, , and and are presented in Tables 1–6. Conclusively, we have noticed that our presented charged compact stellar object in the background of two well-known models obeys all the necessary conditions for the stable equilibrium position and which is also perfectly fit to compose the strange quark star object.
Several few decades ago, an intellectual thinking came in mind of research collaborators why our Universe is much rapidly growing towards large expansion. They tried to find out reasons behind this accelerating growth of Universe. Then, in 1990s, it was found that our Universe is expanding due to two hidden key factors of the nature, in which one of them is the theorized form of the matter known as dark matter and the other is unknown dark energy. Later on, the right picture of these factors was revealed by the international research collaboration team in 1998 by observing supernovae type-Ia [1–9], which was later proved by surveying of the cosmic microwave background radiation [10, 11], huge scale structure [12–17], and Wilkinson Microwave Anisotropy Probe . The above phenomenological factors of accelerating cosmic Universe can be well interpreted in higher order modified gravity theories rather than the concept of general relativity (GR) theory because these gravity theories could easily identify the right cosmological scenario of this mysterious Universe at the higher order curvature scale. For their assessment, several theoretical researchers and astrophysicists have contributed a quite exceptional work in different mathematical advances. To obtain these required mathematical setups, we can simply modify the Einstein–Hilbert action of corresponding to alternative theories of gravity such as [19–22], , [24–26], and  gravity theories.
Exploring the physical stable configuration of relativistic stellar bodies, e.g., black hole, strange quark stars, pulsars, neutron stars, and white dwarfs in modified gravity theories, would be a good task to grasp this issue at the theoretical and the astrophysical gauge for the researchers. In the massive stellar bodies, the analysis of huge gravitational attraction clearly defines the basic differences between GR and its alterations. The formation of huge dense star in an alternative theory of gravity has included several fruitful characteristics to star models [28–36]. Psaltis  investigated that the huge gravitational field can be reviewed as modified theories of gravity. Briscese et al.  have proposed the stable position of objects in gravity as a test of the theory’s consistency. They reported that some paradigms of gravity cannot maintain the stable position of star and are treated unreliable. From the analysis of Tsujikawa et al. , the unreliability regarding the stable position of these stellar bodies can be eluded due to scalar tensor theory. Despite of these outcomes, various concrete solutions have been made in the modeling of neutron stars by employing the theory of gravity [40–48].
In the current investigation, we tried to find out new verity of nonsingular solutions of charged anisotropic relativistic compact stellar models in minimally coupled gravity that were priory propounded by Alcock et al.  and Haensel et al. . In regard to the existence of these objects, many attempts have been performed with different mechanisms during the last decade. Very recently, Shamir and Fayyaz  studied different properties of anisotropic compact celestial objects for Tolman–Kuchowicz spacetime by considering two viable classes of models. The same author and his collaborators  determined the physical aspects of compact stellar bodies for line-element in modified context. Yousaf et al.  investigated the impacts of different viable paradigms on the existence of anisotropic stellar compact objects and found that these paradigms are well behaved at the astrophysical and the theoretical scales. The physical features of anisotropic spherically symmetric strange stars with quintessence field were discussed by Abbas et al.  for the specific pattern of the model. Zubair and Abbas  found various realistic solutions of anisotropic interior compact celestial systems in the framework of gravity. Sussman and Jaime  explored the viable characteristics of the model for nonstatic LTB spacetime in the presence of traceless anisotropic pressure tensor. Shabani and Ziaie  revealed the influence of the particular model on the stability of an emerging Einstein Universe by employing the dynamical and numerical approaches. Garattini and Mandanici  proposed some equilibrium stable formations of different compact stellar systems and analyzed that additional higher order curvature terms emerging from rainbows gravity are likely to support different models of stellar systems. Several results of the anisotropic cosmic evolution in the background of different particular choices of models were suggested by Sahoo et al. [59, 60]. From the literature survey, several phenomenal findings regarding these compact objects have been investigated in different alternative gravity theories with distinct approaches [61–69].
In spite of these consequences, several realistic features of anisotropic compact stellar systems have been examined in during the couple of few decades. Some notable exact solutions for the Einstein field equations with anisotropic source distributions in different backgrounds have been determined by Bayin , Cosenza et al. , and Harko and Mak [72, 73]. Mak and Harko  found a class of exact solutions of gravitational field equations for the physical existence of a compact object made of a strange quark matter. Kalam et al. [75, 76] proposed different analytical solutions of anisotropic compact stellar objects with spacetime. Hossein et al.  analyzed the physical aspects of the anisotropic celestial system in the presence of a cosmological constant. The analytical results of Einstein field equations describing the static anisotropic matter distributions of compact objects were examined by Bhar et al. [78, 79]. Very recently, the new family of exact solutions of the embedding class 1 method for relativistic anisotropic stellar bodies was explored by Singh et al. . The same author and his collaborators  established new exact solutions of Tolman for anisotropic spherically symmetric compact star candidates in the presence of exotic matter nonlinear . Various physical aspects of relativistic anisotropic compact celestial objects with the dark matter density profile were evaluated by Sarkar et al. . Moreover, the physical realistic solutions of the profile for anisotropic relativistic compact stellar bodies were determined by Errehymy and Daoud .
The motivation for introducing an electromagnetic source in a matter distribution can be well justified in light of some theoretical manifests based on new techniques allowing for the appearance of a greater charge in relativistic compact celestial bodies. Particularly, there is a chance of an immense electromagnetic field in compact stellar bodies with a strange quark matter. A concerning problem emerges from the fact that, if star can hold a nonzero amount of charge, the contraction of such star can lead to a Reissner–Nordström black hole. A quite novel study presented by Rosseland  affirmed that the independent electrons of the extremely ionized gas that creates the star can be expelled due to its immense thermal velocities. In fact, a stellar body can hold a large value of electric charge maintaining stability [85, 86]. Rahaman et al.  studied several realistic properties of charged anisotropic compact objects with strange matter . The embedding class 1 solutions of spherically symmetric compact stellar objects with charge distribution were investigated by Maurya et al. . Thirukkanesh and Maharaj  concluded that the stability of a relativistic stellar object boost up with the infusion of an electric charge. Esculpi and Aloma  reported that the charge and anisotropy increases the stability of the object under certain bounds. Eiroa and Simeone  found that the electromagnetic field enhances the region of consistency for both shells and bubbles around a black hole.
The strange matter based on the bag model has a crucial role in the modeling of very massive dense strange quark stars. It has been foreseen that a neutron star is an end state of the gravitationally collapsed object which after consuming all its thermonuclear fuel comes into stabilized position by degenerating pressure. After a short period of the detection of the particle “neutron” by Chadwick, the occurrence of neutron stars was forecasted. Subsequently, the notion got observational assistance with the detection of pulsars . With the progress in our thinking of particle interaction at greater energy, theoretical composition of neutron stars has quite enhanced during the last few decades . The speculation that the quark matter can be the most probable state of hadrons [94, 95] has focused to the debates of a complete new class of celestial objects composed of deconfined , , and quarks, generally says that strange quark stars.
According to above certain particulars, we have attempted to discuss the interior charged sphere distribution in the assistance of strange matter within the context. The realistic features of the physical parameters for the obtained solutions have been comprehensively studied, and its numerical estimation is also obtained. The arrangement of this manuscript as follows: in Section 2, the Einstein–Maxwell field equations for static anisotropic charged sphere case are formulated, and its corresponding solutions with central and surface values are also obtained. Section 3 deals with the unknown arbitrary constants that have been derived from the smooth matching conditions of the interior metric and exterior Reissner–Nordström solution. The estimated values of the physical parameters for our strange star candidates and their physical significance including energy conditions, equation, anisotropy, and stability conditions are thoroughly discussed in Sections 4 and 5, respectively. In subsequent section, we have examined various physical profiles such as effective gravitational mass, compactification factor, and surface redshift for our presented star candidate. Finally, Section 7 comprises the final remarks for our present strange star candidate.
2. Charged Interior Anisotropic Matter Configuration
We considered the static spherically symmetric strange star configuration which is bounded by interiorly charged anisotropic source distribution within the framework of so-called gravity theory. The general formulation of Einstein–Hilbert (EH) action in GR is expressed by
The above action in metric formalism with minimally coupled Maxwell source has the following form:where determines the role of Maxwell invariant, corresponds to the generic function of Ricci scalar , and represents the coupling constant. Therefore, the set of field equations can be acquired by the variation of equation (2) w.r.t metric tensor:
Here, is an electromagnetic tensor, indicates an energy-momentum tensor, is the derivative function of Ricci scalar , corresponds to the D’Alembert operator, and stands for the covariant derivative. The above equation can be rearranged in the form of Einstein tensor which yield aswhere with
The role of as an effective energy-momentum tensor arrives from the theory parameter, which literally describes that the nature of dark energy correlates with early and late time accelerating expansions of the cosmic Universe in the gravitational system. In other words, this factor expresses the fourth order differential geometry to identify the huge curvature at the astrophysical and cosmological backgrounds. Now, we presume the interior static strange star distribution defined by the  line-element and is given bywhere and are the two metric variables that depend on function and positively assumed. It is equivalent to the type solutions where and . Thus, , , and are the arbitrary constants and would be investigated later through matching conditions. The Einstein–Maxwell field equations can be arranged with the help of equations (3) and (6) by employing and read bywhere signifies the total charge inside the matter spheroid of radius . The following specific bag model [49, 94, 95, 97, 98] determines the strange matter distribution in the interior celestial object and is read as
Here, is defined as the bag constant. The difference between the bag constant and mass density of the perturbed and nonperturbed vacuum was investigated by Mak and Harko , and the units of bag constant is derived by Chodos et al. .
2.1. Solution of Modified Field Equations
There are five independent equations together with above given in five unknowns, namely, energy density , pressures ( and ), electric field , and proper charge density . The solution sets of these unknown parameters are determined from equations (7)–(11); by implementing the type ansatz and , it becomeswhile the charge density is evaluated aswhere
The total charge within a compact sphere of radius becomes
2.2. Constraints on Physical Parameters
To examine the regular value (from central singularity free) at the core of the strange stars models for the physical acceptability of energy density and pressure . These physical parameters are analyzed in the form of analytical expressions and are given by
The above solutions ( and ) are free of central singularity and finite (regular) at the core of the different strange star candidates. There is also nonnegative behavior and maximum position at the center of the compact sphere. The numerical values of the said parameters are provided in Tables 1–6 for three different strange stars ( , , and ) with two cosmologically prominent viable models. Moreover, at the core of compact sphere, the electric field must disappear for the requirement of regularity condition. The regularity of the electric field turned out to be
From the above result, we obtain the value of the bag constant .
We used the value of the bag constant in equation (16) and then obtained the parameter in terms of the central density and modified terms which is expressed by
3. Matching Conditions
We explore the three unknown arbitrary constants in terms of , , and by imposing the smooth matching conditions between the interior spacetime and exterior Reissner–Nordström solution at the bounding three-surface (where ) of the fluid sphere. Both interior and exterior metrics at the bounding hypersurface must be continuous. Hence, the exterior line-element of the spherical star was well depicted by the Reissner–Nordström solution [99, 100] and is expressed bywhere , , and are the total charge surrounded within a bounding three-surface, the total mass of the gravitational system, and the radius at the bounding hypersurface (i.e., where exterior and interior spacetimes are smoothly matched). The continuity of the gravitational potentials , , and across the junction interface between the exterior and interior geometries of the sphere provides the following expressions:
Consequently, we found the solutions of these unknown parameters , , and in the form of total charge , mass , and radius for our proposed different strange stars which are quite suitable to describe the graphical evolution. The approximated numerical values of these unknown parameters for three different strange star candidates , , and with two distinct viable models are given in Tables 1–6. Here, it would be very necessary to examine the analytical expression of the charge of the strange star. So, we employed the above equations on the system constraint and is given by
Alternatively, we obtained the numerically solved estimated value of the above result to the ratio of charge and radius for the given compactness of star. In spite of this condition, there is another estimation that should be validated between the charge radius ratio and the compactness as . The maximum allowable limit of the charge to radius ratio is given in Tables 1–6 for different celestial objects with two well-known models for different choices of free parameter . These numerical estimated values signify that our presented celestial objects are very capable to be strange stars instead of neutron stars.
4. Analysis of Physically Estimated Values
Since we have examined the physical bounds of the compact stars along with their corresponding unknown parameters that are well-consistent to our standard observational data, we noticed from this investigation that the compactification of the star is greater than that of a neutron star. This analysis would be very fruitful to obtain an estimation of the physical reliable results, namely, matter density, pressure, charge to radius ratio, and the bag constant. We studied compact star candidates of distinct compactness and computed the relevant unknown constant parameters. The numerical values are given in Tables 1–7. For instance, we observed the estimated values of candidate of mass and radius in the background of the viable model for the value of , and the numerical values of the unknowns are gained as , , and in units of and . Substituting the values of and into the corresponding expressions, afterwards, the values of the physical parameters and the bag constant are , , , and , respectively. Note that for the next increasing values of , the corresponding values of the unknowns, energy density, pressure, and bag constant increase, but the value of charge to radius ratio decreases (Table 1). At similar fashion, one can also check the values of these physical parameters against another well-known viable model . From the constraint of equation (26), the minimum value of the charge to radius ratio is 0.0128 against the minimum compactness of the given star data. Hence, for the star of mass , the relevant maximal radius is . For the feasible reliable investigation of the physical parameters at the interior of the celestial object, we have considered the object and drawn different evolutions of the matter density, pressures (radial and tangential), electric field, energy conditions, equilibrium equation, and stability parameters.
Moreover, in this scenario, the value of escalates with the increment of the compactness (i.e., is the density reliant). An ultradense object wants maximum . The corresponding attentions were determined in , where a density-reliant has been considered to paradigm magnetized strange quark structures. To continue this description, the leading motivation of Farhi and Jaffe  presented that for the stable strange matter configuration and estimated that the must encompass nearly . In comparison of these results, our outcomes indicate that with the incorporation of electromagnetic charge and anisotropy, returns in a greater value. In regard of the study by Farhi and Jaffe , the investigation was identified for a -stable strange quark matter fulfilling the baryon number conservation principle where the uncharged state was considered. The framework of stability was constituted by three factors such as , the mass of the quark particles, and the coupling constant. What comes off if the uncharge state is not implemented is not clear from the study. For a likely immense strange quark matter, there might be a gathering of an absolute positive charge in the interior of the object. Perhaps, in the inclusion of electric charge, to resist the directed outward force produced due to the electromagnetic field, the bag pressure escalates. The point, although, is a problem of more analysis. Thus, when one wants to employ a mathematical consistent and physically reliable exact result to paradigm strange objects, the does not carry a constant. Therefore, it becomes a free parameter that depends on the compactification of the object.
5. Physical Aspects of the Object
5.1. Evolution of Matter Density, Pressure, and Electric Field
In this conjecture, we described suitable aspects of the various physical parameters, namely, energy density, radial and transverse pressures, electric field, and derivative function of the energy density and radial pressure for the proposed candidate in the background of two popular and cosmologically viable classes of models. These types of feasible models are  and , in which one of them is recognized as quadratic curvature formulation of the generic function of Ricci scalar and the other is the modified form of the prior model in terms of cubic corrections. Several reasonable outcomes of these models have been devoted in the literature [53, 103–106] for the physical evaluation of the collapsing star and the stable equilibrium condition of the star at the theoretical as well as the astrophysical backgrounds.
It would be very necessary to discuss on the role of model parameters because they should be quite consistent and right for the stable formation of the interior celestial object. Besides other unknown parameters, these are considered as initial parameters and govern the physical aspects and their graphical evolution of the stable object. Hence, in our first suggested paradigm , the value of is considered as 0.1–0.3 at equal intervals of units, and this similar approach was used for the model , in which .
To obtain the stable configuration of the compact star, there are several feasible features available from which first we analyze matter density and radial and transverse pressures for their physical credibility and graphical evolution. These said parameters must be free of central singularity or regular at the core of the interior compact stellar object as well as nonnegative throughout the whole distribution. Also, the nature should be maximum around the center where and minimum at the boundary surface of the sphere where . The derivatives of the matter density and radial pressure are monotonically decreasing functions of , i.e., their evolution satisfies the bound .
The evolution of the energy density with their respective models for the interior compact object is studied and shown in Figure 1. It is seen from Figure 1 that each plot of the energy density shows nonnegative nature within the entire region of the star, and it suggests maximum behavior at the center and minimum towards the boundary surface of the sphere. Apart from this description, one important point to discuss is that the numerical value of the central density is up most in competition of the surface density of the compact star candidates (Tables 1–6), which is actually the property of massive strange quark stellar objects [107–110]. Figures 2 and 3 show the evolutionary nature of the radial and transverse pressures through graphical analysis by using two well-known models. One can easily understand from Figures 2 and 3 that the radial and tangential pressures with their respective models represent positive evolution in the entire distribution of the star and indicate the utmost regular behavior at the core of the star but low most at the boundary surface where . In particular, we note that the radial pressure sharply dies out at the boundary sphere , but transverse pressure does not disappears promptly at the surface of the sphere which displays the spheroidal nature of the strange celestial star [112–114]. Figure 4 shows the distribution of the derivative function of energy density and radial pressure by employing the numerical values of different unknown parameters. It can be eminently seen from Figure 4 that both combined plots with their respective models indicate monotonically decreasing nature within the entire region of the sphere, i.e., they possess negative behavior at the interior of the star. Moreover, we plot the evolution of the electric field for the proposed object with two viable models (Figure 5). The evolutionary behavior of the electric field in each respective plot suggests positive nature and increases with the increase of of the celestial star.