Effects of Charge on Gravitational Decoupled Anisotropic Solutions in f(R) Gravity

TL;DR

This study explores charged anisotropic solutions in f(R) gravity using gravitational decoupling, demonstrating that charge enhances the stability of self-gravitating systems within this modified gravity framework.

Contribution

It introduces a method to generate charged anisotropic solutions in f(R) gravity using minimal geometric deformation, extending known isotropic solutions with stability analysis.

Findings

Charged solutions satisfy physical and stability conditions.

Charge improves the stability of self-gravitating systems.

Solutions are viable for specific charge and model parameters.

Abstract

This paper is devoted to studying charged anisotropic static spherically symmetric solutions through gravitationally decoupled minimal geometric deformation technique in $f(R)$ gravity. For this purpose, we first consider the known isotropic Krori-Barua solution for $f(R)$ Starobinsky model in the interior of a charged stellar system and then include the effects of two types of anisotropic solutions. The corresponding field equations are constructed and the unknown constants are obtained from junction conditions. We analyze the physical viability and stability of the resulting solutions through effective energy density, effective radial/tangential pressure, energy conditions, and causality condition. It is found that both solutions satisfy the stability range as well as other physical conditions for specific values of charge as well as model parameter and anisotropic constant. We conclude that the modified theory under the influence of charge yields more stable behavior of the self-gravitating system.

Equations

$${\mathcal{I}}_{f(R)}=\int d^{4}x\sqrt{-g}\left[\frac{f(R)}{2\kappa }+{\mathcal{L}}_m\right],$$

$$f_R{R}_{\xi \eta }-\frac{1}{2}{g}_{\xi \eta }f(R)-({\nabla }_{\xi }{\nabla }_{\eta }-g_{\xi \eta }\square )f_R=T_{\xi \eta }^{(m)}+E_{\xi \eta },$$

$$G_{\xi \eta }=R_{\xi \eta }-\frac{1}{2}R{g}_{\xi \eta }=\frac{1}{f_R}{T}_{\xi \eta }^{(tot)},$$

$$T_{\xi \eta }^{(tot)}=T_{\xi \eta }^{(m)}+E_{\xi \eta }+\chi {\Theta }_{\xi \eta }+T_{\xi \eta }^{(D)}.$$

$$T_{\xi \eta }^{(m)}=(\rho +p)U_{\xi }{U}_{\eta }-p{g}_{\xi \eta },$$

$$E_{\xi \eta }=\frac{1}{4\pi }\left(\frac{F^{\alpha \beta }{F}_{\alpha \beta }{g}_{\xi \eta }}{4}-F_{\xi }^{\alpha }{F}_{\eta \alpha }\right),$$

$$T_{\xi \eta }^{(D)}=\left(\frac{f(R)-R{f}_R}{2}\right)g_{\xi \eta }+({\nabla }_{\xi }{\nabla }_{\eta }-g_{\xi \eta }\square )f_R,$$

$$d{s}_{-}^2=e^{\mu (r)}d{t}^2-e^{\lambda (r)}d{r}^2-r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2),$$

$$F_{\xi \eta ;\alpha }=0,F_{;\eta }^{\xi \eta }=4\pi j^{\xi },$$

$$j^{\xi }=\zeta (r)U^{\xi },{\phi }^{\xi }=\phi (r){\delta }_0^{\xi },$$

$${\phi }^{\prime \prime }-\left(\frac{{\mu }^{\prime }+{\lambda }^{\prime }}{2}-\frac{2}{r}\right){\phi }^{\prime }=4\pi \zeta (r)e^{\frac{\mu }{2}+\lambda },$$

$${\phi }^{\prime }=\frac{q{e}^{\frac{\mu +\lambda }{2}}}{r^2},$$

$$\frac{1}{r^2}+e^{-\lambda }\left(\frac{{\lambda }^{\prime }}{r}-\frac{1}{r^2}\right)$$

$$e^{-\lambda }\left(\frac{{\mu }^{\prime }}{r}+\frac{1}{r^2}\right)-\frac{1}{r^2}$$

$$e^{-\lambda }\left(\frac{{\mu }^{\prime \prime }}{2}+\frac{{\mu }^{\prime }}{2r}-\frac{{\lambda }^{\prime }}{2r}+\frac{{\mu }^{\prime 2}}{4}-\frac{{\mu }^{\prime }{\lambda }^{\prime }}{4}\right)$$

$$p^{\prime }+\frac{{\mu }^{\prime }}{2}(\rho +p)-\chi ({\Theta }_1^1{)}^{\prime }+\frac{{\mu }^{\prime }\chi }{2}({\Theta }_0^0-{\Theta }_1^1)+\frac{2\chi }{r}({\Theta }_2^2-{\Theta }_1^1)-\frac{q{q}^{\prime }}{4\pi r^4}=0.$$

$$f(R)=R+\sigma R^2,$$

$$\rho +\sigma F_1+\frac{q^2}{8\pi r^4}+\chi {\Theta }_0^0$$

$$p+\sigma F_2-\frac{q^2}{8\pi r^4}-\chi {\Theta }_1^1$$

$$p+\sigma F_3+\frac{q^2}{8\pi r^4}-\chi {\Theta }_2^2$$

$$F_1$$

$$F_2$$

$$F_3$$

$${\tilde{\rho }}^{eff}={\rho }^{eff}+\chi {\Theta }_0^0,{\tilde{p}}_r^{eff}=p^{eff}-\chi {\Theta }_1^1,{\tilde{p}}_t^{eff}=p^{eff}-\chi {\Theta }_2^2,$$

$${\tilde{\Delta }}^{eff}={\tilde{p}}_t^{eff}-{\tilde{p}}_r^{eff}=\chi ({\Theta }_1^1-{\Theta }_2^2).$$

$$d{s}^2=e^{\alpha (r)}d{t}^2-\frac{d{r}^2}{\nu (r)}-r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2),$$

$$\alpha \mapsto \mu =\alpha +\chi g,\nu \mapsto e^{-\lambda }=\nu +\chi I,$$

$$g\mapsto 0,I\mapsto I^{\ast },$$

$$\alpha \mapsto \mu =\alpha ,\nu \mapsto e^{-\lambda }=\nu +\chi I^{\ast }.$$

$${\rho }^{eff}+\frac{q^2}{8\pi r^4}$$

$$p^{eff}-\frac{q^2}{8\pi r^4}$$

$$p^{eff}+\frac{q^2}{8\pi r^4}$$

$$p^{\prime }+\frac{{\mu }^{\prime }}{2}(\rho +p)-\frac{q{q}^{\prime }}{4\pi r^4}=0.$$

$${\Theta }_0^0$$

Full text

Keywords: Gravitational decoupling; Exact solutions; $f(R)$ gravity.

PACS: 04.20.Jb; 04.40.-b; 04.50.Kd.

1 Introduction

Modified theories of gravity have made a remarkable success to expose invisible effects of dark matter and dark energy. In this regard, Starobinsky [1] proposed the novel concept of higher curvature terms to discuss the inflationary scenario in which the action of general relativity (GR) is modified by $R+\sigma R^{2}$ instead of the Ricci scalar ($R$). The $f(R)$ gravity is the simplest extension of GR developed by considering an arbitrary function instead of $R$ in the Einstein-Hilbert action [2]. A lot of work has been done to discuss the viability as well as stability of this theory through different approaches [3]-[10]. The condition of hydrostatic equilibrium in stellar structure can be employed as a test to analyze the physical acceptability of $f(R)$ gravity. However, there are some functional forms of $f(R)$ which do not exhibit the stable stellar system and are considered unrealistic. During the last few years, many researchers have discussed stable as well as dynamical unstable structure of relativistic objects in the framework of this gravity [11]-[21].

The study of exact spherical solutions for relativistic objects is a difficult problem due to the presence of non-linear terms in the field equations. To resolve this issue, the gravitational decoupling via minimal geometric deformation (MGD) technique has provided appreciable results in finding new physically acceptable solutions for spherical compact configuration. This is a direct, systematic and simple approach to generate new anisotropic results from perfect fluid distribution. The decoupling of gravitational sources via MGD method is a novel concept which displays a large number of interesting constituents in the construction of new spherical solutions. This technique was first presented by Ovalle [22] to obtain new exact solutions of compact stars in the configuration of braneworld. Later, Ovalle and Linares [23] evaluated the exact solution for isotropic spherical stellar system and found that their results show consistency for Tolman-IV solution in the braneworld.

Casadio et al. [24] considered this approach to formulate new exterior spherical solution which yields singular behavior at Schwarzschild radius. Ovalle [25] formulated exact anisotropic spherical solutions from perfect fluid by decoupling the gravitational source through this technique. Ovalle et al. [26] extended interior isotropic solution by including the anisotropic source term for spherical stellar object and examined the graphical description of effective radial pressure, anisotropic factor and gravitational redshift. In the same scenario, Gabbanelli et al. [27] obtained physically viable anisotropic solutions by taking isotropic Durgapal-Fuloria stellar system and discussed the graphical interpretation of matter variables. Graterol [28] deformed the Buchdahl solution to attain analytic solution and calculated the unknown constants in new anisotropic solution via matching conditions. Recently, Panotopoulos and Rincón [29] found exact analytical solutions in 3-dimensional gravity using the MGD approach in a cloud of strings and analyzed the behavior of matter variables graphically.

The presence of electric field in self-gravitating object has significant importance in describing their evolution and stable structure. In the analysis of astrophysical scenarios, it is observed that a star needs a large amount of charge to repel strong gravitational pull. In the context of GR as well as alternative theories, a numerous research has been done to examine the influence of charge on the physical behavior of celestial bodies. It is found that the existence of electric field leads to more stable configuration of stellar systems [30]-[36]. Recently, Sharif and Sadiq [37] explored charged anisotropic spherical solutions through MGD technique by considering Krori-Barua solution as a known charged isotropic solution. They observed the role of physical parameters, stability and energy conditions for different values of charge parameter to analyze the regularity of their solutions.

In this paper, we discuss the influence of charge as well as modified theory on new exact anisotropic spherically symmetric solutions through MGD approach. We consider the well-known Krori-Barua solution as the known charged isotropic solution and analyze the analytic anisotropic solutions in $f(R)$ gravity. The paper is arranged in the following pattern. The next section provides some basics of $f(R)$ field equations with electromagnetic field corresponding to a realistic model and uses MGD method to gain two sets of decoupled equations. Section 3 deals with the Krori-Barua charged isotropic solution whose unknown constants are found through matching conditions. We obtain two new charged anisotropic solutions whose viability is investigated via graphical analysis. Finally, we present our conclusive remarks in the last section.

2 Gravitational Decoupled Field Equations

The modification of the Einstein-Hilbert action in the presence of matter Lagrangian ($\mathcal{L}_{m}$) depending upon the metric tensor ($g_{\xi\eta}$) is presented by the action [2]

[TABLE]

where $f(R)$ represents an arbitrary function of the curvature scalar and $\kappa=1$ indicates the coupling constant. The field equations corresponding to action (1) are

[TABLE]

where $f_{R}=\frac{\partial f}{\partial R}$, $\Box=g^{\xi\eta}\nabla_{\xi}\nabla_{\eta}$ and $\nabla_{\xi}$ stands for covariant derivative. An alternate expression of Eq.(2) can be expressed as

[TABLE]

where

[TABLE]

Here, $T_{\xi\eta}^{(m)}$ represents the standard energy-momentum tensor whose mathematical form corresponding to perfect fluid distribution comprising the four-velocity field $(U_{\xi})$, energy density $(\rho)$ and pressure $(p)$ is as follows

[TABLE]

$E_{\xi\eta}$ denotes the electromagnetic energy-momentum tensor defined as

[TABLE]

$T_{\xi\eta}^{(D)}$ shows the constituents that appear from the contribution of modified terms in the energy-momentum tensor given by

[TABLE]

and $\Theta_{\xi\eta}$ illustrates an extra term that is gravitationally coupled via constant $\chi$ which may comprise new fields (such as scalar, vector and tensor) and may induce anisotropy in relativistic objects [38].

In order to portray the internal configuration of self-gravitating objects, we consider a static spherically symmetric spacetime

[TABLE]

where the metric potentials ($\mu$ and $\lambda$) depend only on radial coordinate $r$ ranging from the center $(r=0)$ to the surface $(r=\mathcal{R})$ of star while the four-velocity yields $U^{\xi}=e^{-\frac{\mu}{2}}\delta^{\xi}_{0}$ for $0\leq r\leq\mathcal{R}$. The Maxwell field equations are

[TABLE]

where $j^{\xi}$ is the four current density. Here, we take a comoving frame in which the charge parameter remains at rest and consequently, no more magnetic field is produced. The four current density and four potential in comoving coordinates obey the following identities

[TABLE]

where $\zeta$ indicates the charge density. The Maxwell field equations corresponding to the metric (6) become

[TABLE]

where prime reveals derivative with respect to radial coordinate. Integration of the above equation leads to

[TABLE]

where $q$ represents charge in the interior region of star. The field equations of $f(R)$ gravity (3) for spherically symmetric spacetime are

[TABLE]

In $f(R)$ gravity, the conservation equation $(\nabla_{\xi}T^{\xi\eta(tot)}=0)$ is also satisfied whose expression corresponding to (6) becomes

[TABLE]

Notice that the standard conservation equation for charged perfect fluid configuration can be recovered for $\chi=0$.

In the analysis of early universe, various inflationary models are developed on scalar fields originating from super-string and super-gravity theories. Starobinsky [1] suggested the first inflation model which corresponds to the conformal deviation in quantum gravity given by

[TABLE]

where $\sigma\in\mathbb{R}$. It is observed that this functional form may lead to the accelerated cosmic expansion due to the impact of $\sigma R^{2}$ term. This model is also found to be consistent with the temperature anisotropies detected in cosmic microwave background and hence can be served as a reliable alternative for the inflationary candidates [39]. The signature of $f_{RR}$ is of fundamental importance as it examines how much this modified theory reaches to the GR limit. The consistency of this model is attained for $\sigma>0$ which is directly related to $f_{RR}>0$. In the analysis of self-gravitating objects, Zubair and Abbas [21] established that the acceptable values of $\sigma$ lie in the range $0<\sigma<6$. The results of GR can be regained from the proposed model for $\sigma=0$. This model has extensively been implemented in literature to narrate various cosmological issues.

The field equations (10)-(LABEL:12) corresponding to the model (14) become

[TABLE]

where $F_{1}$, $F_{2}$ and $F_{3}$ are of the following forms

[TABLE]

Here, we have a system of non-linear differential equations (13) and (15)-(17) which contains eight unknown functions ($\mu$, $\lambda$, $\rho$, $p$, $q$, $\Theta_{0}^{0}$, $\Theta_{1}^{1}$ and $\Theta_{2}^{2}$). In order to close the system, we employ a systematic approach proposed by Ovalle [26]. For the set of equations (15)-(17), the matter variables (effective energy density, effective radial and effective tangential pressures) can be identified as

[TABLE]

where $\rho^{eff}$ and $p^{eff}$ denote $\rho+\sigma F_{1}$ and $p+\sigma F_{2}$, respectively. From these definitions, it is clearly observed that the source $\Theta_{\xi\eta}$ generates anisotropy in the interior of self-gravitating systems. The effective anisotropic factor is defined as follows

[TABLE]

It is worth mentioning here that the anisotropy factor vanishes for $\chi=0$.

2.1 The MGD Approach

In this section, we consider a new technique known as gravitational decoupling through MGD approach to solve a set of non-linear differential equations (15)-(17). This technique is used to transform the field equations in such a way that the source $\Theta_{\xi\eta}$ provides the form of effective equations which may produce an anisotropy. The most fundamental constituent of this technique is the perfect fluid solution ($\alpha$, $\nu$, $\rho$, $p$ and $q$) with the metric

[TABLE]

where $\nu(r)=1-\frac{2m(r)}{r}+\frac{q^{2}}{r^{2}}$ is the usual GR expression that contains the Misner-Sharp mass $m$ and charge $q$. The influence of source $\Theta_{\xi\eta}$ in charged isotropic model can be encoded by the implementation of geometric deformation on the metric potentials ($\alpha$ and $\nu$) through a linear mapping defined as

[TABLE]

where $g$ and $I$ are the corresponding deformations offered to temporal and radial metric ingredients, respectively. It is worthwhile to mention here that the geometric deformations in (21) are entirely radial functions which confirm the spherical symmetry of the solution. Among these deformations, MGD corresponds to

[TABLE]

where $I^{*}$ shows the minimal geometric deformation. In this case, the deformation is applied only on the radial component whereas the temporal one remains unchanged. Thus, the anisotropic source $\Theta_{\xi\eta}$ is purely merged in the radial deformation denoted by

[TABLE]

Inserting Eq.(23) into Eqs.(15)-(17), the system decouples into two sets.

The first set corresponds to $\chi=0$ leading to the following charged perfect fluid matter configuration

[TABLE]

as well as the conservation equation

[TABLE]

The second set of equations comprising the source $\Theta_{\xi\eta}$ yields

[TABLE]

and the conservation equation, $\nabla_{\xi}\Theta_{\xi\eta}=0$, is explicitly expressed as

[TABLE]

From Eqs.(27) and (31), it is clearly shown that there is no change of energy-momentum tensor between the charged perfect fluid distribution and the source $\Theta_{\xi\eta}$ which assures that their interaction is absolutely gravitational. It is noted that the set of Eqs.(28)-(30) are similar to the spherically symmetric field equations for anisotropic matter distribution with source $\Theta_{\xi\eta}$ relative to the metric

[TABLE]

However, the expressions on right-hand side of Eqs.(28)-(30) are not the standard one as they show deviation from anisotropic solution by the factor $\frac{1}{r^{2}}$ which represent the matter constituents as

[TABLE]

Thus, the MGD approach has turned the indefinite system (15)-(17) into a set of equations for charged perfect fluid along with a set of four unknown functions $(I^{*},\Theta_{0}^{0},\Theta_{1}^{1},\Theta_{2}^{2})$ satisfying the anisotropic system (33)-(35). Hence, the system (15)-(17) has been decoupled successfully.

2.2 Junction Conditions

In the evolution of self-gravitating systems, the junction conditions has a dynamical contribution that provide a linear relation between interior as well as exterior metrics at the boundary of star to analyze the physical behavior of stellar objects. In this work, the interior geometry of stellar distribution is obtained through MGD as

[TABLE]

where the internal mass function is $\tilde{m}=m(r)-\frac{\chi r}{2}I^{*}(r)$. For a smooth relation between the geometries (interior and exterior) of star, the general exterior metric is

[TABLE]

The continuity of the first fundamental form of junction conditions over the hypersurface $(\Sigma)$ leads to $[ds^{2}]_{\Sigma}=0$, where $[S]_{\Sigma}\equiv S^{+}(\mathcal{R})-S^{-}(\mathcal{R})$ for any function $S=S(r)$ gives

[TABLE]

Here, $M_{0}=m(\mathcal{R})$, $Q_{0}$ and $I^{*}(\mathcal{R})$ denote the total mass, total charge and deformation at the boundary of star, respectively.

The continuity of the second fundamental form ($[T_{\xi\eta}^{(tot)}K^{\eta}]_{\Sigma}=0$ with $K^{\eta}$ as a unit four-vector in radial direction) yields [26]

[TABLE]

which gives rise to

[TABLE]

where $M$, $\mathcal{Q}$ show the mass as well as charge of exterior geometry and $h^{*}$ denotes the outer radial geometric deformation for Riessner-Nordström (RN) metric in the presence of source $\Theta_{\xi\eta}$ described by

[TABLE]

The necessary and sufficient conditions for a direct relation between MGD interior and RN exterior metrics (filled with source $\Theta_{\xi\eta}$) are provided by the constraints (38) and (39). If we consider the exterior spacetime as the standard RN metric ($h^{*}=0$), then

[TABLE]

3 Anisotropic Solutions

In order to attain the anisotropic solutions for charged stellar system through MGD technique, we need solution of the field equations for charged perfect fluid spherical system in $f(R)$ gravity. In this regard, we consider the Krori-Barua solution that has become a subject of great interest due to its singularity free nature [40]. This solution has attained much attention in analyzing the behavior of charged stellar systems both in GR as well as modified theories. In the background of $f(R)$ gravity, Momeni et al. [20] examined the stellar configuration for this solution without charge by employing extended forms of Tolman-Oppenheimer-Volkoff equations. In the same theory, Zubair and Abbas [21] used this solution to investigate physical characteristics as well as stable structure of anisotropic compact objects.

The Krori-Barua solution yields a consistent as well as realistic method in the analysis of stellar evolution. For charged perfect fluid distribution in $f(R)$ gravity, this solution is defined as

[TABLE]

where the triplet ($\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$) represents unknown constants that can be computed from matching conditions. From the matching between interior and exterior geometries of stellar object, the continuity of metric variables $g_{tt}$, $g_{rr}$ and $g_{tt,r}$ leads to the following forms of $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$

[TABLE]

along with the compactness factor $\frac{2M_{0}}{\mathcal{R}}<\frac{8}{9}$. These expressions assure the continuity of charged isotropic solution (42)-(46) with the exterior RN geometry at star’s surface which will surely be changed with the presence of source $\Theta_{\xi\eta}$ in the interior region.

In order to have anisotropic solution, i.e., for $\chi\neq 0$ in the interior of spherical stellar object, the radial as well as temporal metric constituents are given by Eqs.(23) and (42), respectively. The geometric deformation $(I^{*})$ and the source term ($\Theta_{\xi\eta}$) are related through Eqs.(28)-(30) whose solution will be evaluated by considering some additional constraints. For this purpose, we impose some conditions to derive two physically consistent interior solutions in the following subsections.

3.1 The First Solution

In this section, we adopt a constraint on $\Theta_{1}^{1}$ and find a solution of the field equations for the source term $\Theta_{\xi\eta}$ and deformation function $I^{*}$. From Eq.(41), it is observed that RN exterior geometry shows compatibility with the isotropic interior metric as long as $p^{eff}(\mathcal{R})-\frac{Q_{0}^{2}}{8\pi\mathcal{R}^{4}}\sim\chi(\Theta_{1}^{1}(\mathcal{R}))_{-}$. In order to satisfy this requirement, the direct choice is to consider [37]

[TABLE]

With the help of Eqs.(25) and (29), this leads to

[TABLE]

which yields the radial metric coefficient as

[TABLE]

The metric constituents of interior geometry in Eqs.(42) and (52) illustrate the minimally deformed Krori-Barua solution through the generic anisotropic source $\Theta_{\xi\eta}$. For $\chi\rightarrow 0$, Eq.(52) reduces to the standard spherical solution.

Now, employing junction conditions, the continuity of first fundamental form leads to

[TABLE]

which further gives rise to

[TABLE]

Similarly, the continuity of second fundamental form ($p^{eff}-\frac{Q_{0}^{2}}{8\pi\mathcal{R}^{4}}+\chi(\Theta_{1}^{1}(\mathcal{R}))_{-}\\ =0$) along with Eq.(50) provides

[TABLE]

To evaluate the expression of mass, Eq.(54) leads to

[TABLE]

On inserting the above expression in Eq.(53), we obtain

[TABLE]

where the constant $\mathcal{C}$ can be described in terms of $\mathcal{B}$. The set of equations (55)-(57) presents the necessary and sufficient conditions for smooth matching of interior as well as exterior metrics at star’s surface. In the case of pressure like constraint solution, the anisotropic solution, i.e., the expressions of $\tilde{\rho}^{eff}$, $\tilde{p}_{r}^{eff}$ and $\tilde{p}_{t}^{eff}$ are evaluated in the following forms

[TABLE]

The anisotropic factor in this case is calculated as

[TABLE]

In order to examine the physical properties of stellar system corresponding to the first solution, we analyze their graphical behavior for $\sigma=2$ and two different values of $Q_{0}$. In this regard, we employ the constant $\mathcal{A}$ as presented in Eq.(55) whereas $\mathcal{B}$ is assumed as a free parameter and will be considered from Eq.(48). The structure of self-gravitating objects demands that the nature of energy density as well as radial pressure should be finite, positive, maximum and regular in the interior of compact objects. The physical analysis of effective energy density and effective pressure (radial and tangential) is presented in Figure 1. The effective energy density shows maximum behavior at the center of star and decreases monotonically with increase in $r$. We notice that the larger value of $Q_{0}$ yields smaller $\tilde{\rho}^{eff}$ which indicates that increase in charge makes the sphere less dense. It is also found that the value of $\tilde{\rho}^{eff}$ enhances with increasing $\chi$.

The behavior of $\tilde{p}_{r}^{eff}$ as well as $\tilde{p}_{t}^{eff}$ in the presence of charge is also regular as well as finite similar to that of effective energy density. The value of $\tilde{p}_{r}^{eff}$ becomes zero at the boundary of star’s surface and represents decreasing behavior with increase in $Q_{0}$, $r$ and $\chi$. The physical interpretation of $\tilde{p}_{t}^{eff}$ also reveals monotonically decreasing behavior with respect to $r$. The role of effective anisotropic factor with the inclusion of charge is also examined graphically in Figure 1 which shows that the variation of $\tilde{\Delta}^{eff}$ remains positive. This indicates the existence of a repellent source that permits the evolution of large massive configuration in the interior region of stellar object. This factor depicts a constant behavior for small values of $\chi$ but with the increase in $\chi$, the value of $\tilde{\Delta}^{eff}$ decreases for larger value of $Q_{0}$.

To check physical consistency of the resulting solutions and existence of ordinary matter configuration, there are some physical features known as energy conditions. These conditions are the constraints imposed on the energy-momentum tensor and are categorized into null, strong, weak and dominant energy conditions. In $f(R)$ gravity with the influence of charge, these conditions for anisotropic matter distribution are expressed as

• •

NEC:  $\tilde{\rho}^{eff}+\tilde{p}_{r}^{eff}\geq 0$, $\tilde{\rho}^{eff}+\tilde{p}_{t}^{eff}+\frac{q^{2}}{4\pi r^{4}}\geq 0$,

• •

SEC:  $\tilde{\rho}^{eff}+\tilde{p}_{r}^{eff}+2\tilde{p}_{t}^{eff}+\frac{q^{2}}{4\pi r^{4}}\geq 0$,

• •

WEC: $\tilde{\rho}^{eff}+\frac{q^{2}}{8\pi r^{4}}\geq 0$,  $\tilde{\rho}^{eff}+\tilde{p}_{r}^{eff}\geq 0$, $\tilde{\rho}^{eff}+\tilde{p}_{t}^{eff}+\frac{q^{2}}{4\pi r^{4}}\geq 0$,

• •

DEC:  $\tilde{\rho}^{eff}-\tilde{p}_{r}^{eff}+\frac{q^{2}}{4\pi r^{4}}\geq 0$, $\tilde{\rho}^{eff}-\tilde{p}_{t}^{eff}\geq 0$.

Figure 2 represents that all energy conditions are satisfied which assure the physical viability of the considered charged anisotropic solution.

In astrophysics, the stability of stellar structure has a crucial role in evaluating any physically viable system. We discuss the stability of charged anisotropic solution by means of squared speed of sound ($v^{2}_{s}$) based on Herrera’s cracking concept [41]. The causality condition demands that the squared speed of sound represented by $v^{2}_{s}=dp/d\rho$ must be in the range [0, 1], i.e., $0\leq v^{2}_{s}\leq 1$ in the interior geometry of stars for a physically stable structure. Herrera [41] described the idea of cracking by considering a different technique to obtain potentially stable or unstable regions of compact stars. These regions are evaluated by the difference of squared sound speed in tangential and radial directions as $|v^{2}_{st}-v^{2}_{sr}|\leq 1$, where $v^{2}_{sr}$ and $v^{2}_{st}$ indicate squared sound speed in the transverse and radial directions, respectively. The stability analysis for $Q_{0}=0.1,1$ is shown in Figure 3 which interprets that our resulting charged anisotropic solution is stable for all adopted values of $\sigma$, $Q_{0}$ and $\chi$. It is also observed that there is a very small increase in stability with increase in $Q_{0}$.

3.2 The Second Solution

Here, we consider an alternative form of constraint to attain a second type of physically acceptable charged anisotropic solution. We take a density like constraint [37] presented by

[TABLE]

From Eqs.(24) and (28), we evaluate

[TABLE]

whose solution is obtained as

[TABLE]

where $d_{1}$ shows the integration constant. In order to have a singularity free solution at the center ($r=0$) of stellar object, we suppose that $d_{1}=0$ which leads to

[TABLE]

By adopting the same strategy as applied for the first solution, the matching conditions are given as

[TABLE]

The expressions of charged anisotropic solution in terms of $\tilde{\rho}^{eff}$, $\tilde{p}_{r}^{eff}$ and $\tilde{p}_{t}^{eff}$ for density like constraint are obtained as follows

[TABLE]

The anisotropic factor corresponding to the second solution becomes

[TABLE]

This factor vanishes for $\chi=0$ and our solution reduces to the standard isotropic solution.

In order to investigate the behavior of second solution, we take $\sigma=0.2$ and fix the constant $\mathcal{B}$ which is calculated from Eqs.(49) and (64) whereas $\mathcal{A}$ is a free parameter which will be considered from Eq.(47). The graphical behavior of $\tilde{\rho}^{eff}$, $\tilde{p}_{r}^{eff}$, $\tilde{p}_{t}^{eff}$ and $\tilde{\Delta}^{eff}$ in the presence of charge is represented in Figure 4. It is observed that for the second solution, all physical quantities (effective energy density, effective radial and tangential pressure) decrease with the increasing values of $Q_{0}$. The role of these matter variables is positive, finite, regular and consistent within the interior of stellar object. It is also found that this solution depicts physically viable behavior only for small values of $\chi$ as compared to the first solution. For larger values of $\chi$, this solution does not show physically acceptable behavior.

The effect of anisotropic parameter is found to be negative for the second solution which indicates the less massive distribution in the interior of stellar object. The consistency of energy conditions is shown in Figure 5 which represents that all energy conditions are satisfied for this solution. Our second solution reveals the potentially stable structure of compact stars as presented in Figure 6. It is also observed that the stability of stellar system decreases for the larger value of charge parameter.

4 Concluding Remarks

In the analysis of stellar system, the quest for new spherical solutions has captured thoughts of many researchers. Recently, the gravitational decoupling through MGD method has gained much attention in obtaining the new exact solutions of self-gravitating objects. This technique is implemented to extend the interior isotropic spherical solutions by including the effects of anisotropic gravitational sources. In this paper, we have used this technique in Starobinsky form of $f(R)$ gravity with the inclusion of charge to extend isotropic interior solution by adding the contribution of anisotropic solution comprised in gravitational source. In this regard, we have included a new source in charged isotropic as well as effective energy-momentum tensor which provides the $f(R)$ field equations corresponding to anisotropic matter configuration.

In order to attain anisotropic solutions, we have assumed the well-known charged isotropic Krori-Barua solution in which the unknown constants are calculated via matching conditions. For anisotropic solutions, we have imposed constraints on effective pressure and effective energy density in the presence of charge which yield the first and second solution, respectively. The physical viability of these solutions is examined through the graphical analysis of matter variables, effective anisotropic factor, energy conditions and potential stability corresponding to some specific values of charge as well as model parameter. We have observed that both solutions are physically acceptable and show stable structure of charged stellar object in $f(R)$ gravity. Moreover, we have analyzed that increase in charge parameter enhances the stability of the first solution but decreases the stability of second solution.

Ovalle et al. [26] constructed new anisotropic uncharged spherical solutions using Tolman IV solution as interior solution but the stable structure as well as energy conditions are not analyzed for their solutions. Sharif and Sadiq [27] employed the Krori-Barua solution for charged spherical stellar object and observed that only the first solution, i.e., the pressure like constraint represents stable structure whereas the second solution, i.e., the density like constraint disobeys the physical acceptability. Recently, Sharif and Saba [42] examined the uncharged anisotropic spherical solutions by MGD approach in the context of $f(\mathcal{G})$ gravity using Krori-Barua solution as interior isotropic solution. They deduced that the stability exists only for pressure like constraint solution. We would like to mention here that our both solutions along with the influence of charge show viable behavior and satisfy the required range of squared speed of sound in $f(R)$ gravity. These solutions reduce to the solutions obtained for uncharged case in the same gravity for $q=0$ [43]. We conclude that the $f(R)$ theory with the inclusion of charge provides more stable distribution of stellar system as compared to GR and $f(\mathcal{G})$ gravity.

Acknowledgment

One of us (AW) would like to thank the Higher Education Commission, Islamabad, Pakistan for its financial support through the Indigenous Ph.D. 5000 Fellowship Program Phase-II, Batch-III.