1. Introduction
In recent years, cosmologists have demonstrated a fascinating pattern in the arrangement of astrophysical structures within the universe. Instead of random distribution, these celestial bodies appear to be intricately organized. This observation has sparked significant interest among researchers because it offers insights into the enigmatic phenomenon of accelerated cosmic expansion. Various experiments have pointed out the presence of a mysterious force that opposes gravity, driving the expansion of the cosmos. Termed ‘dark energy’ because of its elusive nature, this force continues to amaze scientists. Although general relativity (${\mathbb{GR}}$) provides some explanation for this expansion, it grapples with issues related to the cosmological constant, leading to the development of multiple modifications. Among these, the $f({\mathbb{R}})$ theory stands out as a straightforward generalization, achieved by altering the Einstein–Hilbert action, effectively interchanging the Ricci scalar $({\mathbb{R}})$ with its generic function. This theory delves into different cosmic epochs, yielding promising results [1–4]. Furthermore, it has opened up avenues to explore self-gravitating systems using various approaches, enhancing our understanding of the physical relevance in this context [5–8].
At that moment, researchers were investigating the geometry of spacetime and the distribution of matter as separate entities [9–15]. In the pioneering work of Bertolami et al [16], they postulated that the properties of self-gravitating bodies could be better understood by interrelating matter configuration and geometry. In pursuit of this novel approach, they introduced the Lagrangian density of fluid dependent on ${\mathbb{R}}$ and explored the ramifications of this interaction within the framework of the $f({\mathbb{R}})$ theory. This groundbreaking insight spurred a wave of research, as scientists sought to comprehend the implications of such coupling on cosmic expansion. Subsequently, it became apparent that the extension of this coupling to the action level could open up new avenues for understanding cosmic evolution. Harko et al [17] led the way by introducing a generalized theory based on ${\mathbb{R}}$ and the trace of the energy–momentum tensor (${\mathbb{EMT}}$), aptly named the $f({\mathbb{R}},{\mathbb{T}})$ gravity.
In the context of this modified theory, it has been noted that the considered system deviates from its conservation principles, giving rise to an additional force influenced by various physical factors (as discussed in [18]). This force effectively alters the trajectories of moving particles within a gravitational field, transitioning them from geodesic paths to a nongeodesic framework. Houndjo [19], in his exploration of a minimal $f({\mathbb{R}},{\mathbb{T}})$ model, delved into cosmic dynamics. His findings support existing observations in the field. Within this scenario, several minimal and nonminimal models have emerged, with the ${\mathbb{R}}+2\varpi {\mathbb{T}}$ model gaining considerable attention for its linear nature. Nashed [20] conducted a comprehensive analysis of different rotating pulsars, solving differential equations with some interesting explicit assumptions. The results indicate that the calculated masses of the considered pulsars aligned closely with the existing values. Furthermore, numerous studies focusing on stellar interiors within this theory have yielded promising and insightful outcomes [21–29].
Solving field equations portraying self-gravitating systems, whether within the framework of ${\mathbb{GR}}$ or in modified theories, remains a fundamental challenge for astrophysicists [30–34]. These equations are highly nonlinear, and although analytical solutions are attainable in some cases, there are situations where formulating such solutions proves elusive, compelling researchers to turn to numerical techniques. It is crucial to note that the scientific community exclusively acknowledges solutions that meet the stringent criteria necessary for modeling physically relevant interiors. Discovering an effective strategy to tackle these differential equations is a monumental endeavor in astrophysics. In this context, the scientific community has put forth a range of methodologies aimed at investigating the intricate nature of interior fluid distribution within compact stars. One approach that has gained notable attention is the concept of gravitational decoupling. This method offers physicists a valuable tool to study stellar systems, taking into account different fluid-dependent factors in a highly efficient manner.
The development of this technique stems from the realization that when equations of motion encompass several physical parameters, gravitational decoupling can be applied to separate these equations into distinct sets, each corresponding to a specific matter source. These segregated systems of equations can be independently addressed. Gravitational decoupling encompasses two key categories of deformation: minimal and extended. In a recent pioneering study by Ovalle [35], the minimal geometric deformation (MGD) approach was introduced, offering solutions that align well with the observed data of compact stars in the context of the brane-world scenario. Building on this, Ovalle and Linares [36] extended the methodology to address isotropic self-gravitating spherical fluid interiors, deriving an exact solution. Casadio et al [37] further extended this approach to attain the vacuum spacetime within the framework of the brane-world theory. Ovalle et al [38] explored isotropic spheres and extended their analysis to anisotropic analogs using the MGD technique. Gabbanelli et al [39] successfully generalized a particular isotropic metric into the anisotropic realm using the MGD method, leading to the formulation of physically realistic compact star models. Similarly, investigations have been conducted on different isotropic ansatzes, yielding favorable results under specific parametric choices [40, 41]. We utilized both uncharged and charged models and examined the influence of matter–geometry interactions and decoupling parameter on these models [42–45].
In this study, we expanded upon the isotropic Finch–Skea metric by introducing multiple anisotropic extensions within the context of modified gravity. Our approach involves a systematic technique for the transition from isotropy to anisotropy. To provide a clearer overview of the structure of this paper, we break it down as follows. Section 2 delves into the fundamental concepts of the modified theory while also identifying the field equations characterizing the entire (including both the pre-existing and newly introduced) fluid source. Section 3 introduces the MGD scheme, which, when applied to the field equations, segregates them into two distinct sets. In section 4 , we focus on the calculation of a Finch–Skea triplet under the junction conditions. Section 5 presents the requirements that must be satisfied for a feasible solution. Finally, in section 6 , we present three novel anisotropic solutions and provide a graphical interpretation. In section 7 , we provide a concise summary of our findings.
2. $f({\mathbb{R}},{\mathbb{T}})$ theory admitting Einstein–Maxwell field equations
The Einstein–Hilbert action for the modified $f({\mathbb{R}},{\mathbb{T}})$ gravity containing an extra fluid distribution is defined as1 ) and obtained
$\begin{eqnarray}S=\int \sqrt{-g}\left[\displaystyle \frac{f({\mathbb{R}},{\mathbb{T}})}{16\pi }+{\unicode{x00141}}_{m}+{\unicode{x00141}}_{{\mathbb{E}}}+\alpha {\unicode{x00141}}_{{\rm{\Upsilon }}}\right]{{\rm{d}}}^{4}x,\end{eqnarray}$
where ${\unicode{x00141}}_{m},\,{\unicode{x00141}}_{{\mathbb{E}}}$, and Łϒ are the Lagrangian densities of the matter source, the electromagnetic field, and the additional gravitationally coupled fluid distribution, respectively. In addition, α serves as a decoupling parameter that plays a vital role in understanding the impact of this new source on the existing matter field. To derive the modified Einstein–Maxwell field equations, we applied the principle of variation on the action ( $\begin{eqnarray}{{\mathbb{G}}}_{\eta \xi }=8\pi {{\mathbb{T}}}_{\eta \xi }^{(\mathrm{tot})}.\end{eqnarray}$
Here, ${{\mathbb{G}}}_{\eta \xi }$ is known as the Einstein tensor, which encapsulates the geometric properties of the structure, whereas ${{\mathbb{T}}}_{\eta \xi }^{(\mathrm{tot})}$ characterizes the total fluid distribution within the considered spacetime. The latter term can be further subdivided into three distinct components as follows: $\begin{eqnarray}{{\mathbb{T}}}_{\eta \xi }^{(\mathrm{tot})}=\displaystyle \frac{1}{{f}_{{\mathbb{R}}}}\left({{\mathbb{T}}}_{\eta \xi }+{{\mathbb{E}}}_{\eta \xi }+\alpha {{\rm{\Upsilon }}}_{\eta \xi }\right)+{{\mathbb{T}}}_{\eta \xi }^{(C)},\end{eqnarray}$
where the first three terms enclosed in parentheses represent the components of the usual, charged, and extra matter distributions, respectively. Moreover, the $f({\mathbb{R}},{\mathbb{T}})$ corrections are introduced in the entity ${{\mathbb{T}}}_{\eta \xi }^{(C)}$ defined in the following: $\begin{aligned} \mathbb{T}_{\eta \xi}^{(C)}= & \frac{1}{8 \pi f_{\mathbb{R}}}\left[f_{\mathbb{T}} \mathbb{T}_{\eta \xi}+\left\{\frac{\mathbb{R}}{2}\left(\frac{f}{\mathbb{R}}-f_{\mathbb{R}}\right)-\mathrm{Ł}_{m} f_{\mathbb{T}}\right) g_{\eta \xi}\right. \\ & \left.-\left(g_{\eta \xi} \square-\nabla_{\eta} \nabla_{\xi}\right) f_{\mathbb{R}}+2 f_{\mathbb{T}} g^{\lambda \beta} \frac{\partial^{2} \mathrm{Ł}_{m}}{\partial g^{\eta \xi} \partial g^{\lambda \beta}}\right] \end{aligned}$
where ${f}_{{\mathbb{T}}}$ and ${f}_{{\mathbb{R}}}$ indicate the partial derivatives of the functional f with respect to ${\mathbb{T}}$ and ${\mathbb{R}}$, respectively. In addition, the D’Alembert operator is defined as $\square \equiv \tfrac{1}{\sqrt{-g}}{\partial }_{\eta }\left({g}^{\eta \xi }\sqrt{-g}{\partial }_{\xi }\right)$, and ∇η means the covariant derivative.At the outset, we assumed the spacetime geometry to exhibit isotropic features at an initial moment, characterized by ${\mathbb{EMT}}$ given as follows:2 ) as
$\begin{eqnarray}{{\mathbb{T}}}_{\eta \xi }=(\mu +P){{\mathbb{K}}}_{\eta }{{\mathbb{K}}}_{\xi }+{{Pg}}_{\eta \xi },\end{eqnarray}$
where ${{\mathbb{K}}}_{\eta },\,\mu $, and P correspond to the four-velocity, energy density, and an isotropic pressure, respectively. Furthermore, we derived the trace of equation ( $\begin{eqnarray*}\begin{array}{l}2f+{\mathbb{T}}({f}_{{\mathbb{T}}}+1)+\alpha {\rm{\Upsilon }}-{\mathbb{R}}{f}_{{\mathbb{R}}}-3{{\rm{\nabla }}}^{\eta }{{\rm{\nabla }}}_{\eta }{f}_{{\mathbb{R}}}\\ \qquad -\ 4{f}_{{\mathbb{T}}}{\unicode{x00141}}_{m}+2{f}_{{\mathbb{T}}}{g}^{\lambda \beta }{g}^{\eta \xi }\displaystyle \frac{{\partial }^{2}{\unicode{x00141}}_{m}}{\partial {g}^{\lambda \beta }\partial {g}^{\eta \xi }}=0.\end{array}\end{eqnarray*}$
In the scenario where we considered vacuum, the interaction was nullified. This resulted in the theoretical framework transitioning into the $f({\mathbb{R}})$ configuration. Notably, the modified correction terms caused a deviation from the energy–momentum conservation. This nonconservation phenomenon offers compelling evidence for the presence of an extra force. This additional force can be mathematically expressed by $\begin{eqnarray}\begin{array}{rcl}{{\rm{\nabla }}}^{\eta }{{\mathbb{T}}}_{\eta \xi } & = & \displaystyle \frac{{f}_{{\mathbb{T}}}}{8\pi -{f}_{{\mathbb{T}}}}\left[({{\mathbb{T}}}_{\eta \xi }+{{\rm{\Upsilon }}}_{\eta \xi }){{\rm{\nabla }}}^{\eta }\mathrm{ln}{f}_{{\mathbb{T}}}+{{\rm{\nabla }}}^{\eta }{{\rm{\Theta }}}_{\eta \xi }\right.\\ & & -\ \left.\displaystyle \frac{8\pi }{{f}_{{\mathbb{T}}}}\left({{\rm{\nabla }}}^{\eta }{{\mathbb{E}}}_{\eta \xi }+\alpha {{\rm{\nabla }}}^{\eta }{{\rm{\Upsilon }}}_{\eta \xi }\right)-\displaystyle \frac{1}{2}{g}_{\lambda \beta }{{\rm{\nabla }}}_{\eta }{{\mathbb{T}}}^{\lambda \beta }\right],\end{array}\end{eqnarray}$
where ${{\rm{\Theta }}}_{\eta \xi }={g}_{\eta \xi }{\unicode{x00141}}_{m}-2{{\mathbb{T}}}_{\eta \xi }-2{g}^{\lambda \beta }\tfrac{{\partial }^{2}{\unicode{x00141}}_{m}}{\partial {g}^{\eta \xi }\partial {g}^{\lambda \beta }}$.In the realm of self-gravitating systems, the geometry is typically divided into two distinct regions, the interior and exterior, separated by a boundary known as the hypersurface. The interior spacetime is mathematically described by the following metric as5 )) takes the form
$\begin{eqnarray}{{\rm{d}}{s}}^{2}=-{{\rm{e}}}^{{b}_{1}}{{\rm{d}}{t}}^{2}+{{\rm{e}}}^{{b}_{2}}{{\rm{d}}{r}}^{2}+{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\right),\end{eqnarray}$
where b1 = b1(r) and b2 = b2(r). In relation with this metric, the four-velocity (given in equation ( $\begin{eqnarray}{{\mathbb{K}}}_{\eta }=-{\delta }_{\eta }^{0}{{\rm{e}}}^{\tfrac{{b}_{1}}{2}}=(-{{\rm{e}}}^{\tfrac{{b}_{1}}{2}},0,0,0).\end{eqnarray}$
To gain a comprehensive understanding of the physical behavior of solutions, it is essential to determine whether they correspond to physically acceptable compact models. This evaluation necessitates consideration of a standard model within the gravitational theory under investigation. The following choice allowed us to effectively apply geometric deformations to the relevant field equations. The specific model we employed can be defined as
$\begin{eqnarray}f({\mathbb{R}},{\mathbb{T}})={f}_{1}({\mathbb{R}})+{f}_{2}({\mathbb{T}})={\mathbb{R}}+2\varpi {\mathbb{T}},\end{eqnarray}$
with ϖ being an arbitrary matter–fluid coupling parameter, and ${\mathbb{T}}=-\mu +3P$. A variety of inflationary potentials have been employed to derive potential slow-roll parameters using this model, leading to results aligned with observed data only within a specific range of ϖ ∈ (−0.37; 1.483) [46]. Furthermore, researchers have successfully applied this linear model to develop the Tolman–Kuchowicz interior, characterizing anisotropic fluid distribution [47]. Using the same model with different constraints, we formulated multiple viable compact interior solutions, as described in [48, 49].Combining equation (7 ) with the modified model (9 ) produce the following field equations:6 ) takes the extended form as13 ) effectively represents the sum of the various physical forces that collectively maintain a self-gravitating system in a state of equilibrium. It is important to recognize that the inclusion of additional components $({{\rm{\Upsilon }}}_{0}^{0},{{\rm{\Upsilon }}}_{1}^{1},{{\rm{\Upsilon }}}_{2}^{2})$ increases the number of unknown quantities in equations (10 )–(12 ). Consequently, a systematic approach [38] becomes imperative to address this inherently underdetermined system.
$\begin{eqnarray}\begin{array}{l}{{\rm{e}}}^{-{b}_{2}}\left(\displaystyle \frac{{b}_{2}^{\prime} }{r}-\displaystyle \frac{1}{{r}^{2}}\right)+\displaystyle \frac{1}{{r}^{2}}\\ =\,8\pi \left(\mu +\displaystyle \frac{{q}^{2}}{8\pi {r}^{4}}-\alpha {{\rm{\Upsilon }}}_{0}^{0}\right)+\varpi \left(3\mu -P\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{{\rm{e}}}^{-{b}_{2}}\left(\displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{{b}_{1}^{{\prime} }}{r}\right)-\displaystyle \frac{1}{{r}^{2}}\\ =\,8\pi \left(P-\displaystyle \frac{{q}^{2}}{8\pi {r}^{4}}+\alpha {{\rm{\Upsilon }}}_{1}^{1}\right)-\varpi \left(\mu -3P\right),\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{e}}}^{-{b}_{2}}}{4}\left[{b}_{1}^{{\prime} 2}-{b}_{2}^{\prime} {b}_{1}^{{\prime} }+2{b}_{1}^{{\prime\prime} }-\displaystyle \frac{2{b}_{2}^{\prime} }{r}+\displaystyle \frac{2{b}_{1}^{{\prime} }}{r}\right]\\ =\,8\pi \left(P+\displaystyle \frac{{q}^{2}}{8\pi {r}^{4}}+\alpha {{\rm{\Upsilon }}}_{2}^{2}\right)-\varpi \left(\mu -3P\right),\end{array}\end{eqnarray}$
where the last entities along with ϖ on the right side of the preceding equations are modified corrections, and prime means $\tfrac{\partial }{\partial r}$. Because two matter sources are now merged in the spherical interior geometry, the Tolman–Opphenheimer–Volkoff equation ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}{P}}{{\rm{d}}{r}}+\displaystyle \frac{{b}_{1}^{{\prime} }}{2}\left(\mu +P\right)+\displaystyle \frac{\alpha {b}_{1}^{{\prime} }}{2}\left({{\rm{\Upsilon }}}_{1}^{1}-{{\rm{\Upsilon }}}_{0}^{0}\right)+\alpha \displaystyle \frac{{\rm{d}}{{\rm{\Upsilon }}}_{1}^{1}}{{\rm{d}}{r}}\\ \qquad +\ \displaystyle \frac{2\alpha }{r}\left({{\rm{\Upsilon }}}_{1}^{1}-{{\rm{\Upsilon }}}_{2}^{2}\right)-\displaystyle \frac{{qq}^{\prime} }{4\pi {r}^{4}}=-\displaystyle \frac{\varpi }{4\pi -\varpi }\left(\mu ^{\prime} -P^{\prime} \right),\end{array}\end{eqnarray}$
where the nonzero rhs indicates the nonconserved nature of this theory. However, it is noteworthy that setting ϖ equal to zero would reinstate the conservation equation. Equation (3. Gravitational decoupling
The idea stands as a highly effective method for partitioning the gravitational equations into separate systems. This transformation allows for independent solutions for distinct systems, significantly simplifying the problem-solving process. To facilitate this transformation, we considered another metric as follows:15 ), therefore, resulted in16 ) to equations (10 )–(12 ), we derived two distinct sets of the field equations. In the following, the first system characterizes the initial source given by17 )–(19 ) offer explicit expressions for the energy density, pressure, and charge, which can be written as
$\begin{eqnarray}{{\rm{d}}{s}}^{2}=-{{\rm{e}}}^{{b}_{3}(r)}{{\rm{d}}{t}}^{2}+\displaystyle \frac{1}{{b}_{4}(r)}{{\rm{d}}{r}}^{2}+{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\right).\end{eqnarray}$
The transformation equations can be expressed by $\begin{eqnarray}{b}_{3}\to {b}_{1}={b}_{3}+\alpha {{\rm{t}}}_{1},\quad {b}_{4}\to {{\rm{e}}}^{-{b}_{2}}={b}_{4}+\alpha {{\rm{t}}}_{2}.\end{eqnarray}$
Here, t1 and t2 represent the time and radial deformation functions, respectively. We adopted the MGD strategy in the current case, which exclusively transformed the grr potential while leaving the gtt component unchanged, i.e. t1 → 0, t2 → T. The transformation ( $\begin{eqnarray}{b}_{3}\to {b}_{1}={b}_{3},\quad {b}_{4}\to {{\rm{e}}}^{-{b}_{2}}={b}_{4}+\alpha {\rm{T}},\end{eqnarray}$
where T = T(r). Importantly, these transformations did not disturb the spherical symmetry of the system. By applying equation ( $\begin{eqnarray}{{\rm{e}}}^{-{b}_{2}}\left(\displaystyle \frac{{b}_{2}^{\prime} }{r}-\displaystyle \frac{1}{{r}^{2}}\right)+\displaystyle \frac{1}{{r}^{2}}=8\pi \mu +\displaystyle \frac{{q}^{2}}{{r}^{4}}+\varpi \left(3\mu -P\right),\end{eqnarray}$
$\begin{eqnarray}{{\rm{e}}}^{-{b}_{2}}\left(\displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{{b}_{1}^{{\prime} }}{r}\right)-\displaystyle \frac{1}{{r}^{2}}=8\pi P-\displaystyle \frac{{q}^{2}}{{r}^{4}}-\varpi \left(\mu -3P\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{{\rm{e}}}^{-{b}_{2}}}{4}\left[{b}_{1}^{{\prime} 2}-{b}_{2}^{\prime} {b}_{1}^{{\prime} }+2{b}_{1}^{{\prime\prime} }-\displaystyle \frac{2{b}_{2}^{\prime} }{r}+\displaystyle \frac{2{b}_{1}^{{\prime} }}{r}\right]\\ \quad =8\pi P+\displaystyle \frac{{q}^{2}}{{r}^{4}}-\varpi \left(\mu -3P\right).\end{array}\end{eqnarray}$
As this system represents an isotropic matter as the primary source, the first two field equations provide a complete description of its interior. However, the interior charge is still an unknown quantity. Equations ( $\begin{eqnarray}\begin{array}{rcl}\mu & = & \displaystyle \frac{{{\rm{e}}}^{-{b}_{2}}}{32{r}^{2}\left({\varpi }^{2}+6\pi \varpi +8{\pi }^{2}\right)}\left[2\left\{(\varpi +4\pi )\left(2{{\rm{e}}}^{{b}_{2}}-{r}^{2}{b}_{1}^{{\prime\prime} }-2\right)\right.\right.\\ & & +\,\left.(7\varpi +20\pi ){{rb}}_{2}^{{\prime} }\right\}\\ & & -\,\left.(\varpi +4\pi ){r}^{2}{b}_{1}^{{\prime} 2}+{{rb}}_{1}^{{\prime} }\left\{6\varpi +(\varpi +4\pi ){{rb}}_{2}^{{\prime} }+8\pi \right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}P & = & \displaystyle \frac{{{\rm{e}}}^{-{b}_{2}}}{32{r}^{2}\left({\varpi }^{2}+6\pi \varpi +8{\pi }^{2}\right)}\left[2\left\{(\varpi -4\pi ){{rb}}_{2}^{{\prime} }+(\varpi +4\pi )\left({r}^{2}{b}_{1}^{{\prime\prime} }+2{{\rm{e}}}^{{b}_{2}}-2\right)\right\}\right.\\ & & +\ \left.(\varpi +4\pi ){r}^{2}{b}_{1}^{{\prime} 2}+{{rb}}_{1}^{{\prime} }\left\{10\varpi -(\varpi +4\pi ){{rb}}_{2}^{{\prime} }+24\pi \right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{q}^{2}=\displaystyle \frac{{{\rm{e}}}^{-{b}_{2}}{r}^{2}}{8}\left[2{r}^{2}{b}_{1}^{\prime\prime} +{r}^{2}{b}_{1}^{{\prime} 2}-{{rb}}_{1}^{{\prime} }\left({{rb}}_{2}^{{\prime} }+2\right)-2{{rb}}_{2}^{{\prime} }+4{{\rm{e}}}^{{b}_{2}}-4\right].\end{eqnarray}$
On the other hand, the second set characterizes the additional fluid source, and its components are given as20 )–(22 ), five unknowns (μ, P, q, b1, and b2) are encountered. These can be addressed by establishing two known factors through certain constraints. In contrast, the system defined by equations (23 )–(25 ) also involves four unknowns, specifically (${\rm{T}},{{\rm{\Upsilon }}}_{0}^{0},{{\rm{\Upsilon }}}_{1}^{1},\,\mathrm{and}\,{{\rm{\Upsilon }}}_{2}^{2}$), and only requires one constraint to yield a unique solution. As we contemplated the inclusion of an additional fluid source to induce anisotropy within the original perfect distribution, it became imperative to define the matter determinants as follows:
$\begin{eqnarray}8\pi {{\rm{\Upsilon }}}_{0}^{0}=\displaystyle \frac{{{\rm{T}}}^{{\prime} }}{r}+\displaystyle \frac{{\rm{T}}}{{r}^{2}},\end{eqnarray}$
$\begin{eqnarray}8\pi {{\rm{\Upsilon }}}_{1}^{1}={\rm{T}}\left(\displaystyle \frac{{b}_{1}^{{\prime} }}{r}+\displaystyle \frac{1}{{r}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}8\pi {{\rm{\Upsilon }}}_{2}^{2}=\displaystyle \frac{{\rm{T}}}{4}\left(2{b}_{1}^{{\prime\prime} }+{b}_{1}^{{\prime} 2}+\displaystyle \frac{2{b}_{1}^{{\prime} }}{r}\right)+{{\rm{T}}}^{{\prime} }\left(\displaystyle \frac{{b}_{1}^{{\prime} }}{4}+\displaystyle \frac{1}{2r}\right).\end{eqnarray}$
One intriguing aspect of the MGD scheme lies in its stringent preservation of both energy and momentum between the original and seed sources. This ensures the independent conservation of both fluid distributions. We can solve these equations individually and subsequently combine their solutions in a specific manner to derive the solution for the entire fluid system. In the system described by equations ( $\begin{eqnarray}\tilde{\mu }=\mu -\alpha {{\rm{\Upsilon }}}_{0}^{0},\quad {\tilde{P}}_{r}=P+\alpha {{\rm{\Upsilon }}}_{1}^{1},\quad {\tilde{P}}_{\perp }=P+\alpha {{\rm{\Upsilon }}}_{2}^{2}.\end{eqnarray}$
Moreover, we introduced the anisotropy specified by $\begin{eqnarray}\tilde{{\rm{\Pi }}}={\tilde{P}}_{\perp }-{\tilde{P}}_{r}=\alpha ({{\rm{\Upsilon }}}_{2}^{2}-{{\rm{\Upsilon }}}_{1}^{1}).\end{eqnarray}$
Notably, when the influence of an extra fluid source was removed (i.e. α = 0), the system reverted to its original isotropic state.4. Finch–Skea ansatz and matching criteria
As discussed previously, the determination of the four unknowns becomes straightforward when specific constraints or metric ansatzes are adopted. In this regard, we employed the Finch–Skea ansatz [50], described as28 ) transforms equations (20 )–(22 ) into
$\begin{eqnarray}\begin{array}{l}{{\rm{d}}{s}}^{2}=-\displaystyle \frac{1}{4}{\left(2{c}_{1}+{c}_{2}\sqrt{{c}_{3}}{r}^{2}\right)}^{2}{{\rm{d}}{t}}^{2}\\ +\left({c}_{3}{r}^{2}+1\right){{\rm{d}}{r}}^{2}+{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\right),\end{array}\end{eqnarray}$
depending on a triplet of parameters (c1, c2, and c3). They have dimensions of null, $\tfrac{1}{{\ell }}$, and $\tfrac{1}{{{\ell }}^{2}}$, respectively. Several remarkable studies have been conducted using this metric [51–60]. Notably, metric ( $\begin{eqnarray}\begin{array}{rcl}\mu & = & \displaystyle \frac{1}{8(\varpi +2\pi )(\varpi +4\pi ){\left({c}_{3}{r}^{2}+1\right)}^{2}\left(2{c}_{1}+{c}_{2}\sqrt{{c}_{3}}{r}^{2}\right)}\left[2{c}_{1}{c}_{3}\left\{8(\varpi +3\pi )+(\varpi +4\pi ){c}_{3}{r}^{2}\right\}\right.\\ & & +\,\left.{c}_{2}\sqrt{{c}_{3}}\left\{4\varpi +{c}_{3}{r}^{2}\left(14\varpi +(\varpi +4\pi ){c}_{3}{r}^{2}+32\pi \right)\right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}P & = & \displaystyle \frac{1}{8(\varpi +2\pi )(\varpi +4\pi ){\left({c}_{3}{r}^{2}+1\right)}^{2}\left(2{c}_{1}+{c}_{2}\sqrt{{c}_{3}}{r}^{2}\right)}\left[{c}_{2}\sqrt{{c}_{3}}\left\{4(3\varpi +8\pi )-(\varpi +4\pi ){c}_{3}^{2}{r}^{4}+2(5\varpi +8\pi ){c}_{3}{r}^{2}\right\}\right.\\ & & -\,\left.2{c}_{1}{c}_{3}\left\{(\varpi +4\pi ){c}_{3}{r}^{2}+8\pi \right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}{q}^{2}=\displaystyle \frac{1}{2{\left({c}_{3}{r}^{2}+1\right)}^{2}\left(2{c}_{1}+{c}_{2}\sqrt{{c}_{3}}{r}^{2}\right)}\left[{r}^{6}{c}_{3}^{\tfrac{3}{2}}\left\{{c}_{2}\left({c}_{3}{r}^{2}-2\right)+2\sqrt{{c}_{3}}{c}_{1}\right\}\right].\end{eqnarray}$
To determine the values of these three constants, it is imperative to establish a smooth connection between the interior spherical spacetime and an appropriate exterior metric at the interface (Σ: r = R). In this regard, junction conditions play a vital role in comprehending the overall structure of the geometrical system. The interior geometry is described by equation (28 ), and for the exterior region, we need a suitable metric. The selection of the latter geometry depends on the requirement that its basic properties must be compatible with the inner spacetime, such as the presence/absence of charge, static/nonstatic nature, etc. Given that the interior geometry is affected by an electromagnetic field, the Reissner–Nordström metric is the most suitable choice in this regard. It is important to stress here that in the $f({\mathbb{R}})$ theory, the junction conditions are different from those in ${\mathbb{GR}}$ because the higher-order geometric terms appear in the former gravity scenario [61, 62]. For example, the Starobinsky model is represented by the function $f({\mathbb{R}})={\mathbb{R}}+\alpha {{\mathbb{R}}}^{2}$, with α being a restricted constant. Nonetheless, in the current scenario, the first term of the considered model (9 ) represents ${\mathbb{GR}}$, and the second factor has a null contribution when vacuum is considered. Therefore, the outer geometry remains the same as that of ${\mathbb{GR}}$.
The Reissner–Nordström metric is thus considered as33 ) lead to34 )–(36 ) simultaneously, we obtained33 ), in addition to the condition of vanishing pressure at the boundary (i.e. P∣Σ = 0). The combination of these three constraints results in distinct values for the triplet, which can also be used for graphical interpretation.
$\begin{eqnarray}{{\rm{d}}{s}}^{2}=-\displaystyle \frac{{r}^{2}-2{Mr}+{Q}^{2}}{{r}^{2}}{{\rm{d}}{t}}^{2}+\displaystyle \frac{{r}^{2}}{{r}^{2}-2{Mr}+{Q}^{2}}{{\rm{d}}{r}}^{2}+{r}^{2}\left({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\right),\end{eqnarray}$
where M and Q are the total exterior mass and charge, respectively. It is essential to ensure the continuity of metric potentials and their derivatives across the boundary, which can be mathematically expressed by $\begin{eqnarray}{g}_{{tt}}^{-}{\,}_{=}^{{\rm{\Sigma }}}\,{g}_{{tt}}^{+},\quad {g}_{{rr}}^{-}{\,}_{=}^{{\rm{\Sigma }}}\,{g}_{{rr}}^{+},\quad {g}_{{tt},r}^{-}{\,}_{=}^{{\rm{\Sigma }}}\,{g}_{{tt},r}^{+},\end{eqnarray}$
where the plus sign corresponds to the outer region, and the minus sign indicates the inner spacetime. The constraints given in ( $\begin{eqnarray}\displaystyle \frac{{R}^{2}-2{MR}+{Q}^{2}}{{R}^{2}}=\displaystyle \frac{1}{4}{\left(2{c}_{1}+{c}_{2}\sqrt{{c}_{3}}{R}^{2}\right)}^{2},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{{R}^{2}}{{R}^{2}-2{MR}+{Q}^{2}}={c}_{3}{R}^{2}+1,\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{2M}{{R}^{2}}-\displaystyle \frac{2{Q}^{2}}{{R}^{3}}={c}_{2}\left(2{c}_{1}\sqrt{{c}_{3}}R+{c}_{2}{c}_{3}{R}^{3}\right).\end{eqnarray}$
Solving equations ( $\begin{eqnarray}{c}_{1}=\displaystyle \frac{2\left({Q}^{2}-2{MR}+{R}^{2}\right)-{MR}+{Q}^{2}}{2R\sqrt{{Q}^{2}-2{MR}+{R}^{2}}},\end{eqnarray}$
$\begin{eqnarray}{c}_{2}=\displaystyle \frac{{MR}-{Q}^{2}}{{R}^{2}\sqrt{2{MR}-{Q}^{2}}},\end{eqnarray}$
$\begin{eqnarray}{c}_{3}=\displaystyle \frac{2{MR}-{Q}^{2}}{{R}^{2}\left({Q}^{2}-2{MR}+{R}^{2}\right)}.\end{eqnarray}$
An alternative method for determining these constants involves making assumptions based on the first two constraints provided in equation (5. Reviewing physical constraints admitted by compact models
Within the vast body of literature, one encounters a plethora of solutions in GR and various extended gravity theories. It is natural to wonder whether all of these solutions are significant for astrophysicists. Nonetheless, the answer is not a straightforward affirmative. The research community, as underscored by notable contributions [63–70], has delineated specific restraints that should be applied to the developed models to explore their physical acceptance. Thus, only the solutions that adhere to these critical constraints warrant further exploration. In subsequent sections, we shed light on these pivotal points.
| (a)Addressing the presence of a singularity within a compact structure is a fundamental concern that forms the basis of our investigation. In our quest to validate the singularity-free nature of the Finch–Skea components, we checked the following criteria. In addition, it is imperative to determine their outwardly increasing behavior. Here, we have $\begin{eqnarray*}\begin{array}{l}{{\rm{e}}}^{{b}_{1}(r)}{| }_{r=0}={c}_{1}^{2},\quad {{\rm{e}}}^{{b}_{2}(r)}{| }_{r=0}=1.\end{array}\end{eqnarray*}$ Upon computing the first derivatives of these potentials, we arrived at $\begin{eqnarray*}\begin{array}{l}({{\rm{e}}}^{{b}_{1}(r)})^{\prime} =2{c}_{2}\sqrt{{c}_{3}}r\left(\displaystyle \frac{1}{2}{c}_{2}\sqrt{{c}_{3}}{r}^{2}+{c}_{1}\right),\quad ({{\rm{e}}}^{{b}_{2}(r)})^{\prime} =2{c}_{3}r,\end{array}\end{eqnarray*}$ finding them both zero in the core. This confirms the nondecreasing behavior of metric components and thereby establishes the validity of employing the Finch–Skea ansatz. | |
| (b)Understanding the fundamental properties of matter-related factors is of paramount importance. These elements should maintain finite and positive values across the whole domain. Moreover, they should attain their maximum (minimum) values at the center (outer boundary). Similarly, it can be established that the decrement in these factors while moving outward is assured when the first derivatives of these parameters vanish at r = 0 while exhibiting a negative profile outward. Further, anisotropy also plays a crucial role in increasing or decreasing the collapse rate of a celestial body [71–73]. | |
| (c)The spherical mass function can be determined using two approaches, accounting for its geometric properties and fluid distribution. In this regard, we opted for the latter approach to calculate the corresponding mass. The mathematical representation for this is given as $\begin{eqnarray}m(r)=\displaystyle \frac{1}{2}{\int }_{0}^{R}{s}^{2}\mu {\rm{d}}{s}.\end{eqnarray}$ In the following, we used $\tilde{\mu }$ instead of μ to explore the effects of the modified theory on the mass function. Within the realm of astrophysics, the concept of compactness plays a pivotal role in characterizing the density of mass within a given region. This factor, often denoted as ν(r), is the ratio of mass to size (radius) of a compact celestial body. A high value of compactness signifies an extraordinary concentration of matter within a remarkably small volume, resulting in the propagation of powerful gravitational attraction. When a spherical body is considered, this factor must be less than $\tfrac{4}{9}$ [74]. When self-gravitating objects interact with potent gravitational effects produced by neighboring bodies, their surfaces emit light and electromagnetic radiation. Measuring the increment in their wavelength is known as redshift and can be expressed as follows: $\begin{eqnarray}z(r)={\left\{1-2\nu (r)\right\}}^{\tfrac{-1}{2}}-1{\rm{.}}\end{eqnarray}$ It should be highlighted that the maximum value of this parameter is 5.211 [75], whereas it is comparatively lower in an isotropic configuration. | |
| (d)Exploring energy conditions is indeed an important aspect when investigating compact fluid configurations. This exploration is pivotal for assessing the viability of the internal structure. The essential criterion for confirming the presence of usual matter is to have a positive profile of the matter variables throughout. If any of these conditions is not met, this suggests the existence of exotic fluids within the structure. For the anisotropic interior, they are $\begin{eqnarray}\begin{array}{l}\mu +\displaystyle \frac{{q}^{2}}{8\pi {r}^{4}}\geqslant 0,\quad \mu +{P}_{\perp }+\displaystyle \frac{{q}^{2}}{4\pi {r}^{4}}\geqslant 0,\quad \mu +{P}_{r}\geqslant 0,\\ \mu -{P}_{\perp }\geqslant 0,\quad \mu -{P}_{r}+\displaystyle \frac{{q}^{2}}{4\pi {r}^{4}}\geqslant 0,\quad \mu +2{P}_{\perp }+{P}_{r}+\displaystyle \frac{{q}^{2}}{4\pi {r}^{4}}\geqslant 0.\end{array}\end{eqnarray}$ In practice, the fulfillment of the dominant conditions, such as $\mu -{P}_{r}+\tfrac{{q}^{2}}{4\pi {r}^{4}}\geqslant 0$ and μ − P⊥ ≥ 0, is sufficient, rendering the fulfillment of all other conditions a mere formality. | |
| (e)In the realm of celestial bodies, various factors can induce fluctuations, causing deviations from being in hydrostatic equilibrium state. Such deviations may lead to instability within the fluid distribution of these celestial objects, potentially affecting their long-term existence. To assess the structural stability, multiple methodologies have been proposed, with a particular focus on sound speed and perturbation techniques. An interesting and fundamental condition for the stability of a celestial body is that the speed of light must surpass the speed of sound within it. This can be expressed mathematically as $\begin{eqnarray*}0\lt {v}_{{sr}}^{2}=\displaystyle \frac{{{\rm{d}}{P}}_{r}}{{\rm{d}}\mu },\,{v}_{s\perp }^{2}=\displaystyle \frac{{{\rm{d}}{P}}_{\perp }}{{\rm{d}}\mu }\lt 1,\end{eqnarray*}$ where vsr2 is the radial speed and ${v}_{s\perp }^{2}$ represents the tangential speed of sound [76]. Additionally, Herrera [77] suggested that structural stability is maintained when the total force in the radial direction remains the same. Thus, preventing structural cracking is imperative for establishing a physically stable model. According to him, the factor $| {v}_{s\perp }^{2}-{v}_{{sr}}^{2}| $ must fall within the range of 0 to 1 to reduce the risk of instability of the considered structure. |
6. Some novel anisotropic models
In recent years, there has been a surge of interest within the scientific community dedicated to unraveling the intricate field equations governing matter distribution within celestial bodies. This attempt led to the formulation of various constraints and theoretical frameworks, resulting in the emergence of physically acceptable models for compact celestial structures. To address the system defined by equations (23 )–(25 ) involving four unknowns, the implementation of just one constraint yields a unique solution. In this context, we focused on three noteworthy constraints. In the subsequent subsections, we delve into a comprehensive discussion of the solutions derived from these constraints.
6.1. Model 1
In adherence to density-like restraint, we equated the fluid densities of both seed and additional fluid sources [78]. This holds considerable importance in the context of anisotropic compact interiors within the framework of both ${\mathbb{GR}}$ and extended theories of gravity. Its mathematical form is as follows:20 ) and (24 ) into equation (43 ), a linear-order differential equation was obtained and is expressed as follows:28 ), was simplified to45 ) involving a single unknown, namely the deformation function; thus, it was easily solvable. In some cases, finding exact solutions for differential equations involving complex functions can be challenging; however, we successfully obtained this. The deformation function was determined as
$\begin{eqnarray}\mu ={{\rm{\Upsilon }}}_{0}^{0}.\end{eqnarray}$
After substituting equations ( $\begin{eqnarray}\begin{array}{l}r{{\rm{T}}}^{{\prime} }(r)+{\rm{T}}(r)-\displaystyle \frac{{{\rm{e}}}^{-{b}_{2}}\pi }{4\left({\varpi }^{2}+6\pi \varpi +8{\pi }^{2}\right)}\left[r\left(2(7\varpi +20\pi ){b}_{2}^{{\prime} }-2(\varpi +4\pi ){{rb}}_{1}^{{\prime\prime} }\right.\right.\\ \qquad +\ \left.\left.{b}_{1}^{{\prime} }\left(6\varpi +(\varpi +4\pi ){{rb}}_{2}^{{\prime} }+8\pi \right)-{{rb}}_{1}^{{\prime} 2}(\varpi +4\pi )\right)+4(\varpi +4\pi )\left({e}^{{b}_{2}}-1\right)\right]=0.\end{array}\end{eqnarray}$
This equation, when combined with metric ( $\begin{eqnarray}\begin{array}{l}r{{\rm{T}}}^{{\prime} }(r)+{\rm{T}}(r)-\displaystyle \frac{\pi {r}^{2}}{8(\varpi +2\pi )(\varpi +4\pi ){\left({c}_{3}{r}^{2}+1\right)}^{2}\left(2{c}_{1}+{r}^{2}{c}_{2}\sqrt{{c}_{3}}\right)}\left[2{c}_{1}{c}_{3}\left\{8(\varpi +3\pi )+(\varpi +4\pi ){c}_{3}{r}^{2}\right\}\right.\\ \qquad +\ \left.{c}_{2}\sqrt{{c}_{3}}\left\{4\varpi +{c}_{3}{r}^{2}\left(14\varpi +(\varpi +4\pi ){c}_{3}{r}^{2}+32\pi \right)\right\}\right]=0.\end{array}\end{eqnarray}$
We observed equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{T}}(r) & = & \displaystyle \frac{\pi }{2r\sqrt{{c}_{3}}\left({\varpi }^{2}+6\pi \varpi +8{\pi }^{2}\right)\left(1+{c}_{3}{r}^{2}\right){\left(2{c}_{1}\sqrt{{c}_{3}}-{c}_{2}\right)}^{2}}\\ & & \times \ \left[8\sqrt{2}(3\varpi +4\pi )\sqrt{{c}_{2}}{c}_{3}^{3/4}{c}_{1}^{3/2}\left({c}_{3}{r}^{2}+1\right){\tan }^{-1}\left(\displaystyle \frac{\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}r}{\sqrt{2}\sqrt{{c}_{1}}}\right)-8\sqrt{2}\varpi \right.\\ & & \times \ {c}_{2}^{3/2}\sqrt[4]{{c}_{3}}\sqrt{{c}_{1}}\left({c}_{3}{r}^{2}+1\right){\tan }^{-1}\left(\displaystyle \frac{\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}r}{\sqrt{2}\sqrt{{c}_{1}}}\right)+4{c}_{3}{c}_{1}^{2}\left\{2(\varpi +4\pi ){c}_{3}^{3/2}{r}^{3}\right.\\ & & +\ (5\varpi +12\pi ){c}_{3}{r}^{2}{\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)-(5\varpi +12\pi )\sqrt{{c}_{3}}r+(5\varpi +12\pi )\\ & & \times \ {\tan }^{-1}\left.\left(\sqrt{{c}_{3}}r\right)\right\}-8{c}_{2}\sqrt{{c}_{3}}{c}_{1}\left\{(\varpi +4\pi ){c}_{3}^{3/2}{r}^{3}+6(\varpi +2\pi ){c}_{3}{r}^{2}\right.\\ & & \times \ \left.{\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)-(3\varpi +8\pi )\sqrt{{c}_{3}}r+6(\varpi +2\pi ){\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)\right\}\\ & & +\ {c}_{2}^{2}\left\{2(\varpi +4\pi ){c}_{3}^{3/2}{r}^{3}+5(3\varpi +4\pi ){c}_{3}{r}^{2}{\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)-(7\varpi +20\pi )\right.\\ & & \times \ \left.\left.\sqrt{{c}_{3}}r+5(3\varpi +4\pi ){\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)\right\}\right]+\displaystyle \frac{{{\mathbb{C}}}_{1}}{r},\end{array}\end{eqnarray}$
where ${{\mathbb{C}}}_{1}$ represents an integration constant taken to be zero to avoid singularity at the center. Some different but interesting studies are [79–83]. The extended form of the radial component can be written as $\begin{eqnarray}{{\rm{e}}}^{{b}_{2}(r)}=\displaystyle \frac{1+{c}_{3}{r}^{2}}{1+\alpha {\rm{T}}\left(1+{c}_{3}{r}^{2}\right)}.\end{eqnarray}$
Eventually, the effective matter determinants (equation (26 )) are expressed as follows:27 ), (49 ), and (50 ) together.
$\begin{eqnarray}\begin{array}{rcl}\tilde{\mu } & = & \displaystyle \frac{\sqrt{{c}_{3}}(1-\alpha )}{8(\varpi +2\pi )(\varpi +4\pi )\left({c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\left({c}_{3}{r}^{2}+1\right)}^{2}}\left[2{c}_{1}\sqrt{{c}_{3}}\left\{8(\varpi +3\pi )+(\varpi +4\pi ){c}_{3}{r}^{2}\right\}\right.\\ & & +\ {c}_{2}\left.\left\{4\varpi +(\varpi +4\pi ){c}_{3}^{2}{r}^{4}+2(7\varpi +16\pi ){c}_{3}{r}^{2}\right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\tilde{P}}_{r} & = & \displaystyle \frac{1}{128{\left({c}_{3}{r}^{2}+1\right)}^{2}}\left[\displaystyle \frac{\alpha \left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right)\left({c}_{3}{r}^{2}+1\right)}{\sqrt{{c}_{3}}{r}^{3}\left({\varpi }^{2}+6\pi \varpi +8{\pi }^{2}\right){\left({c}_{2}-2{c}_{1}\sqrt{{c}_{3}}\right)}^{2}\left({c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right)}\right.\\ & & \times \,\left\{8\sqrt{2}(3\varpi +4\pi )\sqrt{{c}_{2}}{c}_{3}^{3/4}{c}_{1}^{3/2}\left({c}_{3}{r}^{2}+1\right){\tan }^{-1}\left(\displaystyle \frac{\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}r}{\sqrt{2}\sqrt{{c}_{1}}}\right)-8\sqrt{2}\varpi {c}_{2}^{3/2}\right.\\ & & \times \,\sqrt[4]{{c}_{3}}\sqrt{{c}_{1}}\left({c}_{3}{r}^{2}+1\right){\tan }^{-1}\left(\displaystyle \frac{\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}r}{\sqrt{2}\sqrt{{c}_{1}}}\right)+4{c}_{3}{c}_{1}^{2}\left(2(\varpi +4\pi ){c}_{3}^{3/2}{r}^{3}+{c}_{3}{r}^{2}\right.\\ & & \times \,(5\varpi +12\pi ){\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)+(5\varpi +12\pi ){\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)-(5\varpi +12\pi )\\ & & \times \,\left.\sqrt{{c}_{3}}r\right)-8{c}_{2}\sqrt{{c}_{3}}{c}_{1}\left((\varpi +4\pi ){c}_{3}^{3/2}{r}^{3}+6(\varpi +2\pi ){c}_{3}{r}^{2}{\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)\right.\\ & & -\,\left.(3\varpi +8\pi )\sqrt{{c}_{3}}r+6(\varpi +2\pi ){\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)\right)+{c}_{2}^{2}\left(2(\varpi +4\pi ){c}_{3}^{3/2}{r}^{3}\right.\\ & & +\,5(3\varpi +4\pi ){c}_{3}{r}^{2}{\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)-(7\varpi +20\pi )\sqrt{{c}_{3}}r+5(3\varpi +4\pi )\\ & & \times \,\left.\left.{\tan }^{-1}\left(\sqrt{{c}_{3}}r\right)\right)\right\}+\displaystyle \frac{2}{(\varpi +2\pi )(\varpi +4\pi )\left({c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right)}\\ & & \times \,\left\{{c}_{2}\sqrt{{c}_{3}}\left(4(3\varpi +8\pi )-(\varpi +4\pi ){c}_{3}^{2}{r}^{2}+2(5\varpi +8\pi ){c}_{3}{r}^{2}\right)-2{c}_{1}{c}_{3}\right.\\ & & \times \,\left.\left.\left((\varpi +4\pi ){c}_{3}{r}^{2}+8\pi \right)\right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\tilde{P}}_{\perp } & = & \displaystyle \frac{1}{256{r}^{3}\sqrt{{c}_{3}}(\varpi +2\pi )(\varpi +4\pi ){\left(2{c}_{1}+{r}^{2}{c}_{2}\sqrt{{c}_{3}}\right)}^{2}{\left({c}_{2}-2{c}_{1}\sqrt{{c}_{3}}\right)}^{2}{\left({c}_{3}{r}^{2}+1\right)}^{2}}\\ & & \times \ \left[{c}_{2}^{4}{c}_{3}\left(4(\varpi +4\pi )(4\alpha -1){c}_{3}^{5/2}{r}^{5}+25(3\varpi +4\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{2}{r}^{4}\right.\right.\\ & & +\,(4\pi (33\alpha +16)+\varpi (59\alpha +40)){c}_{3}^{3/2}{r}^{3}+50(3\varpi +4\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}{r}^{2}\\ & & +\,\left.(\varpi (48-11\alpha )-4\pi (25\alpha -32))\sqrt{{c}_{3}}r+25(3\varpi +4\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right)\right){r}^{4}\\ & & -\ 40\sqrt{2}\varpi \alpha {\tan }^{-1}\left(\displaystyle \frac{r\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}}{\sqrt{2}\sqrt{{c}_{1}}}\right)\sqrt{{c}_{1}}{c}_{2}^{7/2}{c}_{3}^{5/4}{\left({c}_{3}{r}^{2}+1\right)}^{2}{r}^{4}-8{c}_{1}{c}_{2}^{3}\sqrt{{c}_{3}}\left(2{c}_{3}^{7/2}{r}^{7}\right.\\ & & \times \,(\varpi +4\pi )(4\alpha -1)+30(\varpi +2\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{3}{r}^{6}+2(10\pi (3\alpha +2)\\ & & +\ \varpi (14\alpha +11)){c}_{3}^{5/2}{r}^{5}+5(9\varpi +20\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{2}{r}^{4}+(\pi (56-80\alpha )\\ & & +\ \varpi (14-17\alpha )){c}_{3}^{3/2}{r}^{3}+20\pi \alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}{r}^{2}+\sqrt{{c}_{3}}r(\varpi (5\alpha -12)\\ & & +\ \left.4\pi (5\alpha -8))-5(3\varpi +4\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right)\right){r}^{2}-32\sqrt{2}(3\varpi +4\pi )\alpha \\ & & \times \,{\tan }^{-1}\left(\displaystyle \frac{r\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}}{\sqrt{2}\sqrt{{c}_{1}}}\right){c}_{1}^{7/2}\sqrt{{c}_{2}}{c}_{3}^{3/4}{\left({c}_{3}{r}^{2}+1\right)}^{2}+32\sqrt{2}\alpha {\tan }^{-1}\left(\displaystyle \frac{r\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}}{\sqrt{2}\sqrt{{c}_{1}}}\right)\\ & & \times \,{c}_{1}^{5/2}{c}_{2}^{3/2}\sqrt[4]{{c}_{3}}{\left({c}_{3}{r}^{2}+1\right)}^{2}\left(2(3\varpi +4\pi ){c}_{3}{r}^{2}+\varpi \right)+8\sqrt{2}{\tan }^{-1}\left(\displaystyle \frac{r\sqrt{{c}_{2}}\sqrt[4]{{c}_{3}}}{\sqrt{2}\sqrt{{c}_{1}}}\right)\\ & & \times \,\alpha {c}_{1}^{3/2}{c}_{2}^{5/2}{c}_{3}^{3/4}{\left({c}_{3}{r}^{3}+r\right)}^{2}\left(5{r}^{2}(3\varpi +4\pi ){c}_{3}-8\varpi \right)-16{c}_{1}^{4}{c}_{3}\left(4(\varpi +4\pi ){c}_{3}^{5/2}{r}^{5}\right.\\ & & +\,(5\varpi +12\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{2}{r}^{4}+(\pi (32-52\alpha )-19\varpi \alpha ){c}_{3}^{3/2}{r}^{3}+2{c}_{3}{r}^{2}\\ & & \times \,(5\varpi +12\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right)-(5\varpi +12\pi )\alpha \sqrt{{c}_{3}}r+(5\varpi +12\pi )\alpha \\ & & \times \,\left.{\tan }^{-1}\left(r\sqrt{{c}_{3}}\right)\right)+32{c}_{1}^{3}{c}_{2}\sqrt{{c}_{3}}\left(2(\varpi +4\pi )(2\alpha -1){c}_{3}^{7/2}{r}^{7}+(5\varpi +12\pi )\alpha \right.\\ & & \times \,{\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{3}{r}^{6}+4(4\pi (3\alpha +1)+\varpi (4\alpha +3)){c}_{3}^{5/2}{r}^{5}+4(4\varpi +9\pi )\alpha \\ & & \times \,{\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{2}{r}^{4}+(\varpi (12-13\alpha )-8\pi (5\alpha -6)){c}_{3}^{3/2}{r}^{3}+(17\varpi +36\pi )\alpha \\ & & \times \,\left.{\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}{r}^{2}-(3\varpi +8\pi )\alpha \sqrt{{c}_{3}}r+6(\varpi +2\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right)\right)\\ & & +\ 4{c}_{1}^{2}{c}_{2}^{2}\left(4(\varpi +4\pi )(4\alpha -1){c}_{3}^{9/2}{r}^{9}+5(5\varpi +12\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{4}{r}^{8}\right.\\ & & +\ (4\pi (11\alpha +32)+\varpi (37\alpha +56)){c}_{3}^{7/2}{r}^{7}-2(23\varpi +36\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{3}{r}^{6}\\ & & -\ (\varpi (121\alpha +36)+\pi (412\alpha -48)){c}_{3}^{5/2}{r}^{5}-2(91\varpi +172\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}^{2}{r}^{4}\\ & & +\ (4\pi (47\alpha -72)+\varpi (53\alpha -96)){c}_{3}^{3/2}{r}^{3}-2(63\varpi +116\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right){c}_{3}{r}^{2}\\ & & +\,\left.\left.(7\varpi +20\pi )\alpha \sqrt{{c}_{3}}r-5(3\varpi +4\pi )\alpha {\tan }^{-1}\left(r\sqrt{{c}_{3}}\right)\right)\right].\end{array}\end{eqnarray}$
The anisotropic factor corresponding to this model can be obtained by joining equations (To illustrate this solution graphically, we employed the estimated masses and radii of four different compact star candidates (table 1). In our analysis, we considered the specific values of different parameters as α = 0.3, 0.4, ϖ = 0.1, 0.9, and Q = 0.1, 0.6. This exploration allowed us to observe variations in the matter sector and the newly developed metric components for the obtained model. Notably, the function described by equation (46 ) remained independent of α. However, in figure 1, we observed a consistent increasing trend outward for every choice of ϖ. We also illustrated the behavior of the grr component (equation (47 )) in the same figure, which exhibited a starting value of 1 and an increasing trend throughout.
Figure 1. Deformation function ( |
Graphical representations of the effective matter determinants (48 )–(50 ), along with the anisotropic factor associated with the density-like constraint, can be observed in figure 2. Notably, increasing parameters α, ϖ, and charge Q resulted in a decrease in the energy density, revealing a clear inverse relationship. However, when it comes to the radial and transverse pressures, a strikingly different behavior emerged for specific values of α. The lower right plot in figure 2 shows that the anisotropy was null at the center and increased as we moved outward. We observed that higher values of α contributed to stronger anisotropy within the self-gravitating interior. As shown in figure 3, the mass function exhibited an increasing trend from 0 to R for both values of charge. Furthermore, as we decreased the parametric values, the considered structures became denser. The compactness and surface redshift were consistently below their maximum allowable ranges. Figures 4 and 5 present the energy conditions and stability analysis, establishing the viability and stability of our proposed model across all parametric values.
Figure 2. Physical determinants (in km−2) versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 1. |
Figure 3. Mass (in km), compactness, and surface redshift versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 1. |
Figure 4. Energy bounds (in km−2) versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 1. |
Figure 5. Stability analysis versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 1. |
6.2. Model 2
Here, we applied a pressure-like constraint [87] that set the pressures of both the primary and supplementary fluid sources equal to each other. This constraint is a commonly used approach in the scientific literature to facilitate the extension of solutions. The representation of this restriction can be expressed as51 ), when combined with equations (21 ) and (25 ), resulted in28 )
$\begin{eqnarray}P={{\rm{\Upsilon }}}_{1}^{1}.\end{eqnarray}$
Equation ( $\begin{eqnarray}\begin{array}{l}4{\rm{T}}(r)\left({{rb}}_{1}^{{\prime} }+1\right)-\displaystyle \frac{\pi {{\rm{e}}}^{-{b}_{2}}}{{\varpi }^{2}+6\pi \varpi +8{\pi }^{2}}\left[r\left\{2(\varpi +4\pi ){{rb}}_{1}^{{\prime\prime} }+{b}_{1}^{{\prime} }\left(10\varpi +24\pi -(\varpi +4\pi ){{rb}}_{2}^{{\prime} }\right)\right.\right.\\ \qquad +\ \left.\left.(\varpi +4\pi ){{rb}}_{1}^{{\prime} 2}+2(\varpi -4\pi ){b}_{2}^{\prime} \right\}-4(\varpi +4\pi )\left({{\rm{e}}}^{{b}_{2}}-1\right)\right]=0,\end{array}\end{eqnarray}$
taking the form after joining with metric ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{\left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\rm{T}}(r)}{\pi {r}^{2}}+\displaystyle \frac{1}{(\varpi +2\pi )(\varpi +4\pi ){\left({c}_{3}{r}^{2}+1\right)}^{2}}\left[2{c}_{1}{c}_{3}\left\{(\varpi +4\pi ){c}_{3}{r}^{2}\right.\right.\\ \qquad +\ \left.\left.8\pi \right\}-{c}_{2}\sqrt{{c}_{3}}\left\{4(3\varpi +8\pi )-(\varpi +4\pi ){c}_{3}^{2}{r}^{4}+2(5\varpi +8\pi ){c}_{3}{r}^{2}\right\}\right]=0.\end{array}\end{eqnarray}$
This led to the deformation function given by $\begin{eqnarray}\begin{array}{rcl}{\rm{T}}(r) & = & \displaystyle \frac{\pi {r}^{2}}{(\varpi +2\pi )(\varpi +4\pi )\left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\left({c}_{3}{r}^{2}+1\right)}^{2}}\left[{c}_{2}\sqrt{{c}_{3}}\left\{4(3\varpi +8\pi )\right.\right.\\ & & -\ \left.\left.(\varpi +4\pi ){c}_{3}^{2}{r}^{4}+2(5\varpi +8\pi ){c}_{3}{r}^{2}\right\}-2{c}_{1}{c}_{3}\left\{(\varpi +4\pi ){c}_{3}{r}^{2}+8\pi \right\}\right].\end{array}\end{eqnarray}$
The following equation provides the deformed grr potential by virtue of constraint (51 ) as26 ) and (27 ) for this model are as follows:
$\begin{eqnarray}\begin{array}{rcl}{{\rm{e}}}^{{b}_{2}(r)} & = & \left[(\varpi +2\pi )(\varpi +4\pi )\left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\left({c}_{3}{r}^{2}+1\right)}^{2}\right]\left[(\varpi +2\pi )(\varpi +4\pi )\right.\\ & & \times \,\left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right)\left({c}_{3}{r}^{2}+1\right)+\alpha \pi {r}^{2}\left\{{c}_{2}\sqrt{{c}_{3}}\left(4(3\varpi +8\pi )-(\varpi +4\pi ){c}_{3}^{2}{r}^{4}\right.\right.\\ & & +\ {\left.\left.\left.2(5\varpi +8\pi ){c}_{3}{r}^{2}\right)-2{c}_{1}{c}_{3}\left((\varpi +4\pi ){c}_{3}{r}^{2}+8\pi \right)\right\}\right]}^{-1}.\end{array}\end{eqnarray}$
Consequently, the effective physical factors defined in equations ( $\begin{eqnarray}\begin{array}{rcl}\tilde{\mu } & = & \displaystyle \frac{\sqrt{{c}_{3}}}{8(\varpi +2\pi )(\varpi +4\pi )\left({c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right)}^{2}{\left({c}_{3}{r}^{2}+1\right)}^{3}}\\ & & \times \,\left[2{c}_{1}{c}_{2}^{2}\sqrt{{c}_{3}}{r}^{2}\left\{40\varpi -32(3\varpi +8\pi )\alpha +\left(4(\varpi (95-2\alpha )+\pi (58\alpha +230))\right.\right.\right.\\ & & +\,{c}_{3}{r}^{2}\left.\left.\left(87\varpi \alpha +375\varpi +4\pi (33\alpha +265)+(\varpi +4\pi )(3\alpha +35){c}_{3}{r}^{2}\right)\right){c}_{3}{r}^{2}\right\}\\ & & +\,8{c}_{1}^{3}\sqrt{{c}_{3}}\left\{8(\varpi +3\pi (\alpha +1))+{c}_{3}{r}^{2}\left(\varpi (5\alpha +9)+4\pi (3\alpha +7)+(\varpi +4\pi )\right.\right.\\ & & \times \,\left.\left.(\alpha +1){c}_{3}{r}^{2}\right)\right\}+4{c}_{1}^{2}{c}_{2}\left\{4(\varpi -9\varpi \alpha -24\pi \alpha )+{c}_{3}{r}^{2}\left(98\varpi -38\varpi \alpha \right.\right.\\ & & +\ \left.\left.16\pi (\alpha +17)+{c}_{3}{r}^{2}\left(\varpi (17\alpha +105)+4\pi (79-9\alpha )-(\varpi +4\pi )(\alpha -11){c}_{3}{r}^{2}\right)\right)\right\}\\ & & +\ 5{c}_{2}^{3}{c}_{3}{r}^{4}\left\{{c}_{3}{r}^{2}\left(6\varpi (\alpha +15)+16\pi (3\alpha +10)+(\alpha +5){c}_{3}{r}^{2}\left(15\varpi +(\varpi +4\pi ){c}_{3}{r}^{2}+36\pi \right)\right)\right.\\ & & -\,4\left.\left.(\varpi (3\alpha -5)+8\pi \alpha )\right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\tilde{P}}_{r} & = & \displaystyle \frac{-\sqrt{{c}_{3}}}{8(\varpi +2\pi )(\varpi +4\pi )\left({c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\left({c}_{3}{r}^{2}+1\right)}^{2}}\\ & & \times \,\left[2(\alpha +1){c}_{1}\sqrt{{c}_{3}}\left\{(\varpi +4\pi ){c}_{3}{r}^{2}+8\pi \right\}-{c}_{2}\left\{4(3\varpi +8\pi )(\alpha +1)+2{c}_{3}{r}^{2}\right.\right.\\ & & \times \ \left.\left.(5\varpi +8\pi )(\alpha +1)-(\varpi +4\pi ){c}_{3}^{2}\left({r}^{2}+\alpha {r}^{4}\right)\right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}{\tilde{P}}_{\perp } & = & \displaystyle \frac{-\sqrt{{c}_{3}}}{4(\varpi +2\pi )(\varpi +4\pi )\left({c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right)}^{2}{\left({c}_{3}{r}^{2}+1\right)}^{3}}\\ & & \times \,\left[2{c}_{1}{c}_{2}^{2}\sqrt{{c}_{3}}{r}^{2}\left\{{c}_{3}{r}^{2}\left(8\pi (26\alpha -35)-2\varpi (21\alpha +110)+{c}_{3}{r}^{2}\left(68\varpi \alpha -65\varpi \right.\right.\right.\right.\\ & & +\ \left.\left.\left.168\pi \alpha +(\varpi +4\pi )(12\alpha +35){c}_{3}{r}^{2}+180\pi \right)\right)-4(3\varpi +8\pi )(7\alpha +10)\right\}\\ & & +\,8{c}_{1}^{3}\sqrt{{c}_{3}}\left\{8\pi (\alpha +1)+{c}_{3}{r}^{2}\left(2\varpi \alpha +\varpi +(\varpi +4\pi ){c}_{3}{r}^{2}+12\pi \right)\right\}+4{c}_{1}^{2}{c}_{2}\\ & & \times \,\left\{{c}_{3}{r}^{2}\left(-2\varpi (4\alpha +11)+8\pi (7\alpha +4)+{c}_{3}{r}^{2}\left(18\varpi \alpha +\varpi +11(\varpi +4\pi )\right.\right.\right.\\ & & \times \,\left.\left.\left.{c}_{3}{r}^{2}+108\pi \right)\right)-4(3\varpi +8\pi )(\alpha +1)\right\}+5{c}_{2}^{3}{c}_{3}{r}^{4}\left\{{c}_{3}{r}^{2}\left({c}_{3}{r}^{2}\left(24\pi \alpha -45\varpi \right.\right.\right.\\ & & +\,\left.\left.(\varpi +4\pi )(4\alpha +5){c}_{3}{r}^{2}-60\pi \right)-2(\varpi (23\alpha +55)+24\pi (\alpha +5))\right)\\ & & -\ \left.\left.4(3\varpi +8\pi )(4\alpha +5)\right\}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}\tilde{{\rm{\Pi }}} & = & \displaystyle \frac{\alpha {r}^{2}{c}_{3}}{8(\varpi +2\pi )(\varpi +4\pi )\left({c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right){\left(5{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}\right)}^{2}{\left({c}_{3}{r}^{2}+1\right)}^{3}}\\ & & \times \,\left[8{c}_{3}{c}_{1}^{3}\left\{12\pi -\varpi +(\varpi +4\pi ){c}_{3}{r}^{2}\right\}+4{c}_{2}\sqrt{{c}_{3}}{c}_{1}^{2}\left\{{c}_{3}{r}^{2}\left(11(\varpi +4\pi ){c}_{3}{r}^{2}-17\varpi +108\pi \right)\right.\right.\\ & & -\ \left.2(7\varpi +12\pi )\right\}+2{c}_{2}^{2}{c}_{1}\left\{{c}_{3}{r}^{2}\left({c}_{3}{r}^{2}\left(23(\varpi +4\pi ){c}_{3}{r}^{2}+12\pi -133\varpi \right)\right.\right.\\ & & -\,\left.\left.2(89\varpi +244\pi )\right)-12(3\varpi +8\pi )\right\}+5{c}_{2}^{3}\sqrt{{c}_{3}}{r}^{2}\left\{{c}_{3}{r}^{2}\right.\\ & & \times \,\left.\left.\left({c}_{3}{r}^{2}\left({c}_{3}{r}^{2}(\varpi +4\pi )-45\varpi -84\pi \right)-64(\varpi +3\pi )\right)-4(3\varpi +8\pi )\right\}\right].\end{array}\end{eqnarray}$
The deformation function is depicted in figure 6, showing a valid pattern, starting from null at the center, gradually increasing until reaching its peak somewhere between r = 4 and 5, and subsequently decreasing as it approaches the boundary. Similarly, in the same figure, the extended radial component (55 ) exhibited an increasing trend toward the spherical boundary. Figure 7 provides a clear representation of the required behavior, as discussed in section 5 , of the effective physical determinants (56 )–(58 ). These parameters play a crucial role in governing the interior of compact objects. Their behavior with respect to parameters α, ϖ, and Q mirrors that of model 1. Furthermore, there is no (maximum) anisotropy in principal pressure in the core (boundary), possessing increasing behavior throughout (last plot).
Figure 6. Deformation function ( |
Figure 7. Physical determinants (in km−2) versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 2. |
The mass function for the considered compact candidates based on model 2 is presented in figure 8. Notably, these findings are aligned with the observed mass measurements. Furthermore, we observed favorable trends of the surface redshift and compactness, both of which are indicative of an acceptable solution. Our investigation also considered energy bounds, specifically, $\tilde{\mu }-{\tilde{P}}_{r}+\tfrac{{q}^{2}}{4\pi {r}^{4}}\geqslant 0$ and $\tilde{\mu }-{\tilde{P}}_{\perp }\geqslant 0$, which are graphically represented in figure 9. These bounds exhibited profiles that supported the presence of the usual matter. Finally, figure 10 shows an assessment of the stability criteria, and all graphs ensure the stability of this model across various parametric choices.
Figure 8. Mass (in km), compactness, and surface redshift versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 2. |
Figure 9. Energy bounds (in km−2) versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 2. |
Figure 10. Stability analysis versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 2. |
6.3. Model 3
We now introduce the simplest equation of state [88] to establish a connection between the fluid sectors of a new gravitating source. This relationship serves as an additional constraint, enabling us to obtain a unique solution for the system (equations (23 )–(25 )). The equation of state is expressed as24 ), (25 ), and (60 ), we arrived at the following first-order differential equation:60 ), doing so results in a more complex second-order differential equation that may not have a straightforward solution. Substituting metric (28 ) into equation (61 ), the differential equation takes the form
$\begin{eqnarray}{{\rm{\Upsilon }}}_{1}^{1}(r)={\epsilon }_{1}{{\rm{\Upsilon }}}_{0}^{0}(r)+{\epsilon }_{2},\end{eqnarray}$
where we assumed ε1 and ε2 as arbitrary constants. It is worth noting that the choice of their values can significantly influence the resulting solution. Combining equations ( $\begin{eqnarray}\displaystyle \frac{{\epsilon }_{1}{{\rm{T}}}^{{\prime} }(r)}{r}-\left(\displaystyle \frac{{b}_{1}^{{\prime} }}{r}+\displaystyle \frac{1}{{r}^{2}}-\displaystyle \frac{{\epsilon }_{1}}{{r}^{2}}\right){\rm{T}}(r)+8\pi {\epsilon }_{2}=0.\end{eqnarray}$
Note that although the third factor ${{\rm{\Upsilon }}}_{2}^{2}$ can also be incorporated into equation ( $\begin{eqnarray}\begin{array}{l}{\rm{T}}(r)\left\{\displaystyle \frac{4\sqrt{{c}_{3}}{c}_{2}}{{c}_{2}\sqrt{{c}_{3}}{r}^{2}+2{c}_{1}}-\displaystyle \frac{{\epsilon }_{1}}{{r}^{2}}+\displaystyle \frac{1}{{r}^{2}}\right\}-\displaystyle \frac{{\epsilon }_{1}{{\rm{T}}}^{{\prime} }(r)}{r}-8\pi {\epsilon }_{2}=0.\end{array}\end{eqnarray}$
The preceding equation allowed us to explore two different approaches for solving it. Initially, exact integration provided us with a solution expressed in terms of hypergeometric functions. However, the complexity of these expressions poses a challenge when attempting to graphically visualize the corresponding physical quantities. To overcome this challenge, numerical integration has become a valuable tool when solving equation (62 ) for T(r). We initiated this numerical solution with the boundary condition T(r)∣r=0 = 0 and set values for the parameters ε1 = 0.7 and ε2 = 0.006. In figure 11, we present the deformation function whose variation with respect to ϖ revealed a continuous increase within the range of 0 to R. We also observed that the parameter ϖ has a negligible effect on the extended grr component, owing to constraint (60 ), which primarily relied on the additional sector. However, for all values of α, we observed a consistently increasing trend. We proceeded to interpret the corresponding matter triplet and anisotropy in figure 12, exhibiting acceptable behavior in relation with α, ϖ, and charge.
Figure 11. Deformation function and extended component versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 3. |
Figure 12. Physical determinants (in km−2) versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 3. |
Notably, as the anisotropic factor increased, it exerted an outward force that played a crucial role in maintaining the stability of our model. By examining figure 13, it became evident that the mass exhibited an outward increase, with the model being denser when α = 0.3 and ϖ = 0.1. Furthermore, the remaining factors were within their acceptable limits. As shown in figure 14, it is clear that ordinary matter must be present in the interior region, as the dominant energy conditions were met. Finally, figure 15 reveals that model 3 exhibited stability everywhere for all parameter values.
Figure 13. Mass (in km), compactness, and surface redshift versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 3. |
Figure 14. Energy bounds (in km−2) versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 3. |
Figure 15. Stability analysis versus r (in km) for α = 0.3 and ϖ = 0.1 (solid), α = 0.3 and ϖ = 0.9 (dotted–dashed), and α = 0.4 and ϖ = 0.9 (dashed) along with Q = 0.1 (left) and Q = 0.6 (right) corresponding to model 3. |
7. Conclusions
This study is dedicated to the development of three unique anisotropic extensions of the charged perfect Finch–Skea fluid solution using the gravitational decoupling technique within the framework of the $f({\mathbb{R}},{\mathbb{T}})={\mathbb{R}}+2\varpi {\mathbb{T}}$ gravity. To achieve this, we adopted a static spherical geometry to describe the interior distribution of a self-gravitating fluid. Our initial configuration was assumed to be isotropic, in which we introduced an additional Lagrangian density that acted as a new fluid source, inducing anisotropy in the initial source. When introducing an additional source, it is crucial to consider that both fluid sources are mutually interconnected through gravitational interactions. As a result, the field equations become intertwined, characterizing both interior distributions, leading to an increase in the unknown quantities. This leads to a systematic strategy for finding their solutions. Leveraging the MGD strategy, we applied a radial component transformation to the field equations, effectively separating them into two distinct sets characterizing their parent sources (the initial perfect and additional fluids), allowing us to solve them independently.
First, we addressed the three independent field equations relevant to the perfect fluid configuration. Notably, within the isotropic system, there were five unknowns to contend with. To solve these equations, we considered the Finch–Skea metric, which has the form23 )–(25 ), which included an additional source, introduced four more unknowns, encompassing a fluid triplet and a deformation function. We addressed these equations by imposing three distinct constraints on the ϒ-sector, each leading to a unique solution. Following the solution of both sets, we derived the fluid variables and assessed the degree of anisotropy through the relationships defined in equations (26 ) and (27 ).
$\begin{eqnarray*}{{\rm{e}}}^{{b}_{1}(r)}=\displaystyle \frac{1}{4}{\left(2{c}_{1}+{c}_{2}\sqrt{{c}_{3}}{r}^{2}\right)}^{2},\quad {{\rm{e}}}^{{b}_{2}(r)}={c}_{3}{r}^{2}+1,\end{eqnarray*}$
that required us to determine the values of the triplet (c1, c2, and c3). Furthermore, using a smooth connection between the interior metric and the Reissner–Nordström exterior spacetime, we computed these constants at the spherical interface. On a separate note, field equations (To ensure the physical validity of our models, we conducted a graphical analysis. Specifically, we explored different values for the parameters, such as ϖ = 0.1 and 0.9, α = 0.3 and 0.4, and Q = 0.1 and 0.6, to examine how they impact the interior fluid distribution. We observed that the fluid determinants remained acceptable across every choice of parameter. Focusing on the four specific candidates listed in table 1, we observed a consistent agreement between the calculated and observed masses of the considered stars. Our analysis also included an evaluation of energy conditions, which ensured the existence of ordinary matter within the interiors of the developed models. We also applied causality conditions and cracking criterion to the obtained solutions. Our findings indicate that all the resulting models are stable for all considered parameter choices. Additionally, models 2 and 3 are consistent with [89] and [90], respectively. Finally, when ϖ was set to zero, our results were reduced to ${\mathbb{GR}}$.