Effects of Charge and Gravitational Decoupling on Complexity and Isotropization of Anisotropic Models

General theory of relativity ($\mathrm{GR}$) has gained significant acceptance throughout the scientific community as it is considered the most desirable tool for understanding the nature of gravitational field. This theory relates the matter configuration with the geometry of a spacetime structure characterized by the energy-momentum tensor ($\mathrm{EMT}$) and the Einstein tensor, respectively through Einstein’s field equations. The exact/numerical formulation of their solutions has become an interesting topic among physicists through which they study the structural properties of self-gravitating systems. Schwarzschild pioneered this work by calculating the analytical solution of such equations that prompted researchers to extend this in the context of $\mathrm{GR}$ and other modified theories. For the first time in $1916$, he started with the assumption of a spherical geometry possessing uniform density and formulated the corresponding exterior [1] and interior solutions [2]. The literature offers a large number of solutions in recent times describing geometrical structures coupled with different fluid distributions.

Among all solutions of the field equations, those which can be used to model physically acceptable compact bodies have made remarkable achievements in the literature. However, it is rather difficult to calculate such solutions because the field equations involve multiple geometric terms that make them highly non-linear. Such non-linearity forced researchers to develop certain approaches which can be used to solve these equations and produce physically relevant results. The novel and innovative strategy among them is the gravitational decoupling which is empowered to tackle with several factors reigning the interior distribution such as anisotropy, dissipation flux, expansion scalar and shear, etc. This technique leads the field equations in a new reference frame where each fluid source is characterized by an individual system of equations, and hence, each set can be solved easily. Further, decoupling offers two different types, namely minimal ($\mathrm{MGD}$) and extended geometric deformations, transforming radial and both temporal/radial metric potentials, respectively.

Ovalle [3] recently pioneered the $\mathrm{MGD}$ scheme in the braneworld ($\mathrm{BW}$) scenario and discussed compact objects characterized by exact solutions. Following this, Ovalle and Linares [4] assumed Tolman IV spacetime as the isotropic solution and developed its corresponding anisotropic analog in the context of $\mathrm{BW}$ through the same technique. The study of different phenomena have been made possible through this technique, such as the derivation of solutions corresponding to spherical [5] interior, originating acceptable inner fluid solution [6, 7], studying microscopic properties of black holes [8, 9] and discussing the impact of Weyl stresses [10]. The perturbation was applied to the field equations in $\mathrm{BW}$ and bulk, and results were found compatible with each other [11]-[15]. The $\mathrm{MGD}$ technique was then extended by transforming both $g_{tt}$ and $g_{rr}$ metric coefficients of a static sphere to obtain a modified form of the Schwarzschild geometry in $\mathrm{BW}$ [16].

Initially, a self-gravitating geometry was considered to be coupled with the isotropic fluid in which the pressure acts the same in each direction, usually radial and tangential. However, the pioneering work by Jeans made some revolutions in the literature, suggesting that anisotropy may occur in the interior due to the presence of multiple factors [17]. A theoretical analysis has been done by Ruderman [18] from which he found that the heavily stellar models (i.e., with energy density not less than $10^{15}~{}g/cm^{3}$) may possess anisotropy. A compact star surrounded by a high magnetic field [19]-[22] as well as certain other elements [23]-[25] must contain anisotropy in its interior. Ovalle et al. [26] considered that a spherical fluid and Schwarzschild vacuum spacetime do not exchange energy and momentum with each other, and developed a new anisotropic solution by using $\mathrm{MGD}$ approach. Anisotropic extensions to different isotropic models such as Heintzmann, Duragpal-Fuloria and Tolman VII spacetimes have also been obtained [27]-[29]. Sharif and his collaborators [30, 31] have done the same analysis for charged/uncharged spacetimes in $\mathrm{GR}$ as well as different extended theories and found acceptable results. We have developed physically relevant anisotropic compact models in a non-minimally coupled gravity for certain values of the corresponding parameters [32].

The inclusion of the electric charge has a significant role in the study of compact structures. Electromagnetic forces are considered as the most significant ingredients that help to to reduce the attractive force of gravity. Therefore, a sufficiently enough amount of the charge is needed to resist the gravitational force and hence, the stability of a star is preserved. Bekenstein [33] considered a dynamical spherical model coupled with the electromagnetic field and deduced that the force produced by the charge has repulsive behavior, helping the geometry to maintain its stability. This work has been extended by Esculpi and Aloma [34] for the case of anisotropic fluid from which they observed that both positive anisotropy and charge have repulsive effects. The structure of anisotropic charged stars has been studied by taking a power-law interior charge distribution in terms of exterior charge and radius of the considered compact object, and physically relevant results were obtained [35, 36]. Maurya et al. [37] analyzed the interior distribution of charged stellar models configured with the baryonic fluid via the Karmarkar condition. They solved the corresponding Einstein-Maxwell field equations and observed the resulting matter variables to be dependent on the electromagnetic field.

Herrera [38] recently developed a new definition of the complexity for a static sphere that encounters the factors which complicate the system. Initially, Bel decomposed the curvature tensor into its orthogonal components and found certain scalars, named the structure scalars. Herrera adopted the same formulation for anisotropic fluid to obtain the corresponding scalars that were appeared to be associated with different physical parameters. One of those scalars possesses both anisotropy and inhomogeneous density, came up with his criteria and thus named as the complexity factor. Such a definition has been observed highly acceptable in every field of science. Herrera et al. [39] used this definition to discuss the evolutionary patterns for dynamical dissipative geometry. Yousaf et al. [40, 41] performed the same analysis in a particular non-minimal theory and examined the impact of modified gravity on charged/uncharged spheres and cylinders. The complexity-free models can be obtained once we substitute the corresponding complexity factor equals to zero. This constraint has been employed to static self-gravitating spheres, and several physically relevant objects are modeled [42]-[46].

This article extends the anisotropic spherical solutions to the Einstein-Maxwell framework through the $\mathrm{MGD}$ technique. To do this work, we follow the structure defined in the succeeding lines. Section 2 introduces the influence of charge in Einstein’s field equations for a static spherical interior possessing two sources, i.e., initial anisotropic and newly added fluids. A well-known geometric transformation is applied on the formulated field equations in section 3, resulting into two sets. Different constraints on both systems of equations are employed independently and find the corresponding solutions in sections 4 and 5. Section 6 discusses the influence of the decoupling technique and charge on physical attributes of the resulting models. On a final note, we summarize all our results in section 7.

We present the field equations in this section that describe a static spherical spacetime influenced by an electromagnetic field. The spherical geometry $(t,r,\theta,\vartheta)$ is considered to split into two sectors, namely exterior and interior regions over the hypersurface $\Sigma$. In order to perform our analysis, the interior spacetime is given by the line element as

$$ds^{2}=-e^{\xi_{1}}dt^{2}+e^{\xi_{2}}dr^{2}+r^{2}d\Omega^{2},$$

(1)

where $\xi_{1}=\xi_{1}(r),~{}\xi_{2}=\xi_{2}(r)$ and $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\vartheta^{2}$. The presence of Lagrangian corresponding to the additional source and electromagnetic field in the Einstein-Hilbert action produces the field equations given by

$$\mathrm{G}_{\lambda\chi}=\mathrm{R}_{\lambda\chi}-\frac{1}{2}\mathrm{R}g_{\lambda\chi}=8\pi\bar{\mathrm{T}}_{\lambda\chi},$$

(2)

with

$$\bar{\mathrm{T}}_{\lambda\chi}=\mathrm{T}_{\lambda\chi}+\mathrm{E}_{\lambda\chi}+\omega\mathrm{Z}_{\lambda\chi},$$

(3)

where the quantities are expressed as

• •

  $\mathrm{G}_{\lambda\chi}$ is the Einstein tensor that characterizes the geometry,

• •

  $\mathrm{R}_{\lambda\chi},~{}\mathrm{R}$ and $g_{\lambda\chi}$ indicate the Ricci tensor, Ricci scalar and the metric tensor, respectively,

• •

  $\mathrm{T}_{\lambda\chi}$ is the usual matter $\mathrm{EMT}$,

• •

  $\mathrm{E}_{\lambda\chi}$ is the electromagnetic $\mathrm{EMT}$,

• •

  $\mathrm{Z}_{\lambda\chi}$ is the additional matter source gravitationally associated with $\mathrm{T}_{\lambda\chi}$.

Since our aim is to study the complexity of a self-gravitating system, we must assume anisotropy in the interior initially whose $\mathrm{EMT}$ is defined by

$$\mathrm{T}_{\lambda\chi}=(P_{t}+\rho)\mathrm{W}_{\lambda}\mathrm{W}_{\chi}+P_{t}g_{\lambda\chi}-\left(P_{t}-P_{r}\right)\mathrm{X}_{\lambda}\mathrm{X}_{\chi},$$

(4)

where $P_{r},~{}P_{t},~{}\rho,~{}\mathrm{W}_{\chi}$ and $\mathrm{X}_{\chi}$ are the radial/tangential pressure elements, energy density, four-velocity and the four-vector, respectively. We consider a co-moving frame of reference that gives rise to the following quantities in accordance with Eqs.(1) and (4) as

$$\mathrm{X}^{\chi}=(0,e^{\frac{-\xi_{2}}{2}},0,0),\quad\mathrm{W}^{\chi}=(e^{\frac{-\xi_{1}}{2}},0,0,0),$$

fulfilling the relations

$$\mathrm{X}^{\chi}\mathrm{W}_{\chi}=0,\quad\mathrm{X}^{\chi}\mathrm{X}_{\chi}=1,\quad\mathrm{W}^{\chi}\mathrm{W}_{\chi}=-1.$$

The electromagnetic field characterized by the $\mathrm{EMT}$ is expressed as

$$\mathrm{E}_{\lambda\chi}=-\frac{1}{4\pi}\left[\frac{1}{4}g_{\lambda\chi}\mathrm{F}^{\zeta\beta}\mathrm{F}_{\zeta\beta}-\mathrm{F}^{\zeta}_{\chi}\mathrm{F}_{\lambda\zeta}\right].$$

(5)

Here, the Maxwell field tensor is represented by $\mathrm{F}_{\zeta\beta}=\Psi_{\beta;\zeta}-\Psi_{\zeta;\beta}$ with the four-potential $\Psi_{\zeta}=\Psi(r)\delta^{0}_{\zeta}$. The Maxwell equations can concisely be written as follows

$$\mathrm{F}^{\zeta\beta}_{;\beta}=4\pi\jmath^{\zeta},\quad\mathrm{F}_{[\zeta\beta;\nu]}=0,$$

with $\jmath^{\zeta}=\varpi\mathrm{W}^{\zeta}$ and $\varpi$ being the current and charge densities, respectively. The left side of the above equations yields in this framework as

$$\Psi^{\prime\prime}+\frac{1}{2r}\big{\{}4-r(\xi_{1}^{\prime}+\xi_{2}^{\prime})\big{\}}\Psi^{\prime}=4\pi\varpi e^{\frac{\xi_{1}}{2}+\xi_{2}},$$

where ${}^{\prime}=\frac{\partial}{\partial r}$. Integrating the above equation, we have

$$\Psi^{\prime}=\frac{\mathrm{q}}{r^{2}}e^{\frac{\xi_{1}+\xi_{2}}{2}},$$

with $\mathrm{q}=\int_{0}^{r}\varpi e^{\frac{\xi_{2}}{2}}\bar{y}^{2}d\bar{y}$ being the total charge in the interior. The non-vanishing components of the $\mathrm{EMT}s$ (4) and (5) are

	$\displaystyle\mathrm{T}_{00}=\rho e^{\xi_{1}},\quad\mathrm{T}_{11}=P_{r}e^{\xi_{2}},\quad\mathrm{T}_{22}=P_{t}r^{2}=\frac{\mathrm{T}_{33}}{\sin^{2}\theta},$
	$\displaystyle\mathrm{E}_{00}=\frac{\mathrm{q}^{2}e^{\xi_{1}}}{8\pi r^{4}},\quad\mathrm{E}_{11}=-\frac{\mathrm{q}^{2}e^{\xi_{2}}}{8\pi r^{4}},\quad\mathrm{E}_{22}=\frac{\mathrm{q}^{2}}{8\pi r^{2}}=\frac{\mathrm{E}_{33}}{\sin^{2}\theta}.$

The three independent field equations corresponding to a sphere (1) are obtained from Eq.(2) as

	$\displaystyle 8\pi\rho+\frac{\mathrm{q}^{2}}{r^{4}}-8\pi\omega\mathrm{Z}^{0}_{0}$	$\displaystyle=\frac{1}{r^{2}}-e^{-\xi_{2}}\left(\frac{1}{r^{2}}-\frac{\xi_{2}^{\prime}}{r}\right),$		(6)
	$\displaystyle 8\pi{P}_{r}-\frac{\mathrm{q}^{2}}{r^{4}}+8\pi\omega\mathrm{Z}^{1}_{1}$	$\displaystyle=e^{-\xi_{2}}\left(\frac{1}{r^{2}}+\frac{\xi_{1}^{\prime}}{r}\right)-\frac{1}{r^{2}},$		(7)
	$\displaystyle 8\pi{P}_{t}+\frac{\mathrm{q}^{2}}{r^{4}}+8\pi\omega\mathrm{Z}^{2}_{2}$	$\displaystyle=\frac{e^{-\xi_{2}}}{4}\left[\xi_{1}^{\prime 2}-\xi_{1}^{\prime}\xi_{2}^{\prime}+2\xi_{1}^{\prime\prime}-\frac{2\xi_{2}^{\prime}}{r}+\frac{2\xi_{1}}{r}\right].$		(8)

The conservation equation in this case can be obtained by taking the covariant divergence of the total fluid (seed, charge and additional), i.e., $\nabla^{\chi}(\mathrm{T}_{\lambda\chi}+\mathrm{E}_{\lambda\chi}+\omega\mathrm{Z}_{\lambda\chi})=0$ as

		$\displaystyle\frac{dP_{r}}{dr}+\frac{\xi_{1}^{\prime}}{2}\big{(}\rho+P_{r}\big{)}+\frac{\omega\xi_{1}^{\prime}}{2}\big{(}\mathrm{Z}_{1}^{1}-\mathrm{Z}_{0}^{0}\big{)}+\frac{2}{r}\big{(}P_{r}-P_{t}\big{)}$
		$\displaystyle+\omega\frac{d\mathrm{Z}_{1}^{1}}{dr}+\frac{2\omega}{r}\big{(}\mathrm{Z}_{1}^{1}-\mathrm{Z}_{2}^{2}\big{)}-\frac{\mathrm{qq}^{\prime}}{4\pi r^{4}}=0.$		(9)

The above condition must hold for the system to be in a stable equilibrium, called the Tolman-Opphenheimer-Volkoff (TOV) equation. The mass function can be specified in terms of geometry as well as matter distribution. The geometric definition is given by

$\displaystyle\mathrm{m}(r)=\frac{r}{2}\bigg{(}1-\frac{1}{e^{\xi_{2}}}+\frac{\mathrm{q}^{2}}{r^{2}}\bigg{)}.$

(10)

On the other hand, the mass in relation with the energy density and charge can be obtained through Eqs.(6) and (10) as

	$\displaystyle\mathrm{m}(r)$	$\displaystyle=4\pi\int_{0}^{r}\rho\bar{y}^{2}d\bar{y}+\int_{0}^{r}\frac{\mathrm{qq}^{\prime}}{\bar{y}}d\bar{y}+4\pi\omega\int_{0}^{r}\mathrm{Z}_{0}^{0}\bar{y}^{2}d\bar{y}$
		$\displaystyle=4\pi\int_{0}^{r}\rho\bar{y}^{2}d\bar{y}+\frac{1}{2}\int_{0}^{r}\frac{\mathrm{q}^{2}}{\bar{y}^{2}}d\bar{y}+\frac{\mathrm{q}^{2}}{2r}+4\pi\omega\int_{0}^{r}\mathrm{Z}_{0}^{0}\bar{y}^{2}d\bar{y},$		(11)

where the first three terms correspond to the mass of charged fluid distribution and the last term defines the mass related to the new source. We use Eq.(7) to determine the value of $\xi_{1}^{\prime}$ in terms of the mass function (10) as

$\displaystyle\xi_{1}^{\prime}=\frac{2\big{\{}4\pi\big{(}P_{r}+\omega\mathrm{Z}^{1}_{1}\big{)}r^{4}+\mathrm{m}r-\mathrm{q}^{2}\big{\}}}{r\big{(}r^{2}-2\mathrm{m}r+\mathrm{q}^{2}\big{)}}.$

(12)

Switching the above value into Eq.(9), we get

		$\displaystyle\frac{dP_{r}}{dr}+\bigg{[}\frac{4\pi\big{(}P_{r}+\omega\mathrm{Z}^{1}_{1}\big{)}r^{4}+\mathrm{m}r-\mathrm{q}^{2}}{r\big{(}r^{2}-2\mathrm{m}r+\mathrm{q}^{2}\big{)}}\bigg{]}\big{\{}\rho+P_{r}+\omega\big{(}\mathrm{Z}_{1}^{1}-\mathrm{Z}_{0}^{0}\big{)}\big{\}}$
		$\displaystyle+\frac{2}{r}\big{(}P_{r}-P_{t}\big{)}+\omega\frac{d\mathrm{Z}_{1}^{1}}{dr}+\frac{2\omega}{r}\big{(}\mathrm{Z}_{1}^{1}-\mathrm{Z}_{2}^{2}\big{)}-\frac{\mathrm{qq}^{\prime}}{4\pi r^{4}}=0,$		(13)

where $\Pi=P_{t}-P_{r}$ and $\Pi_{\mathrm{Z}}=\mathrm{Z}_{2}^{2}-\mathrm{Z}_{1}^{1}$ are the anisotropic factors corresponding to the seed and additional fluid sources, respectively.

Since we include an additional source in the initial anisotropic fluid, the corresponding field equations now become difficult to solve due to the increment of unknowns, i.e., $(\xi_{1},\xi_{2},\mathrm{q},\rho,P_{t},P_{r},\mathrm{Z}_{0}^{0},\mathrm{Z}_{1}^{1},\mathrm{Z}_{2}^{2})$. Thus, we need to adopt certain approach or constraints and reduce the degrees of freedom to get an exact solution. On that note, we start off with a systematic technique (referred to the gravitational decoupling [26]) whose implementation on the field equations makes it possible to find their solution. The exciting feature of this strategy is that it transforms the temporal/radial metric potentials to a new frame of reference and make the set of equations easy to handle. For this, we consider the following metric as a solution to Eqs.(6)-(8) given by

$$ds^{2}=-e^{\xi_{3}(r)}dt^{2}+\frac{1}{\xi_{4}(r)}dr^{2}+r^{2}d\Omega^{2}.$$

(14)

In this context, the metric components linearly transform as

$$\xi_{3}\rightarrow\xi_{1}=\xi_{3}+\omega\mathrm{f},\quad\xi_{4}\rightarrow e^{-\xi_{2}}=\xi_{4}+\omega\mathrm{t},$$

(15)

along with the temporal $\mathrm{f}$ and radial $\mathrm{t}$ deformation functions.

We choose the $\mathrm{MGD}$ scheme, thus only the $g_{rr}$ component is deformed in the following, while keeping $g_{tt}$ potential remains unchanged, i.e., $\mathrm{t}\rightarrow\bar{\mathrm{t}},~{}\mathrm{f}\rightarrow 0$. Equation (15) now turns into

$$\xi_{3}\rightarrow\xi_{1}=\xi_{3},\quad\xi_{4}\rightarrow e^{-\xi_{2}}=\xi_{4}+\omega\bar{\mathrm{t}},$$

(16)

where $\bar{\mathrm{t}}=\bar{\mathrm{t}}(r)$. It is important to note that the spherical symmetry remains preserved by these linear mappings. We implement the transformation (16) on Eqs.(6)-(8) and obtain two systems. The first set portraying the seed fluid source is acquired for $\omega=0$ as

	$\displaystyle 8\pi\rho+\frac{\mathrm{q}^{2}}{r^{4}}$	$\displaystyle=e^{-\xi_{2}}\bigg{(}\frac{\xi_{2}^{\prime}}{r}-\frac{1}{r^{2}}\bigg{)}+\frac{1}{r^{2}},$		(17)
	$\displaystyle 8\pi{P}_{r}-\frac{\mathrm{q}^{2}}{r^{4}}$	$\displaystyle=e^{-\xi_{2}}\bigg{(}\frac{1}{r^{2}}+\frac{\xi_{1}^{\prime}}{r}\bigg{)}-\frac{1}{r^{2}},$		(18)
	$\displaystyle 8\pi{P}_{t}+\frac{\mathrm{q}^{2}}{r^{4}}$	$\displaystyle=\frac{e^{-\xi_{2}}}{4}\bigg{(}\xi_{1}^{\prime 2}-\xi_{2}^{\prime}\xi_{1}^{\prime}+2\xi_{1}^{\prime\prime}-\frac{2\xi_{2}^{\prime}}{r}+\frac{2\xi_{1}^{\prime}}{r}\bigg{)}.$		(19)

Furthermore, the impact of the newly added source $\mathrm{Z}_{\lambda\chi}$ is encoded by the following set and can obtained for $\omega=1$ as

		$\displaystyle 8\pi\mathrm{Z}_{0}^{0}=\frac{1}{r}\bigg{(}\bar{\mathrm{t}}^{\prime}+\frac{\bar{\mathrm{t}}}{r}\bigg{)},$		(20)
		$\displaystyle 8\pi\mathrm{Z}_{1}^{1}=\frac{\bar{\mathrm{t}}}{r}\bigg{(}\xi_{1}^{\prime}+\frac{1}{r}\bigg{)},$		(21)
		$\displaystyle 8\pi\mathrm{Z}_{2}^{2}=\frac{\bar{\mathrm{t}}}{4}\bigg{(}2\xi_{1}^{\prime\prime}+\xi_{1}^{\prime 2}+\frac{2\xi_{1}^{\prime}}{r}\bigg{)}+\frac{\bar{\mathrm{t}}^{\prime}}{2}\bigg{(}\frac{\xi_{1}^{\prime}}{2}+\frac{1}{r}\bigg{)}.$		(22)

Since we have used the $\mathrm{MGD}$ scheme, both matter sources must be conserved individually because the exchange of energy is not permitted in this case. The following two equations confirm the conservation of these sources as

		$\displaystyle\frac{dP_{r}}{dr}+\frac{\xi_{1}^{\prime}}{2}\big{(}\rho+P_{r}\big{)}+\frac{2}{r}\big{(}P_{r}-P_{t}\big{)}-\frac{\mathrm{qq}^{\prime}}{4\pi r^{4}}=0,$		(23)
		$\displaystyle\frac{d\mathrm{Z}_{1}^{1}}{dr}+\frac{\xi_{1}^{\prime}}{2}\big{(}\mathrm{Z}_{1}^{1}-\mathrm{Z}_{0}^{0}\big{)}+\frac{2}{r}\big{(}\mathrm{Z}_{1}^{1}-\mathrm{Z}_{2}^{2}\big{)}=0.$		(24)

We observe that the system (17)-(19) possesses six unknown quantities ($\rho,P_{r},\\ P_{t},\mathrm{q},\xi_{1},\xi_{2}$), therefore, we need to choose three of them freely to calculate the required analytical solution. Further, there are four unknowns ($\bar{\mathrm{t}},\mathrm{Z}_{0}^{0},\mathrm{Z}_{1}^{1},\mathrm{Z}_{2}^{2}$) in the second set (20)-(22). We shall adopt a constraint on $\mathrm{Z}$-sector to tackle with the second system. We detect the effective forms of the physical determinants as

	$\displaystyle\bar{\rho}$	$\displaystyle=\rho-\omega\mathrm{Z}_{0}^{0},$		(25)
	$\displaystyle\bar{P}_{r}$	$\displaystyle=P_{r}+\omega\mathrm{Z}_{1}^{1},$		(26)
	$\displaystyle\bar{P}_{t}$	$\displaystyle=P_{t}+\omega\mathrm{Z}_{2}^{2},$		(27)

with the total anisotropy given by

$$\bar{\Pi}=\bar{P}_{t}-\bar{P}_{r}=(P_{t}-P_{r})+\omega(\mathrm{Z}_{2}^{2}-\mathrm{Z}_{1}^{1})=\Pi+\Pi_{\mathrm{Z}}.$$

(28)

It must be mentioned that the presence of positive or negative anisotropy can significantly influence the stability of a compact star. The positive anisotropy (when radial pressure is less than the tangential component) can increase the stability of a compact star as it produces outward-directed pressure. This pressure provides a support against the gravitational attraction and prevents a star from collapse. On the other hand, the negative anisotropy (when radial pressure is greater than the tangential component) can lead to instability of the star as the pressure in the outward direction is not produced. Thus, such a star remains stable for a short period of time as compared to that possessing positive anisotropy. Further, the TOV equation (9) can be written as

$\displaystyle\frac{d\bar{P}_{r}}{dr}+\frac{\xi_{1}^{\prime}}{2}\big{(}\bar{\rho}+\bar{P}_{r}\big{)}+\frac{2}{r}\big{(}\bar{P}_{r}-\bar{P}_{t}\big{)}-\frac{\mathrm{qq}^{\prime}}{4\pi r^{4}}=0.$

(29)

which is a combination of different forces. The four terms on the left side of the above equation represent hydrostatic ($f_{h}$), gravitational ($f_{g}$), anisotropic ($f_{a}$) and electromagnetic ($f_{e}$) forces, respectively that must be satisfied to maintain hydrostatic equilibrium of a self-gravitating object. The concise notation of Eq.(29) is

$\displaystyle f_{h}+f_{a}+f_{w}=0,$

(30)

where $f_{w}=f_{g}+f_{e}$.

The anisotropy triggered in a self-gravitating system due to the original charged fluid source is entirely different from the anisotropic factor produced by the total configuration (seed and additional sources). In this section, we provide a brief study that how the anisotropic interior can be converted into the isotropic analog. In other words, we find the conditions under which the considered matter distribution becomes isotropic, i.e., $\bar{\Pi}=0$. Following lines show such structural conversion is being controlled by the decoupling parameter. We observe that the vanishing decoupling parameter ($\omega=0$) leads to the anisotropic system, while $\omega=1$ corresponds to the isotropic framework. Since we are interested in discussing the second case, thus Eq.(28) yields

$$\Pi_{\mathrm{Z}}=-\Pi\quad\Rightarrow\quad\mathrm{Z}_{2}^{2}-\mathrm{Z}_{1}^{1}=P_{r}-P_{t}.$$

(31)

The construction of minimally/extended decoupled isotropic interiors from being anisotropic have been done by taking the above condition into account [43, 47].

We now assume a specific metric ansatz to deal with the extra degrees of freedom in the system (17)-(19) defined by

	$\displaystyle\xi_{1}(r)$	$\displaystyle=\ln\bigg{\{}\mathrm{C}_{2}^{2}\bigg{(}1+\frac{r^{2}}{\mathrm{C}_{1}^{2}}\bigg{)}\bigg{\}},$		(32)
	$\displaystyle\xi_{4}(r)$	$\displaystyle=e^{-\xi_{2}}=\frac{\mathrm{C}_{1}^{2}+r^{2}}{\mathrm{C}_{1}^{2}+3r^{2}},$		(33)

whose substitution makes the matter triplet as

	$\displaystyle\rho$	$\displaystyle=\frac{6r^{4}\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}-\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}\mathrm{q}^{2}}{8\pi r^{4}\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}},$		(34)
	$\displaystyle P_{r}$	$\displaystyle=\frac{\mathrm{q}^{2}}{8\pi r^{4}},$		(35)
	$\displaystyle P_{t}$	$\displaystyle=\frac{3r^{6}-\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}\mathrm{q}^{2}}{8\pi r^{4}\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}},$		(36)

with $\mathrm{C}_{1}^{2}$ and $\mathrm{C}_{2}^{2}$ being unknown functions and the junction conditions are used in the following to make them known. It is important to stress that the metric ansatz (32) and (33) correspond only to the tangential pressure in the uncharged case while the radial component disappears [43]. However, they both appear in the current scenario due to the presence of an electromagnetic field. The same metric potentials have also been employed in the study of circular-like motion of different particles in their field of gravitation [48].

Junction conditions are a subject of great discussion of all time for astrophysicists which assist the study of multiple physical factors in a self-gravitating interior at the hypersurface, i.e., $\Sigma:r=\mathcal{R}$. The smooth matching requires an interior and exterior metrics representing the corresponding spacetime regions of the considered geometry. Since we have already defined the interior metric in Eq.(1), a suitable exterior geometry in this regard is the Reissner-Nordström line element (a solution to the charged vacuum spacetime) given by

$$ds^{2}=-\left(1-\frac{2\mathcal{M}}{r}+\frac{\mathcal{Q}^{2}}{r^{2}}\right)dt^{2}+\frac{1}{\left(1-\frac{2\mathcal{M}}{r}+\frac{\mathcal{Q}^{2}}{r^{2}}\right)}dr^{2}+r^{2}d\Omega^{2},$$

(37)

with $\mathcal{Q}$ is the total charge and $\mathcal{M}$ symbolizes the corresponding mass. We now obtain the two constants ($\mathrm{C}_{1}^{2},~{}\mathrm{C}_{2}^{2}$) by equating the temporal/radial metrics components of the metrics (1) and (37) as

	$\displaystyle\mathrm{C}_{1}^{2}$	$\displaystyle=$	$\displaystyle\frac{\mathcal{R}^{2}\big{(}2\mathcal{R}^{2}+3\mathcal{Q}^{2}-6\mathcal{M}\mathcal{R}\big{)}}{2\mathcal{M}\mathcal{R}-\mathcal{Q}^{2}},$		(38)
	$\displaystyle\mathrm{C}_{2}^{2}$	$\displaystyle=$	$\displaystyle\frac{2\mathcal{R}^{2}+3\mathcal{Q}^{2}-6\mathcal{M}\mathcal{R}}{2\mathcal{R}^{2}}.$		(39)

The radius ($\mathcal{R}=9.1\pm 0.4~{}km$) and mass ($\mathcal{M}=1.58\pm 0.06$ times the sun’s mass) of a specific compact model $4U~{}1820-30$ is considered to plot the resulting solutions in the following sections [49]. Further, we observe that one extra unknown is still present in the system (34)-(36), thus $\mathrm{q}^{2}(r)=\xi_{5}r^{6}$ is assumed with $\xi_{5}$ as a constant [35]. Joining this with the restriction (31) and field equations, a differential equation is obtained as

		$\displaystyle r\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}\big{[}2r^{3}\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}\big{(}2\xi_{5}\mathrm{C}_{1}^{4}+12\xi_{5}\mathrm{C}_{1}^{2}r^{2}+18\xi_{5}r^{4}-3\big{)}-\big{(}\mathrm{C}_{1}^{2}+2r^{2}\big{)}$
		$\displaystyle\times\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}\bar{\mathrm{t}}^{\prime}(r)\big{]}+2\big{(}\mathrm{C}_{1}^{4}+2\mathrm{C}_{1}^{2}r^{2}+2r^{4}\big{)}\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}\bar{\mathrm{t}}(r)=0,$		(40)

whose exact solution provides $\bar{\mathrm{t}}(r)$ as

$$\bar{\mathrm{t}}(r)=\frac{r^{2}\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}}{\mathrm{C}_{1}^{2}+2r^{2}}\bigg{\{}\mathrm{D}_{1}+2\xi_{5}r^{2}+\frac{2\xi_{5}\mathrm{C}_{1}^{2}}{3}+\frac{1}{\mathrm{C}_{1}^{2}+3r^{2}}\bigg{\}},$$

(41)

with $\mathrm{D}_{1}$ as an integration constant whose dimension is $\frac{1}{\ell^{2}}$. It is well-known that the radial pressure vanishes at the spherical junction, i.e., $\bar{P}_{r}(\mathcal{R})=0$. Hence, Eq.(26) along with (35) and (41) provides $\mathrm{D}_{1}$ as

$$\mathrm{D}_{1}=-2\xi_{5}\mathcal{R}^{2}-\frac{2\xi_{5}\mathrm{C}_{1}^{2}}{3}-\frac{1}{\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}}-\frac{\xi_{5}\mathcal{R}^{2}\big{(}\mathrm{C}_{1}^{2}+2\mathcal{R}^{2}\big{)}}{\omega\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}.$$

(42)

Equation (41) now takes the form

$$\bar{\mathrm{t}}(r)=\frac{r^{2}\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}}{\mathrm{C}_{1}^{2}+2r^{2}}\bigg{\{}2\xi_{5}\big{(}r^{2}-\mathcal{R}^{2}\big{)}-\frac{3\big{(}r^{2}-\mathcal{R}^{2}\big{)}}{\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}-\frac{\xi_{5}\mathcal{R}^{2}\big{(}\mathrm{C}_{1}^{2}+2\mathcal{R}^{2}\big{)}}{\omega\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}\bigg{\}},$$

(43)

and the deformed $g_{rr}$ metric component becomes

	$\displaystyle e^{\xi_{2}}=\xi_{4}^{-1}$	$\displaystyle=\big{[}r^{2}\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}\big{\{}-\xi_{5}\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}\big{(}\mathcal{R}^{2}(2\omega+1)-2r^{2}\omega\big{)}+2\mathcal{R}^{2}$
		$\displaystyle\times\big{(}\mathcal{R}^{2}(3\omega+1)-3r^{2}\omega\big{)}\big{)}-3\omega\big{(}r^{2}-\mathcal{R}^{2}\big{)}\big{\}}+\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+2r^{2}\big{)}$
		$\displaystyle\times\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}\big{]}^{-1}\big{[}\big{(}\mathrm{C}_{1}^{2}+2r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}\big{]}.$		(44)

Finally, the decoupled solution to the system (6)-(8) is expressed by the line element as follows

$$ds^{2}=-\mathrm{C}_{2}^{2}\bigg{(}1+\frac{r^{2}}{\mathrm{C}_{1}^{2}}\bigg{)}dt^{2}+\frac{\mathrm{C}_{1}^{2}+3r^{2}}{\mathrm{C}_{1}^{2}+r^{2}+\omega\bar{\mathrm{t}}(r)\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}}dr^{2}+r^{2}d\Omega^{2},$$

(45)

along with the matter triplet given by

	$\displaystyle\bar{\rho}$	$\displaystyle=\frac{\mathrm{C}_{1}^{2}\big{(}6-6\xi_{5}r^{4}\big{)}-\mathrm{C}_{1}^{4}\xi_{5}r^{2}-9\xi_{5}r^{6}+6r^{2}}{8\pi\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}}-\big{[}8\pi r^{2}\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}^{2}\big{\{}\mathrm{C}_{1}^{4}\big{(}\xi_{5}r^{2}$
		$\displaystyle\times\big{(}2r^{2}\omega-\mathcal{R}^{2}(2\omega+1)\big{)}+1\big{)}+\mathrm{C}_{1}^{2}\big{(}2r^{2}+\xi_{5}r^{2}\big{(}6r^{4}\omega-3r^{2}\mathcal{R}^{2}-2\mathcal{R}^{4}$
		$\displaystyle\times(3\omega+1)\big{)}+3\mathcal{R}^{2}\big{)}+3r^{2}\big{(}2\xi_{5}r^{2}\mathcal{R}^{2}\big{(}3r^{2}\omega-\mathcal{R}^{2}(3\omega+1)\big{)}-\omega r^{2}+\mathcal{R}^{2}$
		$\displaystyle\times(\omega+2)\big{)}\big{\}}^{2}\big{]}^{-1}\big{[}\omega\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}\big{\{}\mathrm{C}_{1}^{10}\big{(}\xi_{5}r^{2}\big{(}\mathcal{R}^{2}(2\omega+1)-6r^{2}\omega\big{)}+1\big{)}$
		$\displaystyle+\mathrm{C}_{1}^{8}\big{(}3\big{(}4r^{2}+\mathcal{R}^{2}\big{)}+\xi_{5}r^{2}\big{(}-50r^{4}\omega+r^{2}\mathcal{R}^{2}(7-4\omega)+2\mathcal{R}^{4}(3\omega+1)\big{)}\big{)}$
		$\displaystyle+\mathrm{C}_{1}^{6}r^{2}\big{(}r^{2}(9\omega+39)+\xi_{5}r^{2}\big{(}r^{2}\mathcal{R}^{2}(17-116\omega)+14\mathcal{R}^{4}(3\omega+1)-150r^{4}\omega\big{)}$
		$\displaystyle-3\mathcal{R}^{2}(\omega-12)\big{)}+\mathrm{C}_{1}^{4}r^{4}\big{(}r^{2}(30\omega+44)+\xi_{5}r^{2}\big{(}3r^{2}\mathcal{R}^{2}(7-136\omega)-198r^{4}\omega$
		$\displaystyle+34\mathcal{R}^{4}(3\omega+1)\big{)}+3\mathcal{R}^{2}(2\omega+39)\big{)}+3\mathrm{C}_{1}^{2}r^{6}\big{(}r^{2}(9\omega+4)+2\xi_{5}r^{2}\big{(}-18r^{4}\omega$
		$\displaystyle+3r^{2}\mathcal{R}^{2}(1-31\omega)+7\mathcal{R}^{4}(3\omega+1)\big{)}+\mathcal{R}^{2}(13\omega+44)\big{)}+18r^{8}\big{(}2\xi_{5}r^{2}\mathcal{R}^{2}\big{(}\mathcal{R}^{2}$
		$\displaystyle\times(3\omega+1)-9r^{2}\omega\big{)}+r^{2}\omega+\mathcal{R}^{2}(\omega+2)\big{)}\big{\}}\big{]},$		(46)
	$\displaystyle\bar{P}_{r}$	$\displaystyle=\frac{1}{8\pi}\bigg{[}\xi_{5}r^{2}+\omega\bigg{(}\frac{\mathrm{C}_{1}^{2}+3r^{2}}{\mathrm{C}_{1}^{2}+2r^{2}}\bigg{)}\bigg{\{}2\xi_{5}\big{(}r^{2}-\mathcal{R}^{2}\big{)}-\frac{3\big{(}r^{2}-\mathcal{R}^{2}\big{)}}{\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}$
		$\displaystyle-\frac{\xi_{5}\mathcal{R}^{2}\big{(}\mathrm{C}_{1}^{2}+2\mathcal{R}^{2}\big{)}}{\omega\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}\bigg{\}}\bigg{]},$		(47)
	$\displaystyle\bar{P}_{t}$	$\displaystyle=\frac{r^{2}\big{(}3-\mathrm{C}_{1}^{4}\xi_{5}-6\mathrm{C}_{1}^{2}\xi_{5}r^{2}-9\xi_{5}r^{4}\big{)}}{8\pi\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}}+\frac{\omega r^{2}\big{(}2\mathrm{C}_{1}^{2}+r^{2}\big{)}}{8\pi\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+2r^{2}\big{)}}$
		$\displaystyle\times\bigg{\{}2\xi_{5}\big{(}r^{2}-\mathcal{R}^{2}\big{)}-\frac{3\big{(}r^{2}-\mathcal{R}^{2}\big{)}}{\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}-\frac{\xi_{5}\mathcal{R}^{2}\big{(}\mathrm{C}_{1}^{2}+2\mathcal{R}^{2}\big{)}}{\omega\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}\bigg{\}}-\frac{\omega}{8\pi}$
		$\displaystyle\times\big{[}\big{(}\mathrm{C}_{1}^{2}+r^{2}\big{)}^{3}\big{(}\xi_{5}r^{2}\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}\big{(}2r^{2}\omega-\mathcal{R}^{2}(2\omega+1)\big{)}+2\mathcal{R}^{2}\big{(}3r^{2}\omega$
		$\displaystyle-\mathcal{R}^{2}(3\omega+1)\big{)}\big{)}+\mathrm{C}_{1}^{2}\big{(}\mathrm{C}_{1}^{2}+2r^{2}+3\mathcal{R}^{2}\big{)}+3r^{2}\big{(}\mathcal{R}^{2}(\omega+2)-r^{2}\omega\big{)}\big{)}^{2}\big{]}^{-1}$
		$\displaystyle\times\big{[}\big{(}\mathrm{C}_{1}^{2}+2r^{2}\big{)}\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}\big{\{}-2\mathrm{C}_{1}^{4}r^{2}\big{(}4r^{2}(3\omega+1)-3\mathcal{R}^{2}(\omega-4)\big{)}-\mathrm{C}_{1}^{6}$
		$\displaystyle\times\big{(}r^{2}(6\omega+8)-3\mathcal{R}^{2}(\omega-2)\big{)}-3\mathrm{C}_{1}^{2}r^{4}\big{(}10r^{2}\omega+\mathcal{R}^{2}(\omega+8)\big{)}+\xi_{5}\big{(}\mathrm{C}_{1}^{2}$
		$\displaystyle+3r^{2}\big{)}^{2}\big{(}\mathrm{C}_{1}^{6}\big{(}-\big{(}\mathcal{R}^{2}(2\omega+1)-4r^{2}\omega\big{)}\big{)}+2\mathrm{C}_{1}^{4}\big{(}5r^{4}\omega+r^{2}\mathcal{R}^{2}(4\omega-1)\mathcal{R}^{4}$
		$\displaystyle-\times(3\omega+1)\big{)}+2\mathrm{C}_{1}^{2}r^{2}\big{(}4r^{4}\omega+r^{2}\mathcal{R}^{2}(13\omega-1)-2\mathcal{R}^{4}(3\omega+1)\big{)}+4r^{4}\mathcal{R}^{2}$
		$\displaystyle\times\big{(}6r^{2}\omega-\mathcal{R}^{2}(3\omega+1)\big{)}\big{)}-2\mathrm{C}_{1}^{8}-18r^{8}\omega\big{\}}\big{]}.$		(48)

The anisotropy in the interior of the above developed model is

$\displaystyle\bar{\Pi}$

$\displaystyle=$

$\displaystyle\frac{r^{2}\big{(}3-2\xi_{5}\mathrm{C}_{1}^{4}-12\xi_{5}\mathrm{C}_{1}^{2}r^{2}-18\xi_{5}r^{4}\big{)}}{8\pi\big{(}\mathrm{C}_{1}^{2}+3r^{2}\big{)}^{2}}\big{(}1-\omega\big{)}.$

(49)

It becomes clear from the above equation that the anisotropy fades away for $\omega=1$, hence, the total matter distribution turns into the isotropic interior for this particular value. We can now say that Eqs.(46)-(49) are the analytical solution of the Einstein-Maxwell field equations for $\omega\in[0,1]$. In other words, the variation in this parameter produces the isotropic configuration from being anisotropic and vice versa.

Herrera [38] defined complexity for the first time in such a way that could be suitable for all scientific fields. According to this definition, a uniform/ homogenous system is always complexity-free, implying that the density inhomogeneity and anisotropy in the pressure make the structure complex. Multiple structure scalars, in this context, were obtained corresponding to a static spherical interior through the orthogonal decomposition of the curvature tensor. Both the above-mentioned factors were found in one of the scalars, i.e., $\mathrm{Y}_{TF}$ and thus entitled the complexity factor. This work was also extended for a non-static scenario where some evolutionary patterns have been discussed [39]. For the charged scenario, the factor $\mathrm{Y}_{TF}$ becomes

$$\mathrm{Y}_{TF}(r)=8\pi\Pi+\frac{4\mathrm{q}^{2}}{r^{4}}-\frac{4\pi}{r^{3}}\int_{0}^{r}\bar{y}^{3}\rho^{\prime}(\bar{y})d\bar{y}-\frac{3}{r^{3}}\int_{0}^{r}\frac{\mathrm{qq}^{\prime}}{\bar{y}}d\bar{y}.$$

(50)

Since the current setup (6)-(8) involves two matter sources, thus the corresponding extension of the complexity factor is given by

	$\displaystyle\bar{\mathrm{Y}}_{TF}(r)$	$\displaystyle=$	$\displaystyle 8\pi\bar{\Pi}+\frac{4\mathrm{q}^{2}}{r^{4}}-\frac{4\pi}{r^{3}}\int_{0}^{r}\bar{y}^{3}\bar{\rho}^{\prime}(\bar{y})d\bar{y}-\frac{3}{r^{3}}\int_{0}^{r}\frac{\mathrm{qq}^{\prime}}{\bar{y}}d\bar{y}$		(51)
		$\displaystyle=$	$\displaystyle 8\pi\Pi+\frac{4\mathrm{q}^{2}}{r^{4}}-\frac{4\pi}{r^{3}}\int_{0}^{r}\bar{y}^{3}\rho^{\prime}(\bar{y})d\bar{y}-\frac{3}{r^{3}}\int_{0}^{r}\frac{\mathrm{qq}^{\prime}}{\bar{y}}d\bar{y}$
		$\displaystyle+$	$\displaystyle 8\pi\omega\Pi_{\mathrm{Z}}+\frac{4\pi\omega}{r^{3}}\int_{0}^{r}\bar{y}^{3}\mathrm{Z}{{}_{0}^{0}}^{\prime}(\bar{y})d\bar{y},$

which can also be written as

$\displaystyle\bar{\mathrm{Y}}_{TF}=\mathrm{Y}_{TF}+\mathrm{Y}_{TF}^{\mathrm{Z}}.$

(52)

Here, $\mathrm{Y}_{TF}$ and $\mathrm{Y}_{TF}^{\mathrm{Z}}$ are the complexity factors for the sources (17)-(19) and (20)-(22), respectively. Since we develop the model (46)-(49) by taking $\bar{\Pi}=0$ into account, therefore, Eq.(51) leads to

$\displaystyle\bar{\mathrm{Y}}_{TF}$

$\displaystyle=$

$\displaystyle\frac{4\mathrm{q}^{2}}{r^{4}}-\frac{4\pi}{r^{3}}\int_{0}^{r}\bar{y}^{3}\bar{\rho}^{\prime}(\bar{y})d\bar{y}-\frac{3}{r^{3}}\int_{0}^{r}\frac{\mathrm{qq}^{\prime}}{\bar{y}}d\bar{y}.$

(53)

After substituting the derivative of the effective energy density (46) in the above equation, we get

	$\displaystyle\bar{\mathrm{Y}}_{TF}$	$\displaystyle=\frac{11\xi_{5}r^{2}}{5}+\frac{r^{2}}{5\big{(}\mathrm{C}_{1}^{4}+5\mathrm{C}_{1}^{2}r^{2}+6r^{4}\big{)}^{2}\big{(}\mathrm{C}_{1}^{2}+3\mathcal{R}^{2}\big{)}}\big{[}\mathrm{C}_{1}^{10}(10\xi_{5}\omega+\xi_{5})+2\xi_{5}$
		$\displaystyle\times\mathrm{C}_{1}^{8}\big{(}5(8\omega+1)r^{2}+4(5\omega+1)\mathcal{R}^{2}\big{)}+\mathrm{C}_{1}^{6}\big{(}\xi_{5}(230\omega+37)r^{4}+60\xi_{5}r^{2}\mathcal{R}^{2}$
		$\displaystyle(5\omega+1)+5\big{(}2\xi_{5}(3\omega+1)\mathcal{R}^{4}+6-3\omega\big{)}\big{)}+6\mathrm{C}_{1}^{4}\big{(}10r^{6}(5\xi_{5}\omega+\xi_{5})+26\xi_{5}$
		$\displaystyle\times(5\omega+1)r^{4}\mathcal{R}^{2}+5r^{2}\big{(}2\xi_{5}(3\omega+1)\mathcal{R}^{4}+4-\omega\big{)}+5(3-2\omega)\mathcal{R}^{2}\big{)}+3\mathrm{C}_{1}^{2}$
		$\displaystyle\times r^{2}\big{(}12r^{6}(5\xi_{5}\omega+\xi_{5})+60\xi_{5}(5\omega+1)r^{4}\mathcal{R}^{2}-60(\omega-2)\mathcal{R}^{2}+5r^{2}\big{(}\omega+6$
		$\displaystyle\times\xi_{5}(3\omega+1)\mathcal{R}^{4}+8\big{)}\big{)}+18r^{4}\mathcal{R}^{2}\big{(}6r^{4}(5\xi_{5}\omega+\xi_{5})-5(\omega-4)\big{)}\big{]}.$		(54)