1. Introduction
The extensive research in physics, mathematics, and cosmology has been significantly impacted since Albert Einstein's general relativity (GR) was accomplished in 1915. He suggested the idea that time and space are inextricably interwoven and events that occur at one moment to one observer may occur at a different time to another. The equivalence principle, local causality, spacetime structure, and local coordinate frames are the cornerstones of GR [1]. Through his nonlinear equations of motion, Einstein connected the matter sources (depicted by the energy-momentum tensor) and the metric. The explanation of a wide range of astronomical phenomena, including wormholes [2], black holes [3–5], gravitational collapse [6] and cosmic expansion [7, 8] depends on these equations.
Based on the discovery of quasars in the 1960s and other extremely high energy phenomena in the Universe, such as gamma ray bursts, relativistic astrophysics and gravity theory have seen considerable advancements in recent decades. One of the most important astrophysical events for stars or other bigger cosmological systems is gravitational collapse. The goal behind the gravitational collapse discussions until now has been to address physical queries, such as the significance of collapse in astrophysics and cosmology. Although the origin of stars, the construction of galaxies, and other cosmic processes are still poorly understood, gravitational collapse will undoubtedly play a significant part in these processes. Compact objects [9–14] like neutron stars [15–17], gravastars [18, 19] and pulsars [20] also exhibit unique physical characteristics, where the effects of powerful gravitational fields are thought to play a key role. It has become increasingly obvious that the strong gravitational fields, as described by the general theory of relativity, play an important and much more dominant role when the masses and energy densities involved in the physical phenomena are sufficiently high. Any accurate description of these reported ultra-high energy particles must take gravitational dynamics into account. Consequently, it's crucial to comprehend the dynamics of the collapse.
When the approach of junction condition [21–23] is applied, one of the matched spacetimes, let's say ${ \mathcal V }$, does not typically have matter but may have a cosmological constant, electromagnetic field, or null radiation. The other (say $\bar{{ \mathcal V }}$) can be quite complex. A hypersurface illustrated with the help of the notation Σ connects the two spacetimes. Depending on the situation being studied, the characteristics of the hypersurface vary. The Σ is a time-like hypersurface in the case of voids or when the inside and outside of bubbles are thought of as two coexisting phases of the cosmos. A space-like hypersurface separates the two phases in a quick global phase shift. The idealized description of ${\rm{\Sigma }}^{\prime} s$ behavior that represents light-like hypersurface is typically seen as merely a simplifying approximation to, for instance, the behavior of sufficiently massive bubbles in a sea of false vacuum, which can quickly accelerate towards the speed of light due to an imbalance in normal pressures.
If not taking into account the object's mass, kinematical variables are those important variables that can be utilized to explain a fluid's motion. They are acceleration, shear tensor, vorticity tensor and expansion scalar. The significant addition of this manuscript is an interpretation of the benefits of adopting the expansion-free condition in the cavity evolution modeling. For that reason, we wish to analytically integrate the relevant equations under a number of different conditions, demonstrating that such an integration might be accomplished without too much effort. Such a study could serve to highlight the potential of the expansion-free condition to simulate situations where cavities are anticipated to occur. Numerous scholars [24–27] have proposed this scenario for self-gravitating stars.
The evolution of spherically symmetric matter content following a central explosion is a very intriguing subject that Skripkin [28] explored many years ago. He examined the formation of cavities in non-dissipative isotropic fluid and characterized cavities in fluid distributions with constant energy density without explicitly supposing zero expansion. Herrera et al [26] investigated the dynamical instability of an anisotropic fluid with a spherically symmetric structure that adiabatically collapses when the expansion scalar is zero. Moreover, they examined thoroughly the Newtonian and post-Newtonian regimes. In the framework of f(R, T) gravity, Yousaf and Bhatti [27] discovered a few restrictions relating to the dynamical instability of cylindrically symmetric systems. They obtained the modified gravitational equations and a few additional stellar equations to study the role the expansion scalar played in the celestial model.
In order to simulate an expanding Universe with a vacuum, Bonnor and Chamorro [29] used a spherical Minkowski zone. They presented one growing void model and many non-expanding void models after obtaining a few intriguing findings. Herrera et al [30] have taken into consideration the evolution of spherically symmetric distributions of anisotropic fluids with a central vacuum cavity. These results highlight the benefits in modelling cavity evolution under the expansion-free condition. In addition to that, on both outlined boundary surfaces, certain analytical solutions are found to meet the Darmois junction criteria.
With the aim of unraveling the mysteries surrounding irrotational spherical static neutron stellar structures, Oikonomou [31] set out on a path of investigation that involved the non-minimal coupling of inflationary gravitational potential. Through the utilization of good numerical techniques, he discovered a set of massive compact objects that exhibited exclusive observational consistency, all governed by the Wiringa-Fiks-Fabrocini equation of state (EoS). Motivated by a deep curiosity regarding neutron stars and their behavior within a particular class of cosmic inflationary models, Oikonomou [32] explored the intricacies within the framework of the Einstein frame. Through the effective application of numerical simulation, he successfully solved the corresponding motion equations, unraveling the complex relationship between mass and radius across three distinct EoS. Driven by a desire to uncover the secrets of compact celestial body formation, Oikonomou [33] extended his analysis to include the Skyrme-Lyon EoS, carefully examining its interplay with quadratic and induced inflationary corrections. He revealed the presence of massive neutron stars that captivated observers due to their remarkable adherence to observation, all made possible by the distinctive characteristics of this particular EoS. Odintsov et al [8] investigated a power-law F(R) gravity that can account for the acceleration of the Universe both in the early and late times. Moreover, they used recent observational data to examine this scenario. For realistic compact star configurations, Nashed and Bamba [14] provided a comprehensive explanation of what vierbein is within the context of conformal teleparallel gravity, including its role in describing spacetime geometry. Furthermore, they explored an interior solution for the compact object in this particular gravity. Odintsov and Oikonomou [17] used a large sample of equations of states (EoS) adopting the piecewise polytropic EoS approach to study the static neutron stars phenomenology in relation to various inflationary attractors.
The significance of charge on Herrera's work [30] is observed in this manuscript. After the quick review of the basic formalism of evaluating Einstein-Maxwell (EM) equations, energy momentum and electromagnetic tensor are featured in section 2 . The field equations and dynamical equations in the context of spherical metrics are determined in section 3 . In section 4 , the kinematical variables that depict the fluid's motion and the mass function of the system are evaluated. Section 5 is occupied for determining the complexity factor (CF). In section 6 , the quasi-homologous constraint and transport equation are obtained along with the junction conditions. Section 7 offers the analytical solutions of the charged, non-static, anisotropic spherical matter configurations by means of a few restrictions. Eventually, the consequences of electric force on this system are summed up in the last section.
2. General formalism
The impact of electric force on the 4D gravitational action is stated as follows within the framework of GR6 ), the following differential equations are extracted as7 ), we reached at
$\begin{eqnarray}{I}_{(\mathrm{GR})}=\frac{1}{2\kappa }\int {{\rm{d}}}^{4}x[{\mathfrak{R}}+{{\mathfrak{L}}}_{m}+{{\mathfrak{L}}}_{e}]\sqrt{-g},\end{eqnarray}$
where ${ \mathcal I },\,\sqrt{-g}$ and κ indicate the action integral, magnitude of metric tensor and the coupling constant (whose value is 8π), respectively. The Ricci scalar, Lagrangian of electric charge and matter content are represented by ${\mathfrak{R}},\,{{\mathfrak{L}}}_{e}$ and ${{\mathfrak{L}}}_{m}$, respectively. It is feasible to describe the tensor that represents the electromagnetic field as $\begin{eqnarray}{T}_{\varpi \chi }^{\mathrm{em}}=\displaystyle \frac{1}{4\pi }\left({F}_{\varpi }^{\omega }{F}_{\chi \omega }-\displaystyle \frac{1}{4}{F}^{\omega \eta }{F}_{\omega \eta }{g}_{\varpi \chi }\right),\end{eqnarray}$
where Fχω is electromagnetic field tensor and its expression is Fχω = ψω,χ − φχ,ω. The four potential is depicted as ${\psi }_{\omega }=\psi (r,t){\delta }_{\omega }^{0}$. The aforementioned form of four potential shows the presence of static charge and hence the magnetic field vanishes in this situation. The scalar potential is denoted by ψ(t, r). To analyze anisotropic spherically symmetric evolving matter content, we assume the following relevant interior metric $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -{K}^{2}(t,r){\rm{d}}{t}^{2}+{H}^{2}(t,r){\rm{d}}{r}^{2}+{J}^{2}(t,r){\rm{d}}{\theta }^{2}\\ & & +{J}^{2}(t,r){\sin }^{2}\theta {\rm{d}}{\phi }^{2},\end{array}\end{eqnarray}$
where K and H are dimensionless while J has the dimensions of r. The compact form of EM equations are $\begin{eqnarray}{G}_{\varpi \chi }=8\pi {{ \mathcal T }}_{\varpi }\chi =8\pi \left({{ \mathcal T }}_{\varpi \chi }^{{\rm{m}}}+{{ \mathcal T }}_{\varpi \chi }^{\mathrm{em}}\right),\end{eqnarray}$
where ${G}_{\varpi \chi },\,{{ \mathcal T }}_{\varpi \chi }^{{\rm{m}}}$ and ${{ \mathcal T }}_{\varpi \chi }^{\mathrm{em}}$ illustrate the Einstein tensor, dissipative anisotropic matter and charged matter, respectively. Mathematically, ${{ \mathcal T }}_{\varpi \chi }^{{\rm{m}}}$ is expressed as $\begin{eqnarray}{{ \mathcal T }}_{\varpi \chi }^{{\rm{m}}}=(\mu +P){V}_{\varpi }{V}_{\chi }-{{Pg}}_{\varpi \chi }+{{\rm{\Pi }}}_{\varpi \chi }+q({V}_{\varpi }{\chi }_{\chi }+{V}_{\chi }{\chi }_{\varpi }),\end{eqnarray}$
where μ, q, Πϖχ and P show the energy density, heat flux, anisotropic tensor and pressure, respectively. The pressure and anisotropic tensor are further expressed as $\begin{eqnarray*}\begin{array}{rcl}P & = & \displaystyle \frac{2{P}_{\perp }+{P}_{r}}{3},\quad {{\rm{\Pi }}}_{\varpi \chi }={\rm{\Pi }}\left({K}_{\varpi }{K}_{\chi }-\displaystyle \frac{{h}_{\varpi \chi }}{3}\right),\\ {h}_{\varpi \chi } & = & {g}_{\varpi \chi }+{V}_{\chi }{V}_{\varpi },\quad {\rm{\Pi }}={P}_{r}-{P}_{\perp },\end{array}\end{eqnarray*}$
where the projection tensor, four vectors and anisotropy factor are symbolized as hϖχ, Vϖ, Kϖ and Π, respectively. The perpendicular and radial directions of pressure are denoted by P⊥ and Pr, respectively. In order to evaluate the value of charge, consider the tensorial form of Maxwell equations as $\begin{eqnarray}{F}_{;\chi }^{\varpi \chi }={\mu }_{0}{\unicode{x00237}}^{\varpi },\quad \quad {F}_{[\varpi \chi ;\omega ]}=0,\end{eqnarray}$
where μ0 and Jϖ = ρ(t, r)Vϖ are the magnetic permeability and four-current, respectively. With the use of equation ( $\begin{eqnarray}\psi ^{\prime\prime} +\left(\displaystyle \frac{K^{\prime} }{K}+\displaystyle \frac{H^{\prime} }{H}-2\displaystyle \frac{J^{\prime} }{J}\right)\psi ^{\prime} =4\pi \rho {{KH}}^{2},\end{eqnarray}$
$\begin{eqnarray}\dot{\psi ^{\prime} }-2\displaystyle \frac{\dot{J}}{J}+\displaystyle \frac{\dot{H}}{H}-\left(\displaystyle \frac{\dot{K}}{K}\right)\psi ^{\prime} =0,\end{eqnarray}$
where prime and dots are used to express the derivatives corresponding to r and t, respectively. Upon integration of equation ( $\begin{eqnarray}\psi ^{\prime} =\displaystyle \frac{{KHs}}{{J}^{2}},\end{eqnarray}$
where s is the charge and its mathematical expression is $\begin{eqnarray}s=4\pi {\int }_{0}^{r}\rho {{HJ}}^{2}{\rm{d}}r,\end{eqnarray}$
where ρ shows the charge density. How much electric charge has collected in a specific field is indicated by its charge density.3. Einstein-Maxwell and dynamical equation
The EM field equations are determined with the use of equations (2 )-(4 ) as15 ) and (16 ), are critical for understanding the attributes of a dynamical system. Moreover, nonlinear electromagnetic fields (NEF) can cause faster expansion and other favorable cosmological properties, resulting in good qualitative phenomenology. Equations (11 )-(14 ) are the NEF equations for our spacetime.
$\begin{eqnarray}\begin{array}{l}8\pi \left(\mu +\frac{{s}^{2}}{8\pi {J}^{4}}\right)={K}^{-2}\left(2\frac{\dot{H}}{H}+\frac{\dot{J}}{J}\right)\frac{\dot{J}}{J}-{\left(\frac{1}{H}\right)}^{2}\\ \quad \times \left[2\frac{J^{\prime\prime} }{J}+{\left(\frac{{J}^{^{\prime} }}{J}\right)}^{2}-2\frac{{H}^{^{\prime} }}{H}\frac{{J}^{^{\prime} }}{J}-{\left(\frac{H}{J}\right)}^{2}\right],\end{array}\end{eqnarray}$
$\begin{eqnarray}8\pi q=\displaystyle \frac{2}{{KH}}\left(\displaystyle \frac{{\dot{J}}^{{\prime} }}{J}-\displaystyle \frac{\dot{H}}{H}\displaystyle \frac{{J}^{{\prime} }}{J}-\displaystyle \frac{\dot{J}}{J}\displaystyle \frac{{K}^{{\prime} }}{K}\right),\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-{\left(\displaystyle \frac{1}{K}\right)}^{2}\left[-\displaystyle \frac{\dot{J}}{J}\left(2\displaystyle \frac{\dot{K}}{K}-\displaystyle \frac{\dot{J}}{J}\right)+2\displaystyle \frac{\ddot{J}}{J}\right]\\ \quad +\left(2\displaystyle \frac{{K}^{{\prime} }}{K}+\displaystyle \frac{{J}^{{\prime} }}{J}\right)\displaystyle \frac{{J}^{{\prime} }}{J}-{\left(\displaystyle \frac{1}{J}\right)}^{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-{\left(\displaystyle \frac{1}{K}\right)}^{2}\left[\displaystyle \frac{\ddot{H}}{H}+\displaystyle \frac{\ddot{J}}{J}-\displaystyle \frac{\dot{K}}{K}\left(\displaystyle \frac{\dot{H}}{H}+\displaystyle \frac{\dot{J}}{J}\right)\right.\\ \quad \left.+\displaystyle \frac{\dot{H}}{H}\displaystyle \frac{\dot{J}}{J}\right]+{\left(\displaystyle \frac{1}{H}\right)}^{2}\left[\displaystyle \frac{K^{\prime\prime} }{K}+\displaystyle \frac{J^{\prime\prime} }{J}\right.\\ \quad \left.-\displaystyle \frac{{K}^{{\prime} }}{K}\displaystyle \frac{{H}^{{\prime} }}{H}+\left(\displaystyle \frac{{K}^{{\prime} }}{K}-\displaystyle \frac{{H}^{{\prime} }}{H}\right)\displaystyle \frac{{J}^{{\prime} }}{J}\right].\end{array}\end{eqnarray}$
We will now construct dynamical equations and subsequently a collapse equation. These equations are written as $\begin{eqnarray}\begin{array}{l}\dot{\mu }+2\left(\mu +{P}_{\perp }\right)\displaystyle \frac{\dot{J}}{J}+\left(\mu +{P}_{r}\right)\displaystyle \frac{\dot{H}}{H}+\left(\displaystyle \frac{{K}^{{\prime} }}{K}+\displaystyle \frac{{J}^{{\prime} }}{J}\right)2q\\ \quad \times \,\displaystyle \frac{K}{H}+q^{\prime} \displaystyle \frac{K}{H}=0,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{P}_{r}^{\prime} -\displaystyle \frac{{ss}^{\prime} }{4\pi {J}^{4}}+\left(\mu +{P}_{r}\right)\displaystyle \frac{{K}^{{\prime} }}{K}+2{\rm{\Pi }}\displaystyle \frac{{J}^{{\prime} }}{J}+\dot{q}\displaystyle \frac{H}{K}\\ \quad +\,\left(\displaystyle \frac{\dot{H}}{H}+\displaystyle \frac{\dot{J}}{J}\right)2q\displaystyle \frac{H}{K}=0.\end{array}\end{eqnarray}$
The dynamical equations (The expansion scalar formula and its value for our spacetime is
$\begin{eqnarray}{\rm{\Theta }}={V}_{;\chi }^{\chi }=\left(2\displaystyle \frac{\dot{J}}{J}+\displaystyle \frac{\dot{H}}{H}\right)\displaystyle \frac{1}{K}.\end{eqnarray}$
The motion of the evolving fluid can be assessed by taking into account the kinematical variables. One of the significant kinematical variables is the expansion scalar. The amount by which the volume of the fluid element increases over time is illustrated by the expansion scalar. The imposition of the expansion-free condition in the analysis necessarily gives rise to the production of vacuoles within the system's interior. This further indicates that the relativistic fluid evolves without being compressed. Furthermore, the zero expansion condition implies the presence of pressure anisotropy as well as energy density inhomogeneity. The constraints Θ ≶ displays the Universe's expanding and contracting aspects. It should be emphasized that when the fluid fills the whole sphere, including the center (r = 0), the regularity criterion r = 0 must be applied. However, we do not require such a condition because we are dealing with expansion-free systems. According to Misner and Sharp [34], the formula for a mass function m is designated and evaluated in the presence of charge for the sphere as $\begin{eqnarray}m=-\displaystyle \frac{J}{2}{R}_{232}^{3}=\left[1-{\left(\displaystyle \frac{J^{\prime} }{H}\right)}^{2}+{\left(\displaystyle \frac{\dot{J}}{K}\right)}^{2}+\displaystyle \frac{{s}^{2}}{{J}^{2}}\right]\displaystyle \frac{J}{2}.\end{eqnarray}$
Keeping in mind, the proper time derivative $\left({D}_{T}=\tfrac{1}{K}\tfrac{\partial }{\partial t}\right)$ and the proper radial derivative $\left({D}_{J}=\tfrac{1}{J^{\prime} }\tfrac{\partial }{\partial t}\right)$, we now examine the dynamics of stellar structures. The mass function is related to the proper time derivative under the impact of charge as $\begin{eqnarray}{E}^{2}\equiv {\left(\displaystyle \frac{J^{\prime} }{H}\right)}^{2}=\left(-\displaystyle \frac{2m}{J}-1+{U}^{2}+\displaystyle \frac{{s}^{2}}{{J}^{2}}\right),\end{eqnarray}$
where U is termed as collapsing velocity and its mathematical expression is U = DTJ. We can also describe it as the evolution of the areal radius corresponding to proper time. These tensors have been extensively studied in conjunction with other fluid characteristics [35, 36].4. Complexity factor in terms of conformal scalar
In this section, firstly, we discuss the conformal scalar with the help of the formula of Weyl tensor which is designated as ${W}_{\varpi \chi }^{(e)}={\mathfrak{E}}\left({\chi }_{\varpi }{\chi }_{\chi }-\tfrac{{h}_{\chi \varpi }}{3}\right)$. This displays only the electric portion of the Weyl tensor as the magnetic one diminishes for our considered metric. The symbol ${\mathfrak{E}}$ is used to define the conformal scalar and its value is determined for our metric as20 ) allow us to accomplish
$\begin{eqnarray}\begin{array}{rcl}{\mathfrak{E}} & = & \displaystyle \frac{1}{2{K}^{2}}\left[\displaystyle \frac{\ddot{J}}{J}-\displaystyle \frac{\ddot{H}}{H}-\left(\displaystyle \frac{\dot{J}}{J}-\displaystyle \frac{\dot{H}}{H}\right)\left(\displaystyle \frac{\dot{K}}{K}+\displaystyle \frac{\dot{J}}{J}\right)\right]-\displaystyle \frac{1}{2{J}^{2}}\\ & & +\displaystyle \frac{1}{2{H}^{2}}\left[-\displaystyle \frac{J^{\prime\prime} }{J}+\displaystyle \frac{K^{\prime\prime} }{K}+\left(\displaystyle \frac{{J}^{{\prime} }}{J}+\displaystyle \frac{{H}^{{\prime} }}{H}\right)\left(\displaystyle \frac{{J}^{{\prime} }}{J}-\displaystyle \frac{{K}^{{\prime} }}{K}\right)\right].\end{array}\end{eqnarray}$
The conformal scalar assists in the knowledge of tidal forces operating on an astronomical object in the manifold. Now, we will discuss the CF [37–39] which is significant at the astrophysical scale, and express it in terms of conformal scalar. Many aspects of self-gravitating systems, including their matter density, stability, pressure, mass-radius ratio and brightness have been widely investigated. The primary factors are matter density and pressure in explaining the evolution of compact objects and they play a crucial role in describing the complexity of these objects. Next, we will cover the key steps to achieve our objective, i.e. the evaluation of CF. Herrera et al [40] pioneered the notion of orthogonal splitting of the Riemann tensor (Rϖηχδ) to generate structure scalars. They achieved five scalars in the context of GR by executing orthogonal splitting on the Riemann tensor. These scalars are the trace and trace-free parts of tensors Yϖχ, Xϖχ and Zϖχ. All of these scalars influence directly the realistic features of the matter content. The following tensor is the first step to take into consideration in order to assess CF is Yϖχ = RϖηχδVηVδ, where Vδ illustrates the four velocities. The tensor Yϖχ can be written in the combination of YT and YTF. The EM field equations together with equation ( $\begin{eqnarray}\begin{array}{rcl}{Y}_{T} & = & 4\pi (\mu -2{\rm{\Pi }}+3{P}_{r})+\displaystyle \frac{{s}^{2}}{4\pi {J}^{4}},\\ {Y}_{{TF}} & = & -4\pi {\rm{\Pi }}+{\mathfrak{E}}+\displaystyle \frac{{s}^{2}}{{J}^{4}}.\end{array}\end{eqnarray}$
Eventually, the final expression of the YTF is accomplished after using EM field equations and conformal scalar, as $\begin{eqnarray}\begin{array}{rcl}{Y}_{{TF}} & = & \displaystyle \frac{1}{{K}^{2}}\left[\displaystyle \frac{\ddot{J}}{J}-\displaystyle \frac{\ddot{H}}{H}+\displaystyle \frac{\dot{K}}{K}\left(\displaystyle \frac{\dot{H}}{H}-\displaystyle \frac{\dot{J}}{J}\right)\right]\\ & & +\displaystyle \frac{1}{{H}^{2}}\left[\displaystyle \frac{K^{\prime\prime} }{K}-\displaystyle \frac{{K}^{{\prime} }}{K}\left(\displaystyle \frac{{H}^{{\prime} }}{H}+\displaystyle \frac{{J}^{{\prime} }}{J}\right)\right].\end{array}\end{eqnarray}$
Structure scalar YTF is the most fundamental scalar operation that aid in describing how self-gravitating fluid distributions arise and change over time. It is identified to be connected to fundamental fluid properties including pressure anisotropy and inhomogeneous energy density. Vanishing of YTF is effective for creating various models and along with this informs us about the consequences of electric force on the substantial parameters of astronomical objects. For further details, one can see [41–43].5. Quasi-homologous constraint and junction condition
In order to accomplish the simplest modes of evolution in a dynamical system, we need to evaluate quasi-homologous constraint [44]. For this purpose, equation (12 ) can be written as19 ) to solve equation (23 ) as24 ), we procure
$\begin{eqnarray}{\left(\displaystyle \frac{U}{J}\right)}^{{\prime} }=4\pi {qH}+\sigma \displaystyle \frac{{J}^{{\prime} }}{J}.\end{eqnarray}$
Using equation ( $\begin{eqnarray}U=\tilde{k}(t)J+J{\int }_{0}^{r}\left(\frac{4\pi q}{E}+\frac{\sigma }{J}\right){J}^{{\prime} }{\rm{d}}r,\end{eqnarray}$
where $\tilde{k}(t)$ is the integration function. As the fluid configuration is bounded by a surface using $r\,=\mathrm{constant}\,=\,{r}_{{{\rm{\Sigma }}}^{e}}$ in equation ( $\begin{eqnarray}U=J\frac{{U}_{{{\rm{\Sigma }}}^{e}}}{{J}_{{{\rm{\Sigma }}}^{e}}}-J{\int }_{r}^{{r}_{{{\rm{\Sigma }}}^{e}}}\left(\frac{4\pi q}{E}+\frac{\sigma }{J}\right){J}^{{\prime} }{\rm{d}}r.\end{eqnarray}$
The quasi-homologous restriction implies that $\begin{eqnarray}U=J\displaystyle \frac{{U}_{{{\rm{\Sigma }}}^{e}}}{{J}_{{{\rm{\Sigma }}}^{e}}},\end{eqnarray}$
which gives $\begin{eqnarray}\displaystyle \frac{4\pi q}{E}+\displaystyle \frac{\sigma }{J}=0.\end{eqnarray}$
In the case of dynamical fluids, the system's complexity depends on the development pattern as well as dissipative factors of the fluid distribution, as detailed in [45]. In this regard, an assumption is made that corresponds to one of the simplest patterns of evolution known as homologous evolution. The homologous evolution is for dissipative processes but in non-dissipative processes, another less restricted constraint has been developed known as the quasi-homologous constraint [44]. Now we describe the matching conditions [46, 47], with emphasis on the smooth gluing of the two geometries [48]. We investigate fluid distributions that are spherically symmetric and are constrained from the outside by a spherical surface (Σe). The construction of a cavity surrounding the fluid configuration necessitates the application of the Darmois requirements to each of the limiting hypersurfaces. The Vaidya-Reissner Nordstöm spacetime [49] is the outside metric, as described by $\begin{eqnarray}\begin{array}{rcl}{\rm{d}}{s}^{2} & = & -\left(1-\frac{2M(\vartheta )}{r(\vartheta )}+\frac{{S}^{2}(\vartheta )}{{r}^{2}(\vartheta )}\right){\rm{d}}{\vartheta }^{2}\\ & & -2{\rm{d}}r{\rm{d}}\vartheta +{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}),\end{array}\end{eqnarray}$
where the system's total mass, total charge, and delayed time are denoted by the symbols M, S and ϑ, respectively. We investigate a spherical surface with motion defined by a time-like 3D hypersurface that divides two 4D manifolds into interior and exterior metric [50, 51].The line element at a hypersurface is3 ) with (29 ), we have28 ) with (29 ), we have3 ) with (28 ), we have29 )-(32 ) and (37 ), we have39 ) for M, we have18 ), we obtain29 )-(32 ) and (37 ), we achieve39 ) and then multiplying it by $\tfrac{-2}{J}$, we obtain40 ) and (42 ). If any of the aforementioned matching requirements are not fulfilled, we must assume that the relevant boundary surface is covered by a thin shell.
$\begin{eqnarray}{\left({\rm{d}}{s}^{2}\right)}_{{\rm{\Sigma }}}=-{\rm{d}}{\tau }^{2}+K(\tau )({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}),\end{eqnarray}$
where $\tau$ describes the proper time. Using equations ( $\begin{eqnarray}{\rm{d}}\tau \mathop{=}\limits^{{\rm{\Sigma }}}K{\rm{d}}t\quad \Longrightarrow \quad \frac{{\rm{d}}t}{{\rm{d}}\tau }\mathop{=}\limits^{{\rm{\Sigma }}}{K}^{-1}.\end{eqnarray}$
Using equation ( $\begin{eqnarray}\begin{array}{l}{\rm{d}}\tau \mathop{=}\limits^{{\rm{\Sigma }}}{\left(1-\frac{2M(\vartheta )}{r(\vartheta )}+\frac{2{\rm{d}}r}{{\rm{d}}\vartheta }+\frac{{S}^{2}(\vartheta )}{{r}^{2}(\vartheta )}\right)}^{\tfrac{1}{2}}{\rm{d}}\vartheta \\ \quad \Longrightarrow \frac{{\rm{d}}\vartheta }{{\rm{d}}\tau }\mathop{=}\limits^{{\rm{\Sigma }}}{\left(1-\frac{2M(\vartheta )}{r(\vartheta )}+\frac{2{\rm{d}}r}{{\rm{d}}\vartheta }+\frac{{S}^{2}(\vartheta )}{{r}^{2}(\vartheta )}\right)}^{\tfrac{-1}{2}}.\end{array}\end{eqnarray}$
Using equation ( $\begin{eqnarray}J(t,r)\mathop{=}\limits^{{\rm{\Sigma }}}{r}_{{\rm{\Sigma }}}(\vartheta ).\end{eqnarray}$
The formula for outward unit normal vector is $\begin{eqnarray}{n}_{\varpi }^{\pm }=\pm \displaystyle \frac{h{,}_{\varpi }}{| {g}^{\alpha \beta }h{,}_{\alpha }h{,}_{\beta }{| }^{\tfrac{1}{2}}},\end{eqnarray}$
here the + and − show the exterior and interior metric, respectively. Its components are evaluated using the hypersurface equations, h− = r − rΣ = 0 and h+ = r − rΣ(ϑ) = 0, as $\begin{eqnarray}{n}_{\varpi }^{-}\mathop{=}\limits^{{\rm{\Sigma }}}H(t,r){\delta }_{\varpi }^{1},\end{eqnarray}$
$\begin{eqnarray}{n}_{\varpi }^{+}\mathop{=}\limits^{{\rm{\Sigma }}}{\left(1-\frac{2M}{r}+\frac{{S}^{2}}{{r}^{2}}+\frac{2{\rm{d}}r}{{\rm{d}}\vartheta }\right)}^{-\tfrac{1}{2}}\left(-\frac{{\rm{d}}r}{{\rm{d}}\vartheta }{\delta }_{\varpi }^{0}+{\delta }_{\varpi }^{1}\right).\end{eqnarray}$
The formula for extrinsic curvature is designated as $\begin{eqnarray}\begin{array}{l}{{\mathfrak{K}}}_{\varpi \chi }^{\pm }=-{n}_{\gamma }^{\pm }\left(\displaystyle \frac{{\partial }^{2}{x}_{\pm }^{\gamma }}{\partial {\eta }^{\varpi }\partial {\eta }^{\chi }}+{{\rm{\Gamma }}}_{\mu \nu }^{\gamma }\displaystyle \frac{\partial {x}_{\pm }^{\mu }}{\partial {\eta }^{\varpi }}\displaystyle \frac{\partial {x}_{\pm }^{\nu }}{\partial {\eta }^{\chi }}\right)\\ \quad (\gamma ,\mu ,\nu =0,1,2,3),\end{array}\end{eqnarray}$
where ηϖ and ${x}_{\pm }^{\gamma }$ represent the coordinates on boundary and the coordinates for exterior/interior metrics, respectively. The non-zero components of extrinsic curvature are given as $\begin{eqnarray}\begin{array}{rcl}{{\mathfrak{K}}}_{00}^{-} & = & {\left[-\frac{K^{\prime} }{{KH}}\right]}_{{\rm{\Sigma }}},\\ {{\mathfrak{K}}}_{22}^{-} & = & \frac{{L}_{33}^{-}}{{\sin }^{2}\theta }={\left[\frac{{JJ}^{\prime} }{H}\right]}_{{\rm{\Sigma }}},\\ {{\mathfrak{K}}}_{00}^{+} & = & {\left[\frac{{{\rm{d}}}^{2}\vartheta }{{\rm{d}}{\tau }^{2}}{\left(\frac{{\rm{d}}\vartheta }{{\rm{d}}\tau }\right)}^{-1}-\frac{M}{{r}^{2}}\left(\frac{{\rm{d}}\vartheta }{{\rm{d}}\tau }\right)+\frac{{S}^{2}}{{r}^{3}}\left(\frac{{\rm{d}}\vartheta }{{\rm{d}}\tau }\right)\right]}_{{\rm{\Sigma }}},\\ {{\mathfrak{K}}}_{22}^{+} & = & \frac{{{\mathfrak{K}}}_{33}^{+}}{{\sin }^{2}\theta }\\ & = & {\left[\left(\frac{{\rm{d}}r}{{\rm{d}}\tau }\right)r+\frac{{\rm{d}}\vartheta }{{\rm{d}}\tau }\left(1-\frac{2M}{r}\right)r+\frac{{S}^{2}}{r}\left(\frac{{\rm{d}}\vartheta }{{\rm{d}}\tau }\right)\right]}_{{\rm{\Sigma }}}.\end{array}\end{eqnarray}$
Consider, ${{\mathfrak{K}}}_{22}^{+}\mathop{=}\limits^{{\rm{\Sigma }}}{{\mathfrak{K}}}_{22}^{-}$. Upon substituting the expression $\tfrac{{\rm{d}}r}{{\rm{d}}\tau }=\left(\tfrac{{\rm{d}}r}{{\rm{d}}t}\right)\left(\tfrac{{\rm{d}}t}{{\rm{d}}\tau }\right)$ and using equations ( $\begin{eqnarray}\left(\displaystyle \frac{J^{\prime} }{H}-\displaystyle \frac{\dot{J}}{K}\right)\left(\displaystyle \frac{J^{\prime} }{H}+\displaystyle \frac{\dot{J}}{K}\right)\mathop{=}\limits^{{\rm{\Sigma }}}\left(1-\displaystyle \frac{2M}{J}+\displaystyle \frac{{S}^{2}}{{J}^{2}}\right).\end{eqnarray}$
Solving equation ( $\begin{eqnarray}M\mathop{=}\limits^{{\rm{\Sigma }}}\displaystyle \frac{J}{2}\left(1+\displaystyle \frac{{\dot{J}}^{2}}{{K}^{2}}-\displaystyle \frac{J{{\prime} }^{2}}{{H}^{2}}+\displaystyle \frac{{S}^{2}}{{J}^{2}}\right)\end{eqnarray}$
With the help of equation ( $\begin{eqnarray}m(t,r)\mathop{=}\limits^{{\rm{\Sigma }}}M(\vartheta ).\end{eqnarray}$
Now, using ${{\mathfrak{K}}}_{00}^{+}={{\mathfrak{K}}}_{00}^{-}$, the chain rule $\tfrac{{{\rm{d}}}^{2}\vartheta }{{\rm{d}}{\tau }^{2}}=\tfrac{{\rm{d}}}{{\rm{d}}t}\left(\tfrac{{\rm{d}}\vartheta }{{\rm{d}}\tau }\right)\tfrac{{\rm{d}}t}{{\rm{d}}\tau }$ and the ( $\begin{eqnarray}\begin{array}{l}-\displaystyle \frac{1}{K}\left(\displaystyle \frac{\dot{J}^{\prime} }{H}-\displaystyle \frac{J^{\prime} \dot{H}}{{H}^{2}}+\displaystyle \frac{\ddot{J}}{K}-\displaystyle \frac{\dot{J}\dot{K}}{{K}^{2}}\right)\\ \quad -\displaystyle \frac{M}{{r}^{2}}+\displaystyle \frac{{S}^{2}}{{r}^{3}}\mathop{=}\limits^{{\rm{\Sigma }}}-\displaystyle \frac{K^{\prime} }{{KH}}\left(\displaystyle \frac{J^{\prime} }{H}+\displaystyle \frac{\dot{J}}{K}\right).\end{array}\end{eqnarray}$
Substituting the value of M from equation ( $\begin{eqnarray}q(t,r)\mathop{=}\limits^{{\rm{\Sigma }}}{P}_{r},\quad \quad ({S}^{2}\mathop{=}\limits^{{\rm{\Sigma }}}{s}^{2}).\end{eqnarray}$
The continuity of the first and second fundamental forms [52] allowed us to achieve matching conditions mentioned in equations (6. Solutions free of complexity and expansion
In this section, we will examine two distinct families of solutions under the influence of electric charge. To begin with, we will bear in mind non-geodesic spheres that meet the diminishing CF condition [53–56] and are supplemented by the quasi-homologous condition or some straightforward metric variable assumptions. To accomplish the second family of solutions, geodesic fluids meeting the vanishing CF criterion or the quasi-homologous condition will be taken into consideration.
6.1. Non-geodesic sphere
In this model, we determine solutions for non-geodesic spheres in the presence of charge with some restrictions as
6.1.1. $\dot{J}=\tilde{a}(t){KJ}$
To begin with, we will bear in mind non-geodesic spheres that meet the diminishing CF condition and are supplemented by the quasi-homologous condition. The expansion free constraint yields47 )-(50 ) exhibit that in the presence of charge the non-geodesic sphere become less dense. In addition to that, its radial pressure increases while the tangential pressure lessens. The constraint of diminishing CF gives51 ), we consider51 ) using (52 ) provides54 ) into equation (53 ), we procure55 ) is achieved as56 ), the equation (52 ) becomes57 ) in equations (47 )-(50 ) transform the state determinants and mass function as
$\begin{eqnarray}\displaystyle \frac{\dot{H}}{H}+\displaystyle \frac{2\dot{J}}{J}=0\Rightarrow H=\displaystyle \frac{\alpha }{{J}^{2}}.\end{eqnarray}$
The quasi-homologous constraint reads $\begin{eqnarray}\dot{J}=\tilde{a}(t){KJ}.\end{eqnarray}$
Utilizing the vanishing of expansion scalar and the constraint of quasi-homologous, we accomplish $\begin{eqnarray}H=\displaystyle \frac{\alpha }{{J}^{2}},\end{eqnarray}$
$\begin{eqnarray}K=\displaystyle \frac{\dot{J}}{\tilde{a}J}.\end{eqnarray}$
The state variables are evaluated in the presence of aforementioned conditions and charge, treating $\alpha \,=\mathrm{constant}\,=\,\tilde{a}$, as $\begin{eqnarray}8\pi \left(\mu +\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-3{\tilde{a}}^{2}-\displaystyle \frac{{J}^{4}}{{\alpha }^{2}}\left[\displaystyle \frac{2J^{\prime\prime} }{J}+5{\left(\displaystyle \frac{{J}^{{\prime} }}{J}\right)}^{2}-\displaystyle \frac{{\alpha }^{2}}{{J}^{6}}\right],\end{eqnarray}$
$\begin{eqnarray}4\pi q=\displaystyle \frac{3\tilde{a}}{\alpha }{{JJ}}^{{\prime} },\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-3{\tilde{a}}^{2}+\displaystyle \frac{{J}^{4}}{{\alpha }^{2}}\left[\displaystyle \frac{2{\dot{J}}^{{\prime} }}{\dot{J}}\displaystyle \frac{{J}^{{\prime} }}{J}-{\left(\displaystyle \frac{{J}^{{\prime} }}{J}\right)}^{2}-\displaystyle \frac{{\alpha }^{2}}{{J}^{6}}\right],\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-3{\tilde{a}}^{2}+\displaystyle \frac{{J}^{4}}{{\alpha }^{2}}\left[\displaystyle \frac{\dot{J}^{\prime\prime} }{\dot{J}}+\displaystyle \frac{{\dot{J}}^{{\prime} }}{\dot{J}}\displaystyle \frac{{J}^{{\prime} }}{J}+{\left(\displaystyle \frac{{J}^{{\prime} }}{J}\right)}^{2}\right].\end{eqnarray}$
Equations ( $\begin{eqnarray}-3{\tilde{a}}^{2}+\displaystyle \frac{{J}^{4}}{{\alpha }^{2}}\left[\displaystyle \frac{\dot{J}^{\prime\prime} }{\dot{J}}-\displaystyle \frac{{\dot{J}}^{{\prime} }}{\dot{J}}\displaystyle \frac{{J}^{{\prime} }}{J}-\displaystyle \frac{J^{\prime\prime} }{J}+{\left(\displaystyle \frac{{J}^{{\prime} }}{J}\right)}^{2}\right]=0.\end{eqnarray}$
To accomplish the solution of equation ( $\begin{eqnarray}J=f\left({\gamma }_{1}r+{\gamma }_{2}t+{\gamma }_{3}\right)\equiv f(z),\end{eqnarray}$
where f indicates the arbitrary function while γ1, γ2 and γ3 illustrate the arbitrary constants. The transformation of equation ( $\begin{eqnarray}-3{\tilde{a}}^{2}+\displaystyle \frac{{\gamma }_{1}^{2}{f}^{4}}{{\alpha }^{2}}\left[\displaystyle \frac{\tfrac{{\partial }^{3}f}{\partial {z}^{3}}}{\tfrac{\partial f}{\partial z}}-2\displaystyle \frac{\tfrac{{\partial }^{2}f}{\partial {z}^{2}}}{f}+{\left(\displaystyle \frac{\tfrac{\partial f}{\partial z}}{f}\right)}^{2}\right]=0.\end{eqnarray}$
Next, considering the auxiliary variables $\begin{eqnarray}\displaystyle \frac{\partial f}{\partial z}=\beta ,\quad \displaystyle \frac{{\partial }^{2}f}{\partial {z}^{2}}={\beta }_{f}\beta ,\quad \displaystyle \frac{{\partial }^{3}f}{\partial {z}^{3}}={\beta }^{2}{\beta }_{{ff}}+\beta {\beta }_{f}^{2},\end{eqnarray}$
where βf shows the derivative of β corresponding to f. Substituting, equation ( $\begin{eqnarray}-3{\tilde{a}}^{2}+\displaystyle \frac{{\gamma }_{1}^{2}{f}^{4}}{{\alpha }^{2}}\left[\beta {\beta }_{{ff}}+{\beta }_{f}^{2}-\displaystyle \frac{2\beta {\beta }_{f}}{f}+\displaystyle \frac{{\beta }^{2}}{{f}^{2}}\right]=0,\end{eqnarray}$
where the solution of equation ( $\begin{eqnarray}\beta =\displaystyle \frac{k}{f},\quad k=\displaystyle \frac{\tilde{a}\alpha }{\sqrt{2}{\gamma }_{1}}.\end{eqnarray}$
Utilizing equation ( $\begin{eqnarray}J=\sqrt{2k\left({\gamma }_{1}r+{\gamma }_{2}t+{\gamma }_{3}\right)}.\end{eqnarray}$
The substitution of equation ( $\begin{eqnarray}8\pi \left(\mu +\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{9}{2}{\tilde{a}}^{2}+\displaystyle \frac{1}{2k\left({\gamma }_{1}r+{\gamma }_{2}t+{\gamma }_{3}\right)},\end{eqnarray}$
$\begin{eqnarray}4\pi q=\displaystyle \frac{3}{2}{\tilde{a}}^{2},\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{9}{2}{\tilde{a}}^{2}-\displaystyle \frac{1}{2k\left({\gamma }_{1}r+{\gamma }_{2}t+{\gamma }_{3}\right)},\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=\displaystyle \frac{9}{2}{\tilde{a}}^{2},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}m-\displaystyle \frac{{s}^{2}}{2J}=\displaystyle \frac{{\tilde{a}}^{2}}{4}{\left[\sqrt{2k\left({\gamma }_{1}r+{\gamma }_{2}t+{\gamma }_{3}\right)}\right]}^{3}\\ \quad +\displaystyle \frac{1}{2}\sqrt{2k\left({\gamma }_{1}r+{\gamma }_{2}t+{\gamma }_{3}\right)}.\end{array}\end{eqnarray}$
6.1.2. K = δH, $\delta =\mathrm{constant}$
In this model, we will assume the supplementary constraint K = δH along with the conditions YTF = 0 and Θ = 0, instead of $\dot{J}=\tilde{a}(t){KJ}$. When we put the constraint of K proportional to H in the condition YTF = 0, we obtain63 ) upon substituting equation (64 ) becomes65 ) transforms into66 ), produces67 ) in $y=\tfrac{{f}_{z}}{f}$ and integrating it, we reached at68 ), the significant variables, in the presence of charge, are determined as
$\begin{eqnarray}\displaystyle \frac{3}{{\delta }^{2}}\displaystyle \frac{\ddot{J}}{J}-\displaystyle \frac{2J^{\prime\prime} }{J}+{\left(\displaystyle \frac{2{J}^{{\prime} }}{J}\right)}^{2}=0.\end{eqnarray}$
We consider J as $\begin{eqnarray}J=f\left({b}_{1}r+{b}_{2}t+{b}_{3}\right)\equiv f(z),\end{eqnarray}$
where b1, b2 and b3 are the arbitrary constants. Equation ( $\begin{eqnarray}\left(\displaystyle \frac{3{b}_{2}^{2}}{{\delta }^{2}}-2{b}_{1}^{2}\right)\displaystyle \frac{{f}_{{zz}}}{f}+\displaystyle \frac{4{b}_{1}^{2}{f}_{z}^{2}}{{f}^{2}}=0.\end{eqnarray}$
For convenience, new variables are introduced as $y=\tfrac{{f}_{z}}{f}$. After this, equation ( $\begin{eqnarray}{y}_{z}+{\eta }_{0}{y}^{2}=0,\end{eqnarray}$
where $\begin{eqnarray*}{\eta }_{0}=\displaystyle \frac{3{b}_{2}^{2}+2{b}_{1}^{2}{\delta }^{2}}{3{b}_{2}^{2}-2{b}_{1}^{2}{\delta }^{2}}.\end{eqnarray*}$
Integration of equation ( $\begin{eqnarray}y=\displaystyle \frac{1}{{\eta }_{0}z+{\eta }_{1}},\end{eqnarray}$
where η1 portrays the integration constant. Putting equation ( $\begin{eqnarray}f=J={\left(\displaystyle \frac{{\eta }_{0}z+{\eta }_{1}}{{\eta }_{2}}\right)}^{\tfrac{1}{{\eta }_{0}}},\end{eqnarray}$
where η2 is another integration constant. With the help of equation ( $\begin{eqnarray}\begin{array}{rcl}8\pi \left(\mu +\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right) & = & \displaystyle \frac{{\left(\tfrac{{\eta }_{0}z+{\eta }_{1}}{{\eta }_{2}}\right)}^{\tfrac{4}{{\eta }_{0}}}\left[{b}_{1}^{2}\left(2{\eta }_{0}-7\right)-\tfrac{3{b}_{2}^{2}}{{\delta }^{2}}\right]}{{\alpha }^{2}{\left({\eta }_{0}z+{\eta }_{1}\right)}^{2}}\\ & & +{\left(\displaystyle \frac{{\eta }_{2}}{{\eta }_{0}z+{\eta }_{1}}\right)}^{\tfrac{2}{{\eta }_{0}}},\end{array}\end{eqnarray}$
$\begin{eqnarray}4\pi q=\displaystyle \frac{{\left(\tfrac{{\eta }_{0}z+{\eta }_{1}}{{\eta }_{2}}\right)}^{\tfrac{4}{{\eta }_{0}}}{b}_{1}{b}_{2}\left(5-{\eta }_{0}\right)}{\delta {\alpha }^{2}{\left({\eta }_{0}z+{\eta }_{1}\right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=\displaystyle \frac{{\left(\tfrac{{\eta }_{0}z+{\eta }_{1}}{{\eta }_{2}}\right)}^{\tfrac{4}{{\eta }_{0}}}\left[\tfrac{{b}_{2}^{2}\left(2{\eta }_{0}-7\right)}{{\delta }^{2}}-3{b}_{1}^{2}\right]}{{\alpha }^{2}{\left({\eta }_{0}z+{\eta }_{1}\right)}^{2}}\\ \quad -{\left(\displaystyle \frac{{\eta }_{2}}{{\eta }_{0}z+{\eta }_{1}}\right)}^{\tfrac{2}{{\eta }_{0}}},\end{array}\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=\displaystyle \frac{{\left(\tfrac{{\eta }_{0}z+{\eta }_{1}}{{\eta }_{2}}\right)}^{\tfrac{4}{{\eta }_{0}}}\left(1+{\eta }_{0}\right)\left({b}_{1}^{2}-\tfrac{{b}_{2}^{2}}{{\delta }^{2}}\right)}{{\alpha }^{2}{\left({\eta }_{0}z+{\eta }_{1}\right)}^{2}},\end{eqnarray}$
$\begin{eqnarray}m-\displaystyle \frac{{s}^{2}}{2J}=\displaystyle \frac{{\left(\tfrac{{\eta }_{0}z+{\eta }_{1}}{{\eta }_{2}}\right)}^{\tfrac{7}{{\eta }_{0}}}\left(\tfrac{{b}_{2}^{2}}{{\delta }^{2}}-{b}_{1}^{2}\right)}{2{\alpha }^{2}{\left({\eta }_{0}z+{\eta }_{1}\right)}^{2}}+\displaystyle \frac{1}{2}{\left(\displaystyle \frac{{\eta }_{0}z+{\eta }_{1}}{{\eta }_{2}}\right)}^{\tfrac{1}{{\eta }_{0}}}.\end{eqnarray}$
6.1.3. K = K(r), J = J1(t)J2(r)
In this model, we will insert the values of metric coefficients, i.e. K = K(r), J = J1(t)J2(r) in equation (22 ) as74 ) become6.1 allows one to examine the presence of an electric charge on a non-geodesic sphere, which can lead to complex effects on the realistic features of the self-gravitating systems. The distribution and the strength of the charge influence the mass, density, radial and tangential pressure of the stellar structure under various assumptions. One can observe the previously stated effects with the help of equations (58 )-(62 ), (69 )-(73 ) and (78 )-(82 ).
$\begin{eqnarray}\displaystyle \frac{3}{{K}^{2}}\left(\displaystyle \frac{\ddot{J}}{J}-\displaystyle \frac{2{\dot{J}}^{2}}{{J}^{2}}-\displaystyle \frac{\dot{K}}{K}\displaystyle \frac{\dot{J}}{J}\right)+\displaystyle \frac{1}{{H}^{2}}\left(\displaystyle \frac{{K}^{{\prime} }}{K}\displaystyle \frac{{J}^{{\prime} }}{J}+\displaystyle \frac{K^{\prime\prime} }{K}\right)=0\end{eqnarray}$
The solution of equation ( $\begin{eqnarray}{J}_{1}(t)=\displaystyle \frac{{\xi }_{0}}{t+{\xi }_{1}},\end{eqnarray}$
$\begin{eqnarray}{J}_{2}(r)={\xi }_{2}{K}^{{\xi }_{3}-1},\end{eqnarray}$
$\begin{eqnarray}K={\xi }_{4}{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{1}{\xi /3}},\end{eqnarray}$
where ξ0, ξ1, ξ2, ξ3, ξ4 and ξ5 are constants. Along with this, ξ1, ξ2, ξ4 and ξ5 have dimensions $[r],\,[{r}^{2}],\,[{r}^{\tfrac{1}{{\xi }_{3}}}]$ and [r], respectively. On the other hand, ξ0 and ξ3 will be treated as dimensionless constants. The physical parameters using these aforementioned solutions become $\begin{eqnarray}\begin{array}{l}8\pi \left(\mu +\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=\displaystyle \frac{{\left(t+{\xi }_{1}\right)}^{2}{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{-2{\xi }_{3}+2}{{\xi }_{3}}}}{{\xi }_{0}^{2}{\xi }_{2}^{2}{\xi }_{4}^{2{\xi }_{3}-2}}\\ \quad -\displaystyle \frac{3{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{-2}{{\xi }_{3}}}}{{\xi }_{4}^{2}{\left(t+{\xi }_{1}\right)}^{2}}\\ \quad -\displaystyle \frac{{\xi }_{0}^{4}{\xi }_{4}^{4{\xi }_{3}-4}{\xi }_{2}^{4}\left(5{\xi }_{3}^{2}-12{\xi }_{3}+7\right){\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{2\left({\xi }_{3}-2\right)}{{\xi }_{3}}}}{{\alpha }^{2}{\left(t+{\xi }_{1}\right)}^{4}},\end{array}\end{eqnarray}$
$\begin{eqnarray}4\pi q=\displaystyle \frac{{\xi }_{2}^{2}{\xi }_{0}^{2}\left(4-3{\xi }_{3}\right){\xi }_{4}^{2{\xi }_{3}-3}{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{{\xi }_{3}-3}{{\xi }_{3}}}}{\alpha {\left(t+{\xi }_{1}\right)}^{3}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{5{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{-2}{{\xi }_{3}}}}{{\xi }_{4}^{2}{\left(t+{\xi }_{1}\right)}^{2}}\\ \quad -\displaystyle \frac{{\left(t+{\xi }_{1}\right)}^{2}{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{-2{\xi }_{3}+2}{{\xi }_{3}}}}{{\xi }_{0}^{2}{\xi }_{2}^{2}{\xi }_{4}^{2{\xi }_{3}-2}}\\ \quad +\displaystyle \frac{{\xi }_{0}^{4}{\xi }_{4}^{4{\xi }_{3}-4}{\xi }_{2}^{4}\left({\xi }_{3}^{2}-1\right){\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{2\left({\xi }_{3}-2\right)}{{\xi }_{3}}}}{{\alpha }^{2}{\left(t+{\xi }_{1}\right)}^{4}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)\\ \quad =\displaystyle \frac{{\xi }_{0}^{4}{\xi }_{4}^{4{\xi }_{3}-4}{\xi }_{2}^{4}\left(2{\xi }_{3}^{2}-3{\xi }_{3}+1\right){\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{2\left({\xi }_{3}-2\right)}{{\xi }_{3}}}}{{\alpha }^{2}{\left(t+{\xi }_{1}\right)}^{4}}\\ \quad -\displaystyle \frac{2{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{-2}{{\xi }_{3}}}}{{\xi }_{4}^{2}{\left(t+{\xi }_{1}\right)}^{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{rcl}m-\displaystyle \frac{{s}^{2}}{2J} & = & \displaystyle \frac{{\xi }_{0}^{3}{\xi }_{2}^{3}{\xi }_{4}^{3{\xi }_{3}-5}{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{3{\xi }_{3}-5}{{\xi }_{3}}}}{2{\left(t+{\xi }_{1}\right)}^{5}}\\ & & +\displaystyle \frac{{\xi }_{0}{\xi }_{2}{\xi }_{4}^{{\xi }_{3}-1}{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{{\xi }_{3}-1}{{\xi }_{3}}}}{2\left(t+{\xi }_{1}\right)}\\ & & -\displaystyle \frac{{\xi }_{0}^{7}{\xi }_{2}^{7}{\xi }_{4}^{7\left({\xi }_{3}-1\right)}{\left({\xi }_{3}-1\right)}^{2}{\left({\xi }_{3}r+{\xi }_{5}\right)}^{\tfrac{5{\xi }_{3}-7}{{\xi }_{3}}}}{2{\alpha }^{2}{\left(t+{\xi }_{1}\right)}^{7}}.\end{array}\end{eqnarray}$
The subsection 6.2. Geodesic sphere
A curve that minimizes local length is called a geodesic. It follows a route similar to what a particle would take if it were not accelerating. The concepts of distance and acceleration are impacted by the Riemannian manifold, which also influences the geodesics in space. The geodesics are large circles on the sphere (like the equator). In addition to having many other intriguing characteristics, geodesics maintain a direction on a surface.
6.2.1. K(t, r) = 1
In this model, we have considered the coefficient of the time coordinate constant. Implementing this condition, we evaluated the coefficient of radial coordinate as82 ) in the EM field equations yield84 ), (86 ) and (88 ), we achieve89 ) is formally determined as91 ), the state variables and mass function are calculated as
$\begin{eqnarray}H(t,r)=\displaystyle \frac{\alpha }{{J}^{2}}.\end{eqnarray}$
The substitution of equation ( $\begin{eqnarray}8\pi \left(\mu +\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{3{\dot{J}}^{2}}{{J}^{2}}-\displaystyle \frac{{J}^{4}}{{\alpha }^{2}}\left[2\displaystyle \frac{J^{\prime\prime} }{J}+5{\left(\displaystyle \frac{{J}^{{\prime} }}{J}\right)}^{2}\right]+\displaystyle \frac{1}{{J}^{2}},\end{eqnarray}$
$\begin{eqnarray}4\pi q=\displaystyle \frac{J{\dot{J}}^{{\prime} }}{\alpha }+\displaystyle \frac{2{J}^{{\prime} }\dot{J}}{\alpha },\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{2\ddot{J}}{J}-\displaystyle \frac{{\dot{J}}^{2}}{{J}^{2}}+{\left(\displaystyle \frac{{{JJ}}^{{\prime} }}{\alpha }\right)}^{2}-\displaystyle \frac{1}{{J}^{2}},\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=\displaystyle \frac{\ddot{J}}{J}-\displaystyle \frac{4{\dot{J}}^{2}}{{J}^{2}}+\displaystyle \frac{{J}^{4}}{{\alpha }^{2}}\left[\displaystyle \frac{J^{\prime\prime} }{J}+2{\left(\displaystyle \frac{{J}^{{\prime} }}{J}\right)}^{2}\right].\end{eqnarray}$
With the help of equations ( $\begin{eqnarray}2\pi \left(\mu +{P}_{r}+2{P}_{\perp }+\displaystyle \frac{{s}^{2}}{4\pi {J}^{4}}\right)=-3{\left(\displaystyle \frac{\dot{J}}{J}\right)}^{2}.\end{eqnarray}$
The vanishing of CF for this model produces $\begin{eqnarray}\displaystyle \frac{\ddot{J}}{J}-\displaystyle \frac{2{\dot{J}}^{2}}{{J}^{2}}=0.\end{eqnarray}$
The solution of equation ( $\begin{eqnarray}J=\displaystyle \frac{1}{{c}_{1}(r)t+{c}_{2}(r)}\equiv \displaystyle \frac{1}{{c}_{1}(r)\left[t+\tfrac{{c}_{2}(r)}{{c}_{1}(r)}\right]}.\end{eqnarray}$
Finally, we accomplish $\begin{eqnarray}\begin{array}{rcl}J & = & \displaystyle \frac{\alpha }{\left[t+\alpha {c}_{2}(r)\right]},\\ H & = & \displaystyle \frac{{\left[t+\alpha {c}_{2}(r)\right]}^{2}}{\alpha }.\end{array}\end{eqnarray}$
Utilizing equation ( $\begin{eqnarray}\begin{array}{l}8\pi \left(\mu +\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{3}{{\left(t+\alpha {c}_{2}\right)}^{2}}+\displaystyle \frac{2{\alpha }^{3}{c}_{2}^{\prime\prime} }{{\left(t+\alpha {c}_{2}\right)}^{5}}\\ \quad -\displaystyle \frac{9{\alpha }^{4}{\left({c}_{2}^{{\prime} }\right)}^{2}}{{\left(t+\alpha {c}_{2}\right)}^{6}}+\displaystyle \frac{{\left(t+\alpha {c}_{2}\right)}^{6}}{{\alpha }^{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}4\pi q=\displaystyle \frac{4{\alpha }^{2}{c}_{2}^{{\prime} }}{{\left(t+\alpha {c}_{2}\right)}^{4}},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{5}{{\left(t+\alpha {c}_{2}\right)}^{2}}+\displaystyle \frac{{\alpha }^{4}{\left({c}_{2}^{{\prime} }\right)}^{2}}{{\left(t+\alpha {c}_{2}\right)}^{6}}\\ \quad -\displaystyle \frac{{\left(t+\alpha {c}_{2}\right)}^{2}}{{\alpha }^{2}},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{2}{{\left(t+\alpha {c}_{2}\right)}^{2}}+\displaystyle \frac{4{\alpha }^{4}{\left({c}_{2}^{{\prime} }\right)}^{2}}{{\left(t+\alpha {c}_{2}\right)}^{6}}\\ \quad -\displaystyle \frac{{\alpha }^{3}{c}_{2}^{\prime\prime} }{{\left(t+\alpha {c}_{2}\right)}^{5}},\end{array}\end{eqnarray}$
$\begin{eqnarray}m-\displaystyle \frac{{s}^{2}}{2J}=\displaystyle \frac{{\alpha }^{3}}{2{\left(t+\alpha {c}_{2}\right)}^{5}}-\displaystyle \frac{{\alpha }^{7}{\left({c}_{2}^{{\prime} }\right)}^{2}}{2{\left(t+\alpha {c}_{2}\right)}^{9}}+\displaystyle \frac{\alpha }{2\left(t+\alpha {c}_{2}\right)}.\end{eqnarray}$
6.2.2. Quasi-homologous constraint
In this geodesic model, we will consider the quasi-homologous constraint with the restriction of K = 1. These suppositions lead to yield6.2 , one can analyze the behavior of the physical parameters of the geodesic sphere depending upon the charge distribution. The electric field may exert force on the charged particles within the geodesic sphere, causing them to interact with one another. These interactions contribute to transfer energy within the sphere. Consequently, the composition of charge influences radial pressure, density as well as tangential pressure. Moreover, the charge has no impact on the heat flow. Using equations (92 )-(96 ) and (98 )-(102 ), one can observe the aforementioned impacts of electric charge.
$\begin{eqnarray}J=\displaystyle \frac{h(r)}{t},\quad H=\displaystyle \frac{{t}^{2}}{\alpha },\end{eqnarray}$
where h(r) is an arbitrary function of r. The state variables under these values of coefficients produces $\begin{eqnarray}8\pi \left(\mu +\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{3}{{t}^{2}}-\displaystyle \frac{{\alpha }^{2}}{{t}^{4}}\left[\displaystyle \frac{2h^{\prime\prime} }{h}+{\left(\displaystyle \frac{{h}^{{\prime} }}{h}\right)}^{2}\right]+\displaystyle \frac{{t}^{2}}{{h}^{2}},\end{eqnarray}$
$\begin{eqnarray}4\pi q=-\displaystyle \frac{3\alpha {h}^{{\prime} }}{{t}^{3}h},\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{r}-\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{5}{{t}^{2}}+\displaystyle \frac{{\alpha }^{2}}{{t}^{4}}{\left(\displaystyle \frac{{h}^{{\prime} }}{h}\right)}^{2}-\displaystyle \frac{{t}^{2}}{{h}^{2}},\end{eqnarray}$
$\begin{eqnarray}8\pi \left({P}_{\perp }+\displaystyle \frac{{s}^{2}}{8\pi {J}^{4}}\right)=-\displaystyle \frac{2}{{t}^{2}}+\displaystyle \frac{{\alpha }^{2}}{{t}^{4}}\displaystyle \frac{h^{\prime\prime} }{h},\end{eqnarray}$
$\begin{eqnarray}m-\displaystyle \frac{{s}^{2}}{2J}=\displaystyle \frac{h}{2t}\left[\displaystyle \frac{{h}^{2}}{{t}^{4}}-{\left(\displaystyle \frac{\alpha {h}^{{\prime} }}{{t}^{3}}\right)}^{2}+1\right].\end{eqnarray}$
Based on the subsection 7. Discussion and final remarks
The end state of a complete gravitational collapse, whether or not it is spherically symmetric, maybe a vacuum spacetime containing the rotation and perhaps also the electromagnetic fields connected to the object. It is feasible that the surrounding plasma will swiftly neutralize the charge that an astrophysical object is carrying. It will be interesting to find all EM equation solutions that represent stationary collapsed configurations with charge, though, in any case. Two hypersurfaces are considered to define the boundaries of the matter distribution. The interior one delimits the cavity in which we have Minkowski spacetime and the external one separates the fluid distribution from a Vaidya Reissner Nordstörm spacetime. Since the known Universe cannot be regarded as homogeneous on scales smaller than 150-300 Mpc, it should be obvious that for cavities of the order of 20 Mpc or smaller, the assumption of a spherically symmetric spacetime outside the cavity is perfectly reasonable. However, it should be more appropriate to take into account a larger cavity's embedding in an expanding Friedmann-Lemaître-Robertson-Walker spacetime. As a result, junction conditions for both hypersurfaces must be taken into account. Different types of models are produced depending on whether Darmois requirements are imposed [52] or thin shells are allowed to exist [23, 57].
We are interested in analytical models that, despite being relatively easy to assess, still include some of the key components of a realistic scenario. It should be underlined that we are only interested in the evolution of the cavity after it has already developed, we are not interested in the dynamics and conditions of the formation of the cavity. It is intuitively obvious that the evolution of an expansion-free, spherically symmetric charged matter content is consistent with the presence of a vacuum cavity inside the distribution due to the fact that the expansion scalar characterizes the rate of change of small volumes of the fluid. We may be able to use our solutions as toy models of localized systems, like supernova explosion simulations. More physically meaningful models would probably result from combining the vanishing expansion scalar requirement with numerical integration of the associated equations. It is worth noting that the Kelvin-Helmholtz phase is of particular importance in these scenarios [58].
Nonlinear electromagnetic fields can cause faster expansion and other favorable cosmological properties, resulting in good qualitative phenomenology. For inflationary epochs (anisotropy, high energies), NEF are a valid assumption. With the help of these equations, we deduced the following results
| • | The innermost fluid shell must be located outside of the center to satisfy the expansion-free (Θ = 0) requirement, which causes a cavity to develop as a result. So, one can say that the central region cannot be occupied by the charged fluid. |
| • | The constraint YTF = 0 is helpful for creating alternative models and can provide insight into the consequences of charge on the physical characteristics of astrophysical objects. |
| • | The charged fluid influences density and mass in such a manner that the celestial object becomes more massive and less dense as one can verify from equations ( |
| • | In the presence of electric force, the fluid exerts more pressure in the radial direction while less pressure is exerted in the perpendicular direction. One can witness it from equations ( |
| • | The heat flux has no influence of charge on it. This can be observed from equations ( |
| • | When s = 0 is substituted, all of the results found in this manuscript will be reduced in GR. |
In this manuscript, we have discussed two different families of solutions. One deals with the non-geodesic sphere and the other is the geodesic sphere. In the case of both spheres, we first consider the constraint of vanishing CF and quasi homologous condition in the presence of charge. After that, we apply additional constraints on metric coefficients. Eventually, the mass functions for all of these models are evaluated along with their matter variables. The charge influences the energy density of the spherical systems evolving along geodesic or non-geodesic congruences in such a manner that its presence makes the compact object less dense no matter what type of charge is (positive or negative). It has been seen that the contribution of electric charges in the relativistic spherical matter reduces the impact of energy density and tangential pressure components in its modeling. However, totally reverse behavior is observed for the mass function and radial pressure.
Conflict of interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Data accessibility declaration
This published article contains all of the data examined or studied during this research.