Influence of Electric Charge and Modified Gravity on Density Irregularities

This work aims to identify some inhomogeneity factors for plane symmetric topology with anisotropic and dissipative fluid under the effects of both electromagnetic field as well as Palatini $f(R)$ gravity. We construct the modified field equations, kinematical quantities and mass function to continue our analysis. We have explored the dynamical quantities, conservation equations and modified Ellis equations with the help of a viable $f(R)$ model. Some particular cases are discussed with and without dissipation to investigate the corresponding inhomogeneity factors. For non-radiating scenario, we examine such factors with dust, isotropic and anisotropic matter in the presence of charge. For dissipative fluid, we investigate the inhomogeneity factor with charged dust cloud. We conclude that electromagnetic field increases the inhomogeneity in matter while the extra curvature terms make the system more homogeneous with the evolution of time.


Introduction
The inclusion of higher order curvature invariants in the action for the modifications of general relativity (GR) have a long primordial history. An alternative approach hypothesizes that GR is accurate only on small scales and has to be generalize on large/cosmological distances. The early struggle was mostly due to the scientific curiosity to understand the newly proposed theory and to give some alternative to dark energy model. However, new motivations came from some theoretical aspects of its physics which revived the study of higher order gravity theories [1]. To initiate with, there are various techniques and proposal for modified gravity to deviate from GR. The f (R) theories of gravity [2] are the straightforward generalization of the Einstein Hilbert action, in which the Ricci scalar (R) becomes a generic function of R. In fact, it is relatively simple and compelling alternative to GR, from which some important results have already been obtained in the literature.
It is worth mentioning that one can apply two variational principles to derive f (R) field equations from the modified form of Einstein-Hilbert action. One is the standard metric variation while the second one is dubbed as the Palatini variation in which the connection and metric dealt as independently. More precisely, one has to vary the action with respect to both metric and connection in such a manner that the matter action does not depend upon connection. Accordingly, there would be two versions of f (R) gravity corresponding to which the variational formalism is explored. Here, the Einstein-Hilbert action can be modified through its gravitational part in order to discuss the f (R) theory of gravity as [4] S f (R) = 1 2κ where κ, S M and f (R) are coupling constant, matter action and a non-linear Ricci function, respectively. Applying the variation with metric (g αβ ) and connection (Γ ρ αβ ) in the above action, respectively, one can formulate the following couple of equations of motion as ∇ µ (g αβ √ −gf R (Ȓ)) = 0.
By taking the trace of Eq.(1), we can constitute an analogy between T ≡ g αβ T αβ and R ≡ R(Γ) as which describes the reliance of Ricci scalar on T . To examine a consistent Palatini f (R) gravity with any other classical theory, we have to deal with only such situations where the solution of the above equation exists. With present cosmological value of Ricci invariant, i.e., R =R, Eq.(3) leads to the covariant conservation of metric thereby fixing Γ ρ αβ to Levi-Civita. Consequently, for vacuum cases, Eq.(1) turns out to bȇ whereȒ αβ is called metric Ricci tensor of g αβ and Λ(R) =R/4. This theory would lessen to GR in the presence/absence of cosmological constant depending on a viable f (R) model. One can obtain a single expression for field equations in Palatini f (R) formalism by substituting Γ σ αβ from Eq.(2) in terms of g αβ as follows which can be written in an alternative form as here is the effective energy-momentum tensor in Palatini f (R) terms describing modified gravitational contribution whileG αβ ≡Ȓ αβ − 1 2 g αβȒ ,˘ = ∇ α∇β g αβ , where∇ α shows covariant derivative with respect to Levi-Civita connection. It is interesting to note that f R and f are functions of R(Γ) ≡ g αβ R αβ (Γ). If one disregard the supposition that matter action is independent of connection, then a new version of f (R) gravity is observed, called metric-affine f (R) gravity, which have both Palatini and metric f (R) on its usual limits.
The viability criteria for any gravitational theory include [2,3] stability, correct Newtonian and post-Newtonian limits, a correct cosmological dynamics, cosmological perturbations compatible with large scale structures and cosmic microwave background and the absence of ghosts. Many interesting results emerge from f (R) gravity as they predict the early universe inflation and have a well-posed Cauchy problem. Nojiri and Odintsov [5] studied various modified gravity models as an alternative to dark energy. They investigate that inhomogeneous terms are originated due to modified gravity model of the universe. Guo and Joshi [6] examined the collapse of spherical star due to Starobinsky R 2 model within the framework of f (R) gravity. Here, we would like to discuss the inhomogeneities/irregularities which emerge in the energy density due to Palatini version of f (R) gravity.
Anisotropic effects are leading paradigms in the description of evolutionary mechanisms of stellar collapsing models. It is an established fact the properties of anisotropic models may differ drastically in contrast with the isotropic spheres. Nguyen and Pedraza [7] investigated anisotropic spherical compact model and deduced that anisotropic effects make the system dissipative with the evolution of time. Leon and Sarikadis [8] investigated impact of anisotropy in the framework of modified gravity and concluded different cosmological behaviors in the geometry as compare to isotropic scenarios. Cosenza et al. [9] figured out the role of anisotropy on radiating fluid spheres. Maartens et al. [10] analyzed the anisotropic evolution of the universe during an intermediate transient regime of inflationary expansion.
The anisotropic picture in relativistic fluid configurations can be achieved by many interconnected phenomena like the existence of strong electric and magnetic interactions [11]. A great deal of attention has also been given to the interaction of electromagnetic and gravitational fields. However, a general harmony exist among the relativists that physical objects with a large amount of electric charge does not exist in nature. This view of thought have been challenged by many researchers and a variety of work have been carried out with this background. Ghezzi [12] explored some analytical models of isotropic spherical star in the presence of electromagnetic field in which the charge density is proportional to the rest mass density. He found that the radius of charged stars is larger as compared to the uncharged ones. Varela et al. [13] solved the Einstein-Maxwell field equations for self-gravitating anisotropic spherical system numerically and link their findings with the models of dark matter including massive charged particles as well as charged strange quark stars. The impact of electromagnetic field and other matter variables on the evolutionary behavior of collapsing relativistic self-gravitating systems in cosmos have been investigated in [14] and [15].
A system begins collapsing once it experiences an inhomogeneous stellar state. Penrose and Hawking [16] explored irregularities in the energy density of spherical relativistic stars by means of Weyl invariant. Herrera et al. [17] discussed the role of density inhomogeneities on the structure and evolution of spherically anisotropic objects. Herrera et al. [18] did a systematic study on structure formation of self-gravitating compact stars by means of some scalar functions (trace and trace-free parts) obtained from splitting of the Riemann tensor. These scalars are associated with electric, magnetic as well as second dual of the Riemann tensor and have eventual relationship with fundamental properties of the matter configuration [19]. The inhomogeneity in the universe can be linked with the dipole anisotropy as found by Planck [20]. Herrera [21] investigated different physical factors responsible for the emergence of inhomogeneities in an initial regular spherical collapsing distributions. Sharif and Yousaf [22] described stability of regular energy density in planar matter distribution by taking into account three parametric model form in Palatini f (R) gravity.
The inhomogeneous models can also be used to discuss the SN-data [23]. Geng and Lü [24] presented a class of models describing the isotropic expansion for inhomogeneous universe. The unresolved issues of the dark energy/dark matter on the homogeneity of the collapsing compact star is still a matter of interest for relativists. We will address two main related problems in this paper: 1. We explore inhomogeneity factors for plane symmetric compact object and discuss it with some particular cases by increasing the complexity in the matter distribution.

2.
The role of Palatini f (R) dark source terms through a viable f (R) model as well as electromagnetic field effects will be analyzed. This paper is outlined in following manner. In the next section, we deduce field equations coupled with the source in f (R) gravity under the influence of electromagnetic field. Section 3 investigates the dynamical as well as evolution equations for the systematic analysis of inhomogeneity factors. In section 4, we formulate the irregularity factors with some particular cases of dissipative and non-dissipative matter field. Finally, we conclude our main findings in the last section.

f (R) Gravity Coupled to Matter Source
We choose a non-static planar geometry for the construction of our systematic analysis as [25] while it is filled with dissipative fluid by means of diffusion (heat) as well as free streaming (null radiation) approximations having anisotropic pressure in the interior. Such matter fields are described by the energy-momentum tensor as follows where ε, µ, P ⊥ , P r and q β are the radiation density, energy density, different stress components and heat flux vector, respectively. In comoving coordinate system, the unit four vector l β = 1 A δ β 0 + 1 C δ β 3 , radial four vector, i.e., χ β = 1 C δ β 3 as well as four velocity vector V β = 1 A δ β 0 , satisfies the following relations The expansion rate of matter configuration under Palatini f (R) background is defined by the scalar as where dot indicates the operator ∂ ∂t . The shear scalar for planar in the framework of GR yields [21] Using Eqs. (9) and (10), we can determine a relation between expansion and shear as follows The stress-energy tensor describing the electromagnetic field and satisfying the Maxwell field equations, i.e., F αβ Here J α and µ 0 = 4π represent four current and magnetic permeability, respectively. The four potential and four current are φ α = φδ α 0 , J α = ξV α , under comoving coordinate system while φ, ξ are functions of t and z representing the scalar potential and charge density, respectively. The non-zero components of the Maxwell field equations yields the following couple of equations as Here prime indicates z differentiation. Integration of Eq.(12) with respect to z provides which equivalently satisfies Eq. (13). The non-vanishing components of the electromagnetic stress tensor turns out to be The field equations in the framework of Palatini f (R) gravity corresponding to planar geometry leads to To describe the quantity of matter within the planar system, the mass function can be evaluated through Taub mass formalism in the presence of electromagnetic field as [26] m(t, z) = (g) which can be written in an alternative way using the fluid velocity as Using Eqs.(16)- (19), the temporal and radial variations of mass function leads to where U is the velocity of the collapsing matter defined by ∂r represents radial derivative operator, respectively. Here E denotes electric field intensity. It is interesting to indicate that for planar celestial configuration undergoing collapse, U is chosen to be less than unity. The link between matter variables and mass function can be found through integration of Eq.(23) with Palatini f (R) background as The electric component of the Weyl tensor in terms of unit four velocity and radial four vector is given as is the scalar encapsulating the effects of spacetime curvature. Alternatively, using Eqs. (16) and (18)- (20), we get 3m whereΠ =P z − P ⊥ . The above equation determines the gravitational contribution of planar geometry due to its fluid variables, mass function and f (R) extra curvature terms.

Dynamical and Evolution Equations
In this section, we will establish some scalar functions in the background of a well-consistent f (R) model. We then make a correspondence between fluid parameters and Weyl scalar with Palatini f (R) corrections by constructing modified Ellis equations. In order to discuss the dynamical properties framed within modified cosmology, we take the f (R) model as follows [29] f where µ and R c are positive constants. The values of these free parameters are, where R c is roughly of the same order as the Ricci scalar today, H 0 is the present day value of Hubble constant, and the critical density ρ c ≃ 10 −29 gr/cm 3 ∼ 4.5 × 10 −47 GeV 4 . We categorize the Riemann tensor in terms of second rank tensors, i.e., X αβ and Y αβ , to devise modified form of structure scalars as [27] where left, right and double dual of the Riemann curvature tensor can be respectively written in a standard form as The above tensors can further disintegrate into their trace and trace-free components as By making use of Eqs. (16), (18), (19) and (27)- (29), these scalar functions can be written in terms of fluid variables as where δ µ , δ Pz and δ P ⊥ are the corresponding values of dark source components evaluated by taking into account Eq. (27) and given in the Appendix A. We found that trace part of the second dual of the Riemann tensor has its dependence on the energy density profile of planar geometry with some extra curvature terms due to f (R) Palatini gravity while the remaining scalar functions have their dependence on anisotropic stress tensor. The conservation of energy and momentum from the contracted Bianchi identities with ordinary and effective matter fields T αβ + T αβ ;β = 0 yields the couple of equations as followṡ where the terms D 0 and D 1 emerge due to Palatini f (R) gravity and are mentioned in Appendix A. Next, we continue our investigation by constructing couple of differential equations using the procedure adopted by Ellis [?]. These equations are found by using Eqs.(16)- (19), (22), (23) and (27) and defines a link between matter variables with Palatini f (R) extra curvature terms and Weyl tensor as where δ q is mentioned in Appendix A. The limit f (R) → R in the above equations provides GR Ellis equations.

Irregularities in the Dynamical System
This section explores some fluid variables that are responsible for irregularities in the dynamical system having planar symmetry. This analysis has been carried out from initial homogeneous configuration of compact body by means some particular choices on matter fields with extra curvature terms of Palatini f (R) gravity. We will restrict our analysis on the present day value of cosmological Ricci scalar, i.e., R =R while dealing with bulky system of equations. Finally, we will study the case with zero expansion. We classify our investigation in two scenarios, i.e., dissipative and non-dissipative systems as follows:

Non-radiating Matter
This section deals with non-dissipative choices of matter fields like dust, perfect and anisotropic fluid configurations, respectively, in the Palatini f (R) gravity back ground.

Dust Fluid
In this case, we considerP z = 0 = P ⊥ =q and A = 1 indicating geodesic motion of non-dissipative dust cloud. In this scenario, the two differential equations for Weyl tensor obtained in (36) and (37) reduce to When µ ′ = 0, Eq.(39) leads to The general solution of the above equation is obtained as It is worth noting that Weyl scalar is the only geometric entity responsible for the irregularities in the energy density depending upon the electromagnetic profile. In the absence of electromagnetic field, the Weyl scalar will also vanish showing the importance of charged fields. By making use of Eqs.(11), (34) and (B3)-(B6) in Eq.(38), we founḋ The above equation reveal the relationship of Weyl scalar with shear scalar indicating the shearing motion of dust cloud. It also shows that the system will be homogeneous if it is shear free as well as conformally flat within the Palatini framework of f (R) gravity. Its solution turns out to be The role of expansion can be brought forward while discussing the inhomogeneities on the evolution of dust matter in the collapse scenario. We study the zero expansion case, i.e., Θ P = 0, so that the above equation becomes It shows that the expansion free system will be inhomogeneous due to the presence of Weyl scalar as it produces tidal forces which made the object inhomogeneous with the passage of time thus indicating the importance of time. Moreover, in the absence of tidal forces, the system will be inhomogeneous due to the presence of electromagnetic field. Consequently, an expansion free system will be homogeneous if it is charged free and conformally flat.

Isotropic Fluid
In this case, we introduce a bit of complexity in the previous case by adding the effects of isotropic pressure and determine the inhomogeneity factors. In this scenario, the Ellis equations (36) and (37) turn out to be We see that the second equation is the same as we have evaluated in the above case with dust cloud (see Eq.(38)). Therefore, indicating the Weyl scalar as the factor responsible of irregularities in the matter distribution. By making use of Eqs. (11) and (34), Eq.(44) leads tȯ which on integration turns out to be This indicates the importance of shear on the evolution of inhomogeneous matter configuration with isotropic pressure. We observed that not only shear and pressure, but extra curvature terms due to f (R) gravity are acting on the system to make it inhomogeneous as the evolution proceeds. We can also examine the factors responsible for irregularities over the relativistic system with zero shear. Moreover, we have already obtained a relation between expansion and shear scalar, therefore, we can analyze the those effects when the system is undergoing collapse with zero expansion. With zero expansion condition, Eq.(46) provides E = 3κ It is seen from the above expression that electromagnetic field have also crucial role in the expansion free scenario. The Weyl scalar also plays a key role due to tidal forces making the system more inhomogeneous with the passage of time.

Anisotropic Fluid
This case generalizes the previous one by introducing the complexity in the form of anisotropic stresses while the dissipative effects are assumed to be zero, i.e., Π = 0 andq = 0. In this framework, the two equations obtained in (36) and (37) takes the form as We can found the following couple of equations by using Eqs. (11) and (34) in (49) and (50) with some computation as Using the trace free part of the second dual of Riemann tensor as obtained in Eq.(31), we founḋ The solution of the the above couple of differential equations turn out to be Equation (51) shows a relation of one of the scalar function, from the splitting of the Riemann tensor, with anisotropic pressure and shear scalar. It indicates the importance of these material variables with planar geometry in the discussion of irregular energy distribution. Now, the factor, that controls inhomogeneities over the compact system, is the trace free part of the double dual of the Riamann tensor, which is obtained through the orthogonal splitting of the Riemann tensor in the framework of Palatini f (R) gravity as seen from Eq.(52). It is well known that these scalar functions play a crucial role in the structure formation of the universe. Also, the solution of the field equations in the static case can be written in the form of these scalar functions. We found that in the absence of electromagnetic field, X T F is the factor describing the irregularities in the star configuration. Consequently, if X T F = 0 then the matter distribution in the charged free system will be homogeneous and vice versa. Next, we discuss the case of collapsing matter with zero expansion in the presence of anisotropic pressure. In this scenario, the solution of Eq.(49) becomes which provides a link of the structure scalar with energy density and pressure anisotropy in the arrow of time with extra curvature terms due to Palatini f (R) gravity. Since we know that in the expansion free system, the center is surrounded by another spacetime appropriately matched with the rest of the system.

Radiating Dust Fluid
This section explores the inhomogeneity factors with dissipation in both diffusion and free streaming limit, but with a particular case of charged dust cloud. For this purpose, we take P z = 0 = P ⊥ in the matter field and the motion is considered to be geodesic by assuming A = 1 in the geometric part, which is well justified on the basis of some theoretical advances made in the discussion of inhomogeneous matter distribution. In this framework, Eqs.(36) and (37) yield Consider If we consider the matter distribution to be homogeneous i.e., µ ′ = 0, then from Eq.(55) we obtain the following expression which should be vanishing for the homogeneous fluid distribution over planar geometry. Consequently, for homogeneous universe with planar topology, one should have Ψ = 0 ⇔ µ ′ = 0 with dissipative charged dust cloud. The evolution equation for Ψ can also be evaluated using Eqs. (11) and (34) in Eq.(54) aṡ whose solution leads to This indicates the importance of fluid parameters as the inhomogeneity factor is related to the matter variables, particularly heat flux, as well as kinematical quantities of the system. Since we already found a relation in which the shear scalar is related to expansion scalar. Thus, for the shear free case, we can obtainḂ Using the above equation in Eq.(35), we founḋ Next, the transportation of heat in the system can be analyzed through a casual radiating theory defined by Muller and Israel as a second order thermodynamical theory in diffusion approximation as follows Substituting the value ofq from Eq.(60) in the above equation, we obtain One can identify the relaxation effects by inserting this value in the evolution equation for the inhomogeneity factor in this case, i.e., Ψ as obtained in Eq.(59). Consequently, the effects of electromagnetic field with the relaxation time can also be analyzed.

Discussion
In this paper, we have investigated some inhomogeneity factors for selfgravitating plane symmetric model. We have done this analysis by taking anisotropic matter distribution in the presence of electromagnetic field. The particular attention has been given to examine the role of dark source terms coming from the modification of gravitational field. The modification includes higher order curvature terms explicitly due to Palatini f (R) corrections in the field equations. In order to continue our analysis systematically, first of all we have explored the Palatini f (R)-Maxwell field equations for our compact object and define the mass function using Taub's mass formalism. An expression for the Weyl scalar has been disclosed in terms of matter variables and higher curvature ingredients due to modified gravity. A set of scalar functions have been evaluated using the splitting of Riemann curvature tensor with comoving coordinate system to discuss the irregularities in energy density. These scalar functions are named as structure scalars whose physical significance have been analyzed in the literature previously. Also, it is established that these scalars are used to write down solutions of field equations with static background metric. We have related these scalars in terms of material variables and dark source terms using the field equations and a cosmological f (R) model. Moreover, a couple of equations describing the conservation of energy-momentum in space have been explored. The evolution equations are also investigated using the procedure adopted by Ellis [30]. We have found some factors responsible for inhomogeneities in the matter configuration with some particular cases of fluid distribution.
A particular attention is given to examine the role of electromagnetic field in this framework. It is usually considered in the study of relativistic astrophysics that compact objects have no sufficient internal electric fields.
It is still vivid that stars can have total net charge or large internal electric fields. However, it is well established that angular momentum plays the role of electric charge in rotating collapsing stars. In the present study, we have dropped some light on more realistic astrophysical scenario, i.e., the inhomogeneities/irregularities in the universe model. The galaxy distribution is observed to be inhomogeneous at small scales while, according to theoretical models, it is expected to become spatially homogeneous for r > λ 0 ≈ 10Mpch −1 [28].
On the basis of the results we have obtained, it is clear that the system becomes inhomogeneous as the evolution proceeds with time indicating a crucial role of gravitational arrow of time. In the non-radiating dust cloud case, we have found that the system will be homogeneous in the absence of electromagnetic field as well as tidal forces, which are due to the presence of Weyl scalar. It shows that the Weyl tensor and presence of charge make the distribution of matter more inhomogeneous during evolution of the universe. With the inclusion of isotropic pressure in matter configuration, the Weyl tensor and electric charge behave similarly as in the dust case. In the presence of anisotropic pressure effects in the matter, we have found a particular factor, known as the trace free component of dual of the Riemann tensor, responsible for the irregularities in the planar system. In the radiating dust cloud case, we have found that the system will be homogeneous if the factor Ψ given in Eq.(59) vanishes otherwise it will make our geometric model more inhomogeneous with evolution in time.
All of our results reduces to charge-free case [22] under the limit s = 0 while our results support the analysis made by [21] under the limit f (R) = R

Appendix A
The higher curvature terms D 0 and D 1 of Eqs.(34) and (35) are given as The quantities δ µ , δ Pz , δ P ⊥ and δ q are