Eur. Phys. J. C (2016) 76:219DOI 10.1140/epjc/s10052-016-4064-2

Regular Article - Theoretical Physics

Influence of electric charge and modified gravity on densityirregularities

M. Zaeem Ul Haq Bhattia, Z. Yousaf b

Department of Mathematics, University of the Punjab, Quaid-i-Azam Campus, Lahore 54590, Pakistan

Received: 27 November 2015 / Accepted: 5 April 2016 / Published online: 21 April 2016© The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract This work aims to identify some inhomogene-ity factors for a plane symmetric topology with anisotropicand dissipative fluid under the effects of both electromag-netic field as well as Palatini f (R) gravity. We construct themodified field equations, kinematical quantities, and massfunction to continue our analysis. We have explored thedynamical quantities, conservation equations and modifiedEllis equations with the help of a viable f (R) model. Someparticular cases are discussed with and without dissipation toinvestigate the corresponding inhomogeneity factors. For anon-radiating scenario, we examine such factors as dust, andisotropic and anisotropic matter in the presence of charge. Fora dissipative fluid, we investigate the inhomogeneity factorwith a charged dust cloud. We conclude that the electromag-netic field increases the inhomogeneity in matter while theextra curvature terms make the system more homogeneouswith the evolution of time.

1 Introduction

The inclusion of higher order curvature invariants in theaction for the modifications of general relativity (GR) havea long primordial history. An alternative approach hypothe-sizes that GR is accurate only on small scales and has to begeneralized on large/cosmological distances. The early effortwas mostly due to the scientific curiosity to understand thenewly proposed theory and to find some alternative to darkenergy model. However, new motivations came from sometheoretical aspects of its physics which revived the studyof higher order gravity theories [1–4]. To begin with, thereare various techniques and proposal for modified gravity todeviate from GR. The f (R) theories of gravity [5,6] are thestraightforward generalization of the Einstein–Hilbert action,in which the Ricci scalar (R) becomes a generic function of

a e-mail: mzaeem.math@pu.edu.pkb e-mail: zeeshan.math@pu.edu.pk

R. In fact, it is a relatively simple and compelling alternativeto GR, from which some important results have already beenobtained in the literature.

It is worth mentioning that one can apply two variationalprinciples to derive f (R) field equations from the modifiedform of the Einstein–Hilbert action. One is the standard met-ric variation, while the second one is dubbed the Palatinivariation in which the connection and metric are dealt withindependently. More precisely, one has to vary the actionwith respect to both metric and connection in such a mannerthat the matter action does not depend upon the connection.Accordingly, there would be two versions of f (R) gravity,corresponding to which variational formalism is explored.Here, the Einstein–Hilbert action can be modified throughits gravitational part in order to discuss the f (R) theory ofgravity as [9]

S f (R) = 1

2κ

∫d4x

√−g f (R) + SM ,

where κ, SM , and f (R) are coupling constant, matter actionand a non-linear Ricci function, respectively. Applying thevariation with metric (gαβ ) and the connection (�ρ

αβ ) in theabove action, respectively, one can formulate the followingcouple of equations of motion:

fR(R)Rαβ − [gαβ f (R)]/2 = κTαβ, (1)

∇μ(gαβ√−g fR(R)) = 0. (2)

By taking the trace of Eq. (1), we can constitute an analogybetween T ≡ gαβTαβ and R ≡ R(�), thus

R fR(R) − 2 f (R) = κT, (3)

which describes the dependence of the Ricci scalar on T . Toexamine a consistent Palatini f (R) gravity with any otherclassical theory, we have to deal with only situations wherethe solution of the above equation exists. With present thecosmological value of the Ricci invariant, i.e., R = R, andEq. (3) leads to the covariant conservation of the metric

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thereby fixing �ραβ to Levi-Civita. Consequently, for vacuum

cases, Eq. (1) turns out to be

Rαβ − �(R)gαβ = 0, (4)

where Rαβ is called the metric Ricci tensor of gαβ and�(R) = R/4. This theory would lead to GR in the pres-ence/absence of a cosmological constant depending on aviable f (R) model. One can obtain a single expression forthe field equations in the Palatini f (R) formalism by substi-tuting �σ

αβ from Eq. (2) in terms of gαβ as follows:

1

fR

(∇α∇β − gαβ�

)fR + 1

2gαβ R

+ κ

fRTαβ + 1

2gαβ

(f

fR− R

)

+ 3

2 f 2R

[1

2gαβ(∇ fR)2 − ∇μ fR∇β fR

]− Rαβ = 0, (5)

which can be written in an alternative form as

Gαβ = κ

fR(Tαβ + Tαβ), (6)

where

Tαβ = 1

κ

(∇α∇β − gαβ�

)fR + fR

2κgαβ

(f

fR− R

)

+ 3

2κ fR

[1

2gαβ(∇ fR)2 − ∇α fR∇β fR

]

is the effective energy-momentum tensor in the Palatinif (R) terms describing a modified gravitational contribution,while Gαβ ≡ Rαβ − 1

2gαβ R, � = ∇α∇βgαβ , where∇α shows the covariant derivative with respect to the Levi-Civita connection. It is interesting to note that fR and fare functions of R(�) ≡ gαβ Rαβ(�). If one disregardsthe supposition that the matter action is independent of theconnection, then a new version of f (R) gravity is found,called metric-affine f (R) gravity, which has both Palatiniand metric f (R) on its usual limits. The viability criteria forany gravitational theory include [5–8] stability, and correctNewtonian and post-Newtonian limits, a correct cosmolog-ical dynamics, cosmological perturbations compatible withlarge scale structures and cosmic microwave background,and the absence of ghosts. Many interesting results emergefrom f (R) gravity as one predicts the early universe to haveinflation and to have a well-posed Cauchy problem. Nojiriand Odintsov [10] studied various modified gravity modelsas an alternative to dark energy. They investigated that inho-mogeneous terms originate with a modified gravity modelof the universe. Guo and Joshi [11] examined the collapseof spherical star due to the Starobinsky R2 model withinthe framework of f (R) gravity. Here, we would like to dis-

cuss the inhomogeneities/irregularities which emerge in theenergy density due to the Palatini version of f (R) gravity.

Anisotropic effects are leading paradigms in the descrip-tion of evolutionary mechanisms of stellar collapsing mod-els. It is an established fact that the properties of anisotropicmodels may differ drastically in contrast with the isotropicspheres. Nguyen and Pedraza [12] investigated an anisotropicspherical compact model and deduced that anisotropic effectsmake the system dissipative with the evolution of time. Leonand Sarikadis [13] investigated the impact of anisotropy inthe framework of modified gravity and concluded to differ-ent cosmological behaviors in the geometry as compared toisotropic scenarios. Cosenza et al. [14] figured out the roleof anisotropy on radiating fluid spheres. Maartens et al. [15]analyzed the anisotropic evolution of the universe during anintermediate transient regime of inflationary expansion.

The anisotropic picture in relativistic fluid configurationscan be achieved by many interconnected phenomena likethe existence of strong electric and magnetic interactions[16,17]. A great deal of attention has also been given to theinteraction of electromagnetic and gravitational fields. How-ever, general agreement exists among relativists that physicalobjects with a large amount of electric charge do not existin nature. This line of thought has been challenged by manyresearchers and a variety of works have been carried out withthis background. Ghezzi [18] explored some analytical mod-els of an isotropic spherical star in the presence of an elec-tromagnetic field in which the charge density is proportionalto the rest mass density. He found that the radius of chargedstars is larger as compared to the uncharged ones. Varela et al.[19] solved the Einstein–Maxwell field equations for a self-gravitating anisotropic spherical system numerically and linktheir findings with the models of dark matter including mas-sive charged particles as well as charged strange quark stars.The impact of an electromagnetic field and other matter vari-ables on the evolutionary behavior of collapsing relativisticself-gravitating systems in the cosmos has been investigatedin [20–27].

A system begins to collapse once it experiences an inho-mogeneous stellar state. Penrose and Hawking [28] exploredirregularities in the energy density of spherical relativisticstars by means of the Weyl invariant. Herrera et al. [29] dis-cussed the role of density inhomogeneities in the structureand evolution of spherically anisotropic objects. Herrera et al.[30,31] did a systematic study of the structure formation ofself-gravitating compact stars by means of some scalar func-tions (trace and trace-free parts) obtained from splitting ofthe Riemann tensor. These scalars are associated with electricand magnetic as well as second dual of the Riemann tensorand have an eventual relationship with the fundamental prop-erties of the matter configuration [32]. The inhomogeneity inthe universe can be linked with the dipole anisotropy as foundby Planck [33]. Herrera [34] investigated different physical

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Eur. Phys. J. C (2016) 76 :219 Page 3 of 13 219

factors responsible for the emergence of inhomogeneities inan initial regular spherical collapsing distributions. Sharifand Yousaf [35] described the stability of the regular energydensity in a planar matter distribution by taking into accounta three parametric model form in Palatini f (R) gravity.

The inhomogeneous models can also be used to discuss theSN-data [36]. Geng and Lü [37] presented a class of modelsdescribing the isotropic expansion for inhomogeneous uni-verse. The unresolved issues of the dark energy/dark matteron the homogeneity of the collapsing compact star is still amatter of interest for relativists. We will address two mainrelated problems in this paper:

1. We explore inhomogeneity factors for a plane symmetriccompact object and discuss it with some particular casesby increasing the complexity in the matter distribution.

2. The role of Palatini f (R) dark source terms througha viable f (R) model as well as electromagnetic fieldeffects will be analyzed.

This paper is organized in the following manner. In thenext section, we deduce the field equations coupled withthe source in f (R) gravity under the influence of an elec-tromagnetic field. Section 3 investigates the dynamical aswell as evolution equations for the systematic analysis ofinhomogeneity factors. In Sect. 4, we formulate the irregu-larity factors with some particular cases of dissipative andnon-dissipative matter fields. Finally, we conclude our mainfindings in the last section.

2 f (R) gravity coupled to matter source

We choose a non-static planar geometry for the constructionof our systematic analysis as [38–41]

ds2− = −A2(t, z)dt2 + B2(t, z)(

dx2 + dy2)

+C2(t, z)dz2, (7)

while it is filled with a dissipative fluid by means of diffu-sion (heat) as well as free-streaming (null radiation) approx-imations having an anisotropic pressure in the interior. Suchmatter fields are described by the energy-momentum tensoras follows:

Tαβ = (P⊥ + μ)VαVβ + qβVα + εlαlβ + P⊥gαβ

+ (Pz − P⊥)χαχβ + qαVβ, (8)

where ε, μ, P⊥, Pr , and qβ are the radiation density, energydensity, different stress components, and the heat flux vector,respectively.

In a comoving coordinate system, the unit four vector lβ =1A δ

β0 + 1

C δβ3 , the radial four vector, i.e., χβ = 1

C δβ3 as well

as the four velocity vector V β = 1A δ

β0 , satisfy the following

relations:

V αVα = −1, χαχα = 1, χαVα = 0,

V αqα = 0, lαVα = −1, lαlα = 0.

The expansion rate of the matter configuration for a Palatinif (R) background is defined by the scalar as

�P = Vα;βV β = 2

A

(B

B+ f R

fR+ C

2C

), (9)

where a dot indicates the operator ∂∂t . The shear scalar for a

planar case in the framework of GR yields [34]

9σ 2 = 9

2σμνσμν = W 2

GR, with WGR = 1

A

(C

C− B

B

).

(10)

Using Eqs. (9) and (10), we can determine a relation betweenexpansion and shear as follows:

WGR = �P − 3C

AC− 2 f R

A fR. (11)

The stress-energy tensor describing the electromagneticfield and satisfying the Maxwell field equations, i.e., Fαβ

;β =μ0 Jα, F[αβ;γ ] = 0, is defined as

Eαβ = 1

4π

(Fγ

α Fβγ − 1

4Fγ δFγ δgαβ

),

where Fαβ = −φα,β + φβ,α is the Maxwell strength tensorwhere φβ describes four potential. Here Jα and μ0 = 4π rep-resent four current and magnetic permeability, respectively.The four potential and four current are φα = φδα

0 , Jα =σV α, under comoving coordinate system, while φ, σ arefunctions of t and z representing the scalar potential andcharge density, respectively. The non-zero components ofthe Maxwell field equations yield the following couple ofequations:

∂2φ

∂z2 −(A′

A+ C ′

C− 2B ′

B− 2 f ′

R

fR

)∂φ

∂z= σμ0AC

2, (12)

∂2φ

∂t∂z−

(A

A+ C

C− 2B

B− 2 f R

fR

)∂φ

∂z= 0. (13)

Here a prime indicates z differentiation. Integration of Eq.(12) with respect to z yields

φ′ = s AC

f 2R B

2, where s = μ0

∫ z

0σ f 2

RCB2dz, (14)

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219 Page 4 of 13 Eur. Phys. J. C (2016) 76 :219

which equivalently satisfies Eq. (13). The non-vanishingcomponents of the electromagnetic stress tensor turn out to be

E00 = s2

8π A2B4 , E11 = s2

8πB6 = E22,

E33 = − s2

8πB4C2 . (15)

The field equations in the framework of Palatini f (R)

gravity corresponding to a planar geometry lead to

κ

fR

[A2(μ + ε) + s2

8π A2B4

− A2

κ

{f ′R

C2

(C ′

C+ f ′

R

4 fR− 2B ′

B

)− fR

2

(R − f

fR

)

− f ′′R

C2 +(C

C+ 9 f R

4 fR+ 2B

B

)f RA2

}]

=(B

B

)2

+ 2C B

CB+

{B ′

B

(2C ′

C− B ′

B

)

−2C ′′

C

}(A

C

)2

, (16)

κ

fR

[CA(q + ε) − 1

κ

(f ′R − 5

2

f R f ′R

fR− C f ′

R

C− A′ f R

A

)]

= 2

(B ′B

− A′ BB A

− B ′CBC

), (17)

κ

fR

[P⊥B2 + s2

8πB6

+ B2

κ

{f RA2 − f ′′

R

C2 +(B

B− f R

4 fR− A

A+ C

C

)f RA2 − fR

2

×(R − f

fR

)+

(C ′

C+ f ′

R

4 fR− B ′

B− A′

A

)f ′R

C2

}]

={B

B

(A

A− C

C

)− C

C

+ C A

C A− B

B

}C2

A2

+{A′

A

(B ′

B− C ′

C

)+ B ′′

B− B ′C ′

BC+ A′′

A

}B2

C2 , (18)

κ

fR

[C2(Pz + ε) − s2

8πB4C2

+C2

κ

{f RA2 − f ′

R

C2

(A′

A+ 9 f ′

R

4 fR+ 2B ′

B

)− fR

2

×(R − f

fR

)+ f RA2 +

(2B

B− f R

4 fR− A

A

)}]

={(

2 A

A− B

B

)B

B− 2B

B

}C2

A2

+ B ′

B

(B ′

B+ 2A′

A

). (19)

To describe the quantity of matter within the planar system,the mass function can be evaluated through the Taub massformalism in the presence of an electromagnetic field as [42]

m(t, z) = (g)32

2R 12

12 + s2

2B= B

2

(B2

A2 − B ′2

C2

)+ s2

2B,

(20)

which can be written in an alternative way using the fluidvelocity as

E ≡ B ′

C=

√U 2 − 2m(t, r)

B+ s2

B2 . (21)

Using Eqs. (16)–(19), the temporal and radial variations ofthe mass function lead to

DTm = − κ

2 fR

{U

(Pz − 2πE2 + T33

C2

)+ E

(q − T03

AC

)}B2

+ 8π2B2E(2E B + 3E B), (22)

DBm = κ

2 fR

{μ + T00

A2 + 2πE2 + U

E

(q − T03

CA

)}B2

+ 8πB2EB ′ × (2BE ′ + 3EB ′), (23)

where U is the velocity of the collapsing matter defined byU = DT B where (DT = 1

A∂∂t ) while Pr = Pr + ε, q =

q + ε, μ = μ + ε, and DB = 1B′ ∂

∂r represents the radialderivative operator, respectively. Here E denotes the electricfield intensity. It is interesting to indicate that for a planarcelestial configuration undergoing collapse, U is chosen tobe less than unity. The link between matter variables and themass function can be found through integration of Eq. (23)with a Palatini f (R) background:

3m

B3 = 3κ

2B3

∫ z

0

[1

fR

{μ + 2πE2 + T00

A2 +(q − T03

CA

)U

E

}B2B ′

+ 8πEB2(2BE ′ + 3EB ′)]

dz. (24)

The electric component of the Weyl tensor in terms of theunit four velocity and radial four vector is given as

Eαβ = E[χαχβ − 1

3(gαβ + VαVβ)

],

where

E =[B

B+

(C

C− B

B

)(B

B+ A

A

)− C

C

]1

A2

−[C ′′

C−

(C ′

C+ B ′

B

)(B ′

B− A′

A

)− A′′

A

]1

C2 (25)

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Eur. Phys. J. C (2016) 76 :219 Page 5 of 13 219

is the scalar encapsulating the effects of spacetime curvature.Alternatively, using Eqs. (16) and (18)–(20), we get

3m

B3 = κ

2 fR

(μ − � + T00

A2 − T33

C2 + T11

B2 + 6πE2)

− E,

(26)

where � = Pz − P⊥. The above equation determines thegravitational contribution of planar geometry due to its fluidvariables, mass function, and extra curvature f (R) terms.

3 Dynamical and evolution equations

In this section, we will establish some scalar functions inthe background of a well-consistent f (R) model. We thenshow a correspondence between fluid parameters and theWeyl scalar with Palatini f (R) corrections by constructingmodified Ellis equations. In order to discuss the dynamicalproperties framed within the modified cosmology, we takethe f (R) model as follows [46]:

f (R) = R − μRc tanh

(R

Rc

), (27)

where μ and Rc are positive constants. The values of thesefree parameters are Rc ∼ H2

0 ∼ 8ρc3m2

p� 10−84GeV2, where

Rc is roughly of the same order as the Ricci scalar today,H0 is the present day value of the Hubble constant, and thecritical density ρc � 10−29gr/cm3 ∼ 4.5 × 10−47GeV4.We categorize the Riemann tensor in terms of second ranktensors, i.e., Xαβ and Yαβ , to devise a modified form of thestructure scalars as [43,44]

Xαβ = ∗R∗αμβνV

μV ν = 1

2ηερ

αμR∗ερβνV

μV ν,

Yαβ = RαμβνVμV ν,

where left, right, and double dual of the Riemann curvaturetensor can be, respectively, written in a standard form as

∗Rαβγ δ ≡ 1

2ηαβερR

εργ δ, R∗

αβγ δ ≡ 1

2ηεργ δR

εραβ,

∗R∗αβγ δ ≡ 1

2η

εραβ R∗

εργ δ.

The above tensors can further be split into their trace andtrace-free components as

Xαβ = 1

3XT hαβ + XT F

(χαχβ − 1

3hαβ

), (28)

Yαβ = 1

3YT hαβ + YT F

(χαχβ − 1

3hαβ

). (29)

By making use of Eqs. (16), (18), (19), and (27)–(29), thesescalar functions can be written in terms of fluid variables as

XT = κRc(R2c + R2)(n+1)

Rc(R2c + R2)(n+1) − 2nλRR(2n+2)

c

(μ + δμ

A2

),

(30)

XT F = −E − κRc(R2c + R2)(n+1)

2[Rc(R2

c + R2)(n+1) − 2nλRR(2n+2)c

]

×(

� − 2Wη + δPz

C2 − δP⊥B2

), (31)

YT = κRc(R2c + R2)(n+1)

2[Rc(R2

c + R2)(n+1) − 2nλRR(2n+2)c

]

×(

μ + δμ

A2 + δPz

C2 + 2δP⊥B2 + 3Pr − 2�

), (32)

YT F = E − κRc(R2c + R2)(n+1)

2[Rc(R2

c + R2)(n+1) − 2nλRR(2n+2)c

]

×(

� − 2ηW + δPz

C2 − δP⊥B2

), (33)

where δμ, δPz , and δP⊥ are the corresponding values of darksource components evaluated by taking into account Eq. (27)and given in Appendix A. We found that the trace part ofthe second dual of the Riemann tensor has its dependenceon the energy density profile of planar geometry with someextra curvature terms due to f (R) Palatini gravity whilethe remaining scalar functions have their dependence on theanisotropic stress tensor. The conservation of energy andmomentum from the contracted Bianchi identities with ordi-nary and effective matter fields,

((D)

T αβ + T αβ

)

;β= 0,

((D)

T αβ + T αβ

)

;β= 0,

yields the couple of equations

˙μA

+ q ′

C+ 1

A

(C

C+ f R

2 fR

)(Pz + μ) + 1

A(μ + P⊥)

×(

2B

B+ f R

fR

)+ μ f R

A fR+ q

C

(2A′

A+ 3 f ′

R

fR+ 2B ′

B

)

+ D0(t, r)

κ+ 6πE2 f R

A2 fR= 0, (34)

˙qC

+ Pz′

C+ 1

C

(A′

A+ f ′

R

2 fR

)(μ + Pz)

+(

2B ′

B+ f ′

R

fR

)(Pz − P⊥)

1

C+ Pz f ′

R

C fR

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219 Page 6 of 13 Eur. Phys. J. C (2016) 76 :219

+ q

A

(2C

C+ 3 f R

fR+ 2B

B

)+ D1(t, r)

κ

− 4πE

BC2 (BE ′ + 2EB ′) − 4πE2 f ′

R

C2 fR= 0, (35)

where the terms D0 and D1 emerge due to Palatini f (R)

gravity and they are addressed in Appendix A. Next, we con-tinue our investigation by constructing a couple of differentialequations using the procedure adopted by Ellis [47]. Theseequations are found by using Eqs. (16)–(19), (22), (23), and(27) and define a link between matter variables with Palatinif (R) extra curvature terms and the Weyl tensor, thus

⎡⎣E − κ

2[1 − μsech2

(RRc

)]⎛⎝μ − � + 6πE2

+ δμ

A2 − δPz

C2 + δP⊥B2

⎞⎠

⎤⎦

.

= 3B

B

⎡⎣ κ

2[1 − μsech2

(RRc

)](

μ + P⊥ + δμ

A2 + δP⊥B2

)

+ 4πE2 − E

⎤⎦ + 3κ

2[1 − μsech2

(RRc

)](AB ′

BC

)

×(q − δq

AB

)+ 24π2E

B

(2BE + 3E B

), (36)

⎡⎣E − κ

2[1 − μsech2

(RRc

)]⎛⎝μ − �

+ 6πE2 + δμ

A2 − δPz

C2 + δP⊥B2

⎞⎠

⎤⎦

,1

= −3B ′

B

⎡⎣ κ

2[1 − μsech2

(RRc

)](

μ + 2πE2 + δμ

A2

)− 3m

B3

⎤⎦

− κ

2[1 − μsech2

(RRc

)](BC

AB

)(q − δq

C A

)

+ 24π2E

B

(2E ′B + 3EB ′) , (37)

where δq is shown in Appendix A. The limit f (R) → R inthe above equations provides the GR Ellis equations.

4 Irregularities in the dynamical system

This section explores some fluid variables that are responsi-ble for irregularities in the dynamical system having planarsymmetry. This analysis has been carried out from an initial

homogeneous configuration of a compact body by means ofsome particular choices on the matter fields with extra cur-vature terms of Palatini f (R) gravity. We will restrict ouranalysis to the present day value of the cosmological Ricciscalar, i.e., R = R, while dealing with a bulky system ofequations. Finally, we will study the case with zero expan-sion. We classify our investigations in two scenarios, i.e.,dissipative and non-dissipative systems, as follows.

4.1 Non-radiating matter

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

4.1.1 Dust fluid

In this case, we consider Pz = 0 = P⊥ = q and A = 1indicating geodesic motion of a non-dissipative dust cloud.In this scenario, the two differential equations for the Weyltensor obtained in (36) and (37) reduce to

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×{

μ + 6πE2 − μR

2+ μ

2tanh

(R

Rc

)

×{R tanh

(R

Rc

)− Rc

}}].

= 3B

B

⎡⎣ κ

2[1 − μsech2

(RRc

)] (μ + 4πE2

)⎤⎦

+ 24π2E

B

(2E B + 3E B

), (38)

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×{

μ + 6πE2 − μR

2+ μ

2tanh

(R

Rc

)

×{R tanh

(R

Rc

)− Rc

}}]′

= −3B ′

BE − 6κπE2B ′[

1 − μsech2(

RRc

)]B

+ 24π2E

B

× (2E ′B + 3EB ′) . (39)

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Eur. Phys. J. C (2016) 76 :219 Page 7 of 13 219

When μ′ = 0, Eq. (39) leads to

E ′ + 3B ′

BE = 6πκE[

1 − μsech2(

RRc

)](E ′ − EB ′

B

)

+ 24Eπ2

B

(2E ′B + 3EB ′) . (40)

The general solution of the above equation is obtained:

E = 6π

B3

∫ z

0

⎡⎣ κEB3[

1 − μsech2(

RRc

)]

×(E ′ − EB ′

B

)+ 3πEB2 (

2E ′B + 3E ′B)⎤⎦ dz.

(41)

It is worth noting that the Weyl scalar is the only geometricentity responsible for the irregularities in the energy density,depending upon the electromagnetic profile. In the absenceof an electromagnetic field, the Weyl scalar will also vanishshowing the importance of charged fields. By making use ofEqs. (11), (34), and (A3)–(A6) in Eq. (38), we found

E + 3B

BE = −κμWGR

2[1 − μtanh2

(RRc

)] + 6π2κE[1 − μtanh2

(RRc

)]

×(E + E B

B

)+ 24π2E

B

(2E B + 3E B

).

(42)

The above equation reveals the relationship of the Weyl scalarwith shear scalar indicating the shearing motion of dust cloud.It also shows that the system will be homogeneous if it is shearfree as well as conformally flat within the Palatini frameworkof f (R) gravity. Its solution turns out to be

E = κ

2B3[1 − μsech2

(RRc

)]

×∫ t

0

[−μWGR + 12πE

(E B

B+ E

)]B3dt

+ 12π2

B3

∫ t

0

[EB3 (

2E B + 3E B)]

dt. (43)

The role of expansion can be made clear while discussingthe inhomogeneities on the evolution of dust matter in thecollapse scenario. We study the zero expansion case, i.e.,�P = 0, so that the above equation becomes

E = κ

2B3[1 − μsech2

(RRc

)]

×∫ t

0

[3μB

B+ 12πE

(E B

B+ E

)]B3dt

+ 12π2

B3

∫ t

0

[EB3 (

2E B + 3E B)]

dt.

It shows that the expansion-free system will be inhomo-geneous due to the presence of the Weyl scalar, as it pro-duces tidal forces which make the object inhomogeneouswith the passage of time, thus indicating the importance oftime. Moreover, in the absence of tidal forces, the systemwill be inhomogeneous due to the presence of the electro-magnetic field. Consequently, an expansion-free system willbe homogeneous if it is charge free and conformally flat.

4.1.2 Isotropic fluid

In this case, we introduce a bit of complexity into the previ-ous case by adding the effects of isotropic pressure and wedetermine the inhomogeneity factors. In this scenario, theEllis equations (36) and (37) turn out to be

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×[μ + 6πE2 − μR

2+ μ

2tanh

(R

Rc

)

×{R tanh

(R

Rc

)− Rc

}]]

= 3B

B

⎡⎣ κ

2[1 − μsech2

(RRc

)] (μ + P + 4πE2) − E

⎤⎦

+ 24π2E

B× (

2E B + 3E B), (44)

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×[μ + 6πE2 − μR

2+ μ

2tanh

(R

Rc

)

×{R tanh

(R

Rc

)− Rc

}]]′

= −3B ′

B

⎡⎣E − 2κπE2[

1 − μsech2(

RRc

)]⎤⎦

+ 24π2E

B(2E ′B + 2EB ′). (45)

123

219 Page 8 of 13 Eur. Phys. J. C (2016) 76 :219

We see that the second equation is the same as the one wehave evaluated in the above case with a dust cloud (see Eq.(38)). Therefore, this indicates the Weyl scalar as the factorresponsible of irregularities in the matter distribution. Bymaking use of Eqs. (11) and (34), Eq. (44) leads to

E + 3B

BE = κ

2[1 − μsech2

(RRc

)]

×[−WGR(μ + P)A + 12πE

(E + E B

B

)]

+ 24π2E

B(2E B + 3E B), (46)

which on integration turns out to be

E = κ

2B3[1 − μsech2

(RRc

)]

×∫ t

0

[−WGR(μ + P)A + 12πE

(E + E B

B

)]B3dt

+ 24π2

B3

∫ t

0

[EB2(2E B + 3E B)

]dt. (47)

This indicates the importance of shear on the evolution ofan inhomogeneous matter configuration with isotropic pres-sure. We observed that not only shear and pressure, but extracurvature terms due to f (R) gravity are acting on the systemto make it inhomogeneous as the evolution proceeds. We canalso examine the factors responsible for irregularities overthe relativistic system with zero shear. Moreover, we havealready obtained a relation between expansion and the shearscalar, therefore, we can analyze those effects when the sys-tem is undergoing collapse with zero expansion. Under thezero expansion condition, Eq. (46) yields

E = 3κ

2B3[1 − μsech2

(RRc

)]

×∫ t

0

[B

B

{(μ + P)A + 4πE2

}+ 12πE E

]B3dt

+ 24π2

B3

∫ t

0

[EB2(2E B + 3E B)

]dt. (48)

It is seen from the above expression that electromagnetic fieldhave also a crucial role to play in the expansion-free scenario.The Weyl scalar also plays a key role due to tidal forcesmaking the system more inhomogeneous with the passage oftime.

4.1.3 Anisotropic fluid

This case generalizes the previous one by introducing thecomplexity in the form of anisotropic stresses while the dis-

sipative effects are assumed to be zero, i.e., � �= 0 andq = 0. In this framework, the two equations obtained in (36)and (37) take the form

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×[μ − � + 6πE2 − μR

2+ μ

2tanh

(R

Rc

)

×{R tanh

(R

Rc

)− Rc

}]],0

= 3B

B

⎡⎣ κ

2[1 − μsech2

(RRc

)] {μ + P⊥ + 4πE2

}− E

⎤⎦

+ 24π2E

B(2E B + 3E B), (49)

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×[μ + 6πE2 − � + μR

2+ μ

2tanh

(R

Rc

)

×{R tanh

(R

Rc

)− Rc

}]]′

− 3B ′

B

⎡⎣E + κ

2[1 − μsech2

(RRc

)] [� − 4πE2

]⎤⎦

+ 24π2E

B

(2E ′B + 3EB ′) . (50)

We can find the following couple of equations by using Eqs.(11) and (34) in (49) and (50) with some computation:

⎡⎣E + κ

2[1 − μsech2

(RRc

)] (� − 4πE2)

⎤⎦

.

+ 3B

B

⎡⎣ κ

2[1 − μsech2

(RRc

)] (� − 4πE2) +E]

= −κ

2[1 − μsech2

(RRc

)] (μ + Pz)AWGR

+ 3κπ BE2[1 − μsech2

(RRc

)]B

+ κπ B[1 − μsech2

(RRc

)]B

+ 2κπ E E[1 − μsech2

(RRc

)] ,

⎡⎣E − κ

2[1 − μsech2

(RRc

)] (μ + � − 2πE2)

⎤⎦

′

123

Eur. Phys. J. C (2016) 76 :219 Page 9 of 13 219

= −3B ′

B

⎡⎣ κ

2[1 − μsech2

(RRc

)]

× (� − 4πE2) + E]

+ 24π2E

B(2E ′B + 3EB ′).

Using the trace-free part of the second dual of Riemann tensoras obtained in Eq. (31), we find

XT F + 3XT F B

B= κ[

1 − μsech2( ˜R

Rc

)]{AWGR

2(μ + Pz)

−�B

B+ πE

(3E B

B+ 2E

)},

X ′T F + 3XT F B ′

B= κ

2[1 − μsech2

(RRc

)] [μ′ + 4πEE ′]

+ 24π2E

B(2E ′B + 3EB ′).

The solution of the above couple of differential equationsturns out to be

XT F = − κ[1 − μsech2

(RRc

)]

×∫ t

0

[�B − AWGR

2(μ + Pz)B

−πEB

(3E B

B+ 2E

)]B2dt, (51)

XT F = κ

2[1 − μsech2

(RRc

)]

×∫ z

0(μ′ + 4πEE ′)B3dz + 24π2

B3

∫ z

0EB2

× (2E ′B + 3EB ′)dz. (52)

Equation (51) shows a relation of one of the scalar func-tions, from the splitting of the Riemann tensor, with theanisotropic pressure and shear scalar. It indicates the impor-tance of these material variables with a planar geometry inthe discussion of an irregular energy distribution. Now, thefactor that controls inhomogeneities over the compact sys-tem is the trace-free part of the double dual of the Riemanntensor, which is obtained through the orthogonal splittingof the Riemann tensor in the framework of Palatini f (R)

gravity, as seen from Eq. (52). It is well known that thesescalar functions play a crucial role in the structure forma-tion of the universe. Also, the solution of the field equa-tions in the static case can be written in the form of thesescalar functions. We found that in the absence of an elec-tromagnetic field, XT F is the factor describing the irregu-larities in the star configuration. Consequently, if XT F = 0

then the matter distribution in the charge-free system willbe homogeneous and vice versa. Next, we discuss the caseof collapsing matter with zero expansion in the presence ofanisotropic pressure. In this scenario, the solution of Eq. (49)becomes

XT F = − κ[1 − μsech2

(RRc

)]

×∫ t

0

[�B + 3A

2(μ + Pz)B − πEB

×(

3E B

B+ 2E

)]B2dt, (53)

which yields a link of the structure scalar with the energydensity and pressure anisotropy in the arrow of time withextra curvature terms due to Palatini f (R) gravity. We knowthat in the expansion-free system, the center is surroundedby another spacetime appropriately matched with the rest ofthe system.

4.2 Radiating dust fluid

This section explores the inhomogeneity factors with dis-sipation in both the diffusion and the free-streaming limit,but in the particular case of a charged dust cloud. For thispurpose, we take Pz = 0 = P⊥ in the matter field and themotion is considered to be geodesic by assuming A = 1 inthe geometric part, which is well justified on the basis ofsome theoretical advances made in the discussion of inho-mogeneous matter distribution. In this framework, Eqs. (36)and (37) yield

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×{

μ + 6πE2 − μR

2+ μ

2tanh

(R

Rc

)

×[R tanh

(R

Rc

)− Rc

]}].

= 3B

B

⎡⎣ κ

2[1 − μsech2

(RRc

)] (μ + 4πE2) − E

⎤⎦

+ 3κ

2[1 − μsech2

(RRc

)] q AB ′

BC

+ 24π2E

B(2E B + 3E B), (54)

123

219 Page 10 of 13 Eur. Phys. J. C (2016) 76 :219

⎡⎣E − κ

2[1 − μsech2

(RRc

)]

×{

μ + 6πE2 − μR

2+ μ

2tanh

(R

Rc

)

×[R tanh

(R

Rc

)− Rc

]}]′

= −3B ′

B

⎡⎣E + 2κπE2[

1 − μsech2(

RRc

)]⎤⎦

− 3κ BCq

2AB ′[1 − μsech2

(RRc

)] + 24π2E

B(2E ′B + 3EB ′).

(55)

Consider

� ≡ E + 1

B3

∫ z

0

3κB2C Bq

2[1 − μsech2

(RRc

)]dz. (56)

If we consider the matter distribution to be homogeneousi.e., μ′ = 0, then from Eq. (55) we obtain the followingexpression:

� = 1

B3

∫ z

0

⎡⎣6πEB3E ′

⎧⎨⎩

κ[1 − μsech2

(RRc

)] + 8π

⎫⎬⎭

+ 6πE2B2B ′⎧⎨⎩12π − κ[

1 − μsech2(

RRc

)]⎫⎬⎭

⎤⎦ dz,

(57)

which should be vanishing for the homogeneous fluid dis-tribution over planar geometry. Consequently, for a homo-geneous universe with planar topology, one should have� = 0 ⇔ μ′ = 0 with a dissipative charged dust cloud.The evolution equation for � can also be evaluated usingEqs. (11) and (34) in Eq. (54) as

� − �

B3 = κ

2[1 − μsech2

(RRc

)]

×[−μWGR − q

C+ q B ′

BC

]− 3B�

B

+ 6πEE ′⎛⎝ κ[

1 − μsech2(

RRc

)] + 8π

⎞⎠

+ 6πE2 B

B

⎛⎝12π + κ[

1 − μsech2(

RRc

)]⎞⎠ ,

(58)

whose solution leads to

� = 1

B3

∫ t

0

⎡⎣� + κ

2[1 − μsech2

(RRc

)]

×{−μWGRB − q B

C+ q B ′

C

}+ 6πEB3 E

×⎛⎝ κ[

1 − μsech2(

RRc

)] + 8π

⎞⎠ + 6πE2B2 B

×⎛⎝12π + κ[

1 − μsech2(

RRc

)]⎞⎠

⎤⎦ dt. (59)

This indicates the importance of fluid parameters as the inho-mogeneity factor, related to the matter variables, particularlyheat flux, as well as kinematical quantities of the system. Wealready found a relation in which the shear scalar is related tothe expansion scalar. Thus, for the shear-free case, we obtain

B

B= A�p

3− 2

3

f RfR

.

Using the above equation in Eq. (35), we find

q +[

2μR′

Rcsech2

(R

Rc

)tanh

(R

Rc

)[1 − μsech2

(R

Rc

)]−1]

×(

μ

2C− 4πE2

C2

)

− 4πE

BC2 (BE ′ + 2EB ′) + 2q

3

[2�p + μR

Rcsech2

(R

Rc

)

× tanh

(R

Rc

) [1 − μsech2

(R

Rc

)]−1]

+ D3 = 0. (60)

Next, the transportation of heat in the system can be ana-lyzed through a casual radiating theory defined by Muller andIsrael as a second order thermodynamical theory in diffusionapproximation as follows:

τ q = −1

2ξqK 2

(τ

ξK 2

),0

− ξ

B(AK )′ − q A − 1

2τq A�,

whose independent component yields

q = −q

τ− κ

CτT ′. (61)

123

Eur. Phys. J. C (2016) 76 :219 Page 11 of 13 219

Substituting the value of q from Eq. (60) in the above equa-tion, we obtain

q =[−4πEτ

r2B3 (BE ′ + 2EB ′) − κT ′

r B+ D3s

+[

2μR′

Rcsech2

(R

Rc

)tanh

(R

Rc

)

×[

1 − μsech2(

R

Rc

)]−1](

μτ

2r B− 4πE2τ

r2B2

)]

×[

1 − 4�pτ

3

].

One can identify the relaxation effects by inserting this valuein the evolution equation for the inhomogeneity factor in thiscase, i.e., � as obtained in Eq. (59). Consequently, the effectsof an electromagnetic field with the relaxation time can alsobe analyzed.

5 Discussion

In this paper, we have investigated some inhomogeneity fac-tors for a self-gravitating plane symmetric model. We havedone this analysis by taking an anisotropic matter distributionin the presence of an electromagnetic field. Particular atten-tion has been given to examine the role of dark source termscoming from the modification of the gravitational field. Themodification includes higher order curvature terms explic-itly due to Palatini f (R) corrections in the field equations.In order to continue our analysis systematically, first of allwe have explored the Palatini f (R)-Maxwell field equationsfor our compact object and define the mass function usingTaub’s mass formalism. An expression for the Weyl scalarhas been disclosed in terms of matter variables and highercurvature ingredients due to modified gravity.

A set of scalar functions have been evaluated using thesplitting of Riemann curvature tensor with comoving coor-dinate system to address the irregularities in the energy den-sity. These scalar functions are named structure scalars; theirphysical significance has been analyzed in the literature pre-viously. Also, it is established that these scalars are used towrite down solutions of field equations with a static back-ground metric. We have related these scalars in terms ofmaterial variables and dark source terms using the field equa-tions and a cosmological f (R) model. Moreover, a couple ofequations describing the conservation of energy-momentumin space have been explored. The evolution equations are alsoinvestigated using the procedure adopted by Ellis [47]. Wehave found some factors responsible for inhomogeneities inthe matter configuration with some particular cases of fluiddistribution.

Particular attention is given to examining the role of theelectromagnetic field in this framework. Usually, in the studyof relativistic astrophysics compact objects are considerednot to have sufficient internal electric fields. It is still fea-sible that stars can have a total net charge or large internalelectric fields. However, it is well established that angularmomentum plays the role of electric charge in rotating col-lapsing stars. In the present study, we have shed some lighton a more realistic astrophysical scenario, i.e., the inhomo-geneities/irregularities in the universe model. The galaxy dis-tribution is observed to be inhomogeneous at small scaleswhile, according to the theoretical models, it is expected tobecome spatially homogeneous for r > λ0 ≈ 10Mpch−1

[45].On the basis of the results we have obtained, it is clear

that the system becomes inhomogeneous as the evolutionproceeds with time indicating a crucial role of gravitationalarrow of time. In the non-radiating dust cloud case, we havefound that the system will be homogeneous in the absenceof an electromagnetic field as well as tidal forces, whichare due to the presence of the Weyl scalar. It shows that theWeyl tensor and the presence of charge make the distributionof matter more inhomogeneous during the evolution of theuniverse. With the inclusion of isotropic pressure in the mat-ter configuration, the Weyl tensor and electric charge behavesimilarly to the dust case. In the presence of anisotropic pres-sure effects in the matter, we have found a particular factor,known as the trace-free component of the dual of the Rie-mann tensor, responsible for the irregularities in the planarsystem. In the radiating dust cloud case, we have found thatthe system will be homogeneous if the factor � given in Eq.(59) vanishes; otherwise it will make our geometric modelmore inhomogeneous with evolution in time.

All of our results reduces to the charge-free case [35] inthe limit s = 0, while our results support the analysis madeby [34] in the limit f (R) = R

Acknowledgments This work was partially supported by Universityof the Punjab, Lahore-Pakistan, through a research project in the fiscalyear 2015-2016 (M.Z.B.).

OpenAccess This article is distributed under the terms of the CreativeCommons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution,and reproduction in any medium, provided you give appropriate creditto the original author(s) and the source, provide a link to the CreativeCommons license, and indicate if changes were made.Funded by SCOAP3.

Appendix A

The higher curvature terms D0 and D1 of Eqs. (34) and (35)are given as

123

219 Page 12 of 13 Eur. Phys. J. C (2016) 76 :219

D0 = (−1)

A2

{(f

R− fR

)R

2− f ′′

R

C2 + f RA2

×(C

C+ 9 f R

4 fR+ 2B

B

)− f ′

R

C2

(C ′

C+ f ′

R

4 fR− 2B ′

B

)},0

+ f RfR A

{3 f R2A2 − R

2

(fR − f

R

)

+3 f ′′R

2C2 − f RA2

(3C

2C+ 3 A

2A+ 5B

B+ 6 f R

fR

)

− f ′R

C2

(3A′

2A+ 3C ′

2C− 3B ′

B+ 3 f ′

R

2 fR

)}

+ C

AC

{f ′′R

C2

+ f RA2 − f R

A2

(5 f R2 fR

+ A

A+ 4B

B+ C

C

)

− f ′R

C2

(A′

A+ 5 f ′

R

2 fR+ C ′

C

)}

+ (−1)

C2A

(f ′R − 5

2

f R f ′R

fR− A′

AfR − C

Cf ′R

)

(3A′

A+ C ′

C+ 3 f ′

R

fR+ 2B ′

B

)]

+ A

[1

A2C2

{f ′R − A′

AfR − C

Cf ′R − 5

2

f R f ′R

fR

}]

,1

,

(A1)

D1 = C

{−1

(CA)2

(f ′R − 5 f R f ′

R

2 fR− A′

AfR − C

Cf ′R

)}

,0

+ 1

C

{f RA2 − R

2

×(fR − f

R

)− f R

A2

(A

A+ f R

4 fR+ 2B

B

)

− f ′R

C2

(2B ′

B+ 9 f ′

R

4 fR+ A′

A

)},1

+ A′

CA

{f ′′R

C2 + f RA2 − f R

A2

(5 f R2 fR

+ A

A+ 4B

B+ C

C

)

− f ′R

C2

(5 f ′

R

2 fR+ A′

A

+C ′

C

)}+ f ′

R

fRC

{(f

R− fR

)R

2+ 3 f R

2A2 + 3 f ′′R

2C2

−3 f R2A2

(A

A+ C

C+ f R

fR

+14B

3B

)− f ′

R

C2

(3B ′

B+ 3A′

2A+ 3C ′

2C+ 6 f ′

R

fR

)}

+2B ′

CB

{f ′′R

C2 − f RA2

×( ˙3B

B+ C

C

)− f ′

R

C2

(B ′

B+ 5 f ′

R

2 fR+ C ′

C

)}

+ (−1)

CA2

(− A′

AfR + f ′

R

−5 f R f ′R

2 fR− C

Cf ′R

)(A

A+ 3C

C+ 3 f R

fR

). (A2)

The quantities δμ, δPz , δP⊥ , and δq are

δμ = −A2

κ

[2μR′

C2Rcsech2

(R

Rc

)tanh

(R

Rc

)

×[C ′

C− 2B ′

B+ μR′

2Rcsech2

(R

Rc

)

× tanh

(R

Rc

) {1 − μsech2

(R

Rc

)}−1]

+ μR

2− μ

2tanh

(R

Rc

){R tanh

(R

Rc

)

−Rc} −[1 − μsech2

(RRc

)]

C2

+[C

C+ 2B

B+ 9μR′

2Rcsech2

(R

Rc

)tanh

(R

Rc

)

×{

1 − μsech2(

R

Rc

)}−1]

×[

2μR

Rcsech2

(R

Rc

)tanh

(R

Rc

)]], (A3)

δq = − 1

κ

[[2μR

Rcsech2

(R

Rc

)tanh

(R

Rc

)

×{

1 − μsech2(

R

Rc

)}−1]′

− 10μ2R′ RR2c

× sech4(

R

Rc

)tanh2

(R

Rc

){1 − μsech2

(R

Rc

)}−1

− 2μR′CC Rc

sech2(

R

Rc

)

× tanh

(R

Rc

){1 − μsech2

(R

Rc

)}−1

−2μR A′

ARcsech2

(R

Rc

)tanh

(R

Rc

)

×{

1 − μsech2(

R

Rc

)}−1]

, (A4)

δP⊥ = B2

κ

[[B

B− A

A+ C

C− μR

2Rcsech2

(R

Rc

)

× tanh

(R

Rc

){1 − μsech2

(R

Rc

)}−1]

123

Eur. Phys. J. C (2016) 76 :219 Page 13 of 13 219

× 2μR

A2Rcsech2

(R

Rc

)tanh

(R

Rc

){1 − μsech2

(R

Rc

)}−1

+[1 − μsech2

(RRc

)]..

A2

−[1 − μsech2

(RRc

)]′′

A2 + μR

2− μ

2tanh

(R

Rc

)

×{R tanh

(R

Rc

)− Rc

}

+[C ′

C− B ′

B− A′

A+ μR′

2Rcsech2

(R

Rc

)

× tanh

(R

Rc

){1 − μsech2

(R

Rc

)}−1]

× 2μR′

C2Rcsech2

(R

Rc

)tanh

(R

Rc

)], (A5)

δPz = C2

κ

[μR

2− μ

2tanh

(R

Rc

){R tanh

(R

Rc

)− Rc

}

+[1 − μsech2

(RRc

)]..

A2

−[A′

A+ 2B ′

B+ 9μR′

2Rcsech2

(R

Rc

)tanh

(R

Rc

)

×{

1 − μsech2(

R

Rc

)}−1]

× 2μR′

C2Rcsech2

(R

Rc

)tanh

(R

Rc

)

+ 2μR

A2Rcsech2

(R

Rc

)tanh

(R

Rc

)

× A′

A+ 2B ′

B+ 9μR′

2Rcsech2

(R

Rc

)tanh

(R

Rc

)].

(A6)

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