Dissipative Spherical Collapse of Charged Anisotropic Fluid in f(R) Gravity

This manuscript is devoted to study the combined effect of a viable model, f(R) = R + \alpha R^n, and electromagnetic field on the instability range of gravitational collapse. We assume charged anisotropic fluid that dissipate energy via heat flow and discuss that electromagnetic field, density inhomogeneity, shear and phase transition on astrophysical bodies can be incorporated by locally anisotropic background. Dynamical equations help to investigate the evolution of self-gravitating objects and leads to the conclusion that adiabatic index depend upon the electromagnetic background, mass and radius of the spherical objects.


Introduction
Gravitational collapse is highly dissipative phenomenon. The effects of dissipation describe a wide range of situations. For example, using quasi-static approximation, limiting cases of radiative transport have been studied in [1].
It is found that hydrostatic time scale is very small as compared to the stellar lifetimes for different phases of a star's life. It is of the order of 27 minutes for the sun, 4.5 seconds for a white dwarf and 10 −4 seconds for a neutron star of one solar mass and 10 km radius [2]. The dissipative factors enhances instability range at Newtonian limits but develop more stability at relativistic annexes. The impressions of radiation, anisotropy and shearing viscosity at Newtonian and post-Newtonian eras are inquired in [3]- [6]. Various prospects of collapse phenomenon in account with dark source are worked out in recent past [7]- [12]. Due to high dissipation, matter produce large amount of charge in collapsing phenomenon and so it is well motivated problem to investigate electromagnetic field effects on the gravitational collapse [13].
Chandrasekhar [14] took initiative to workout dynamical instability problem. Dynamical instability is pragmatic in establishing the evolution and formation of stellar objects that must be stable against fluctuations. Generally, adiabatic index Γ is utile to address instability problem. Isotropic spheres of mass M and R radius may be related as Γ ≥ 4 3 + n M r , where number n depends upon the star's structure. Later on instability range for anisotropic, adiabatic, non-adiabatic and shearing viscous fluids has been examined in [15]- [17]. Besides Γ many other matter variables such as dissipation, radiation, shearing stress, anisotropy, expansion-free condition, etc. may also be responsible for dynamical instability and evolution in stars depending upon fluid properties.
Modified theories of gravity have received enormous attention in recent years. Inclusion of higher order curvature invariants and coupled scalar fields have become a paradigm in alternative gravity theories. For this purpose, various alterations are made in Einstein-Hilbert (EH) action [18]- [20]. The elementary and likely modification is to include curvature terms that are of type f (R) having combinations of Ricci scalar R. In this way, gravity tends to modify on large scales that reveals enormous observational signatures like modified galaxy clustering spectrum [21,22], weak lensing [23,24] and cosmic microwave background [25,26].
The most studied and simplest models in f (R) theory are f (R) = R+σ µ 4 R and f (R) = R + αR 2 , where σ = ±1, α is a positive real number and µ is a parameter with units of mass. Usually, the positive values of scalar curvature depicts standard cosmological corrections leading to de-Sitter space [27] whereas negative values help to discuss accelerating universe due to dark energy [28]. The effects of these f (R) models on the dynamical instability of gravitational collapse has been discussed in recent papers [29,30]. In the same context, Sharif and Yousaf established the range of instability for charged expansion-free, dissipative collapse for spherical and cylindrical symmetries in f (R) gravity [31]- [33]. In this paper, we adopted f (R) = R + αR n to discuss dynamical instability of gravitational collapse in the background of electromagnetic field. The manuscript is arranged as follows. In section 2, energy-momentum tensor of matter distribution along with Maxwell's and Einstein's field equations is given. Section 3 provide the knowledge about adopted f (R) model and perturbation scheme. In the same section, instability range would be discussed for Newtonian and post-Newtonian regimes in the form of Γ. Final section 4 provides summary of the paper and followed by an appendix.

Evolution Equations
We have chosen timelike three dimensional spherical boundary surface Σ that delimitates four dimensional line element into two realms termed as exterior and interior region. Interior region inside the boundary is while line element for exterior region [33] is considered as Here ν corresponds to retarded time, M is the total mass and Q indicates the total charge of fluid. The generalized EH action for f (R) gravity in account with Maxwell source modifies to Here κ stands for coupling constant and ̥ = 1 4 F uv F uv is the Maxwell invariant. We use metric approach to recover field equations by varying above action with g uv as follows where f R ≡ df (R)/dR, ∇ u denote covariant derivative, = ∇ u ∇ v , T uv is minimally coupled stress-energy tensor and E uv is electromagnetic tensor. Above field equations can also be written as denoting effective stress-energy tensor. The usual matter is anisotropic and adiabatic in nature representing dissipative collapse in the form of heat flux q and is given by [15,34] where µ depicts density, p r to the radial pressure, p ⊥ to the tangential pressure, V u to the four-velocity of the fluid and χ u corresponds to the radial four vector. In co-moving coordinates, following pattern is accompanied The electromagnetic energy-momentum tensor is written as [35] (2.9) Here F uv = ϕ v,u − ϕ u,v denotes electromagnetic field tensor while ϕ u = ϕ(t, r)δ 0 u stands for four potential. The Maxwell field equations are given by where  u = µ(t, r)V u is four current, µ 0 is magnetic permeability and µ represents charge density. The electromagnetic field equations turn out to be Herein derivatives with respect to t and r are expressed by dot and prime respectively. Applying integration on Eq.(2.11), we have The total charge q interior to radius r with electric field intensity E has the form (2.14) For interior spacetime, the components on the right hand side of the field equations (2.5) are given as follows, whereas the components of Einstein tensor are present in [10] G The development in collapsing phenomenon with the passage of time can be described by dynamical equations. These dynamical evolution equations for usual matter, effective and Maxwell's energy-momentum tensor carrying higher order curvature invariants are formed by employing Bianchi identities as which turn out to bė Here P 1 (r, t) and P 2 (r, t) corresponds to dark source terms provided in Appendix in the form of Eqs.(5.1) and (5.2) respectively.

f (R) Model and Perturbation Scheme
The f (R) model under consideration is The configuration of second order derivative decides whether the model is viable or not. Any f (R) is supposed to be suitable in GR and Newtonian limits if f ′′ (R) > 0. For our proposed form of f (R), n > 2 and α is a positive real number, in order to fulfill stability criterion and demonstrate accelerated expansion of the universe. It is a fact that general solution of gravitational field equations is yet unknown because these are highly complicated non-linear differential equations. Perturbation theory can be employed to somehow incorporate these discrepancies, so that dynamical equations become linear in the form of metric and material variables. Evolution can be explored by using Eulerian or Lagrangian schemes, i.e., by using fixed or co-moving coordinates respectively. We have applied perturbation with the assumption that initially all the metric and material functions are in static equilibrium and with the passage of time perturbed quantities have both radial and time dependence. Carrying 0 < ε ≪ 1, functions may be written in the following pattern Assuming C 0 (r) = r as Schwarzschild coordinate, static configuration of the field equations (2.15)-(2.18) take the following form where P 2s depicts static part of P 2 (r, t) and provided in Appendix as Eq.(5.3). Perturbed configuration of Evolution equations (2.20) and (2.21) reaḋ where P 1p and P 2p denote perturbed part of P 1 and P 2 respectively and given in Appendix. Elimination ofq from perturbed equation (2.16) implies On substitution ofq and its radial derivative in Eq.(3.18) lead to an equation from whichρ can be extracted. Integrating thisρ with respect to "t", we get where P 3 (r) is presented in Appendix. By second law of thermodynamics, ρ andp r can be related as ratio of specific heat with assumption of Harrison-Wheeler type equation of state, expressed in following expression [36,37] The adiabatic index Γ is a measure to recognize pressure variation with changing density. Putting Eq.(3.21) in above equation, we arrive at When we perturb Ricci scalar curvature, we obtain the following differential equationT (t) − P 4 (r)T (t) = 0.

Newtonian Regime
In this approximation, we assume that ρ 0 ≫ p r0 , ρ 0 ≫ p ⊥0 and A 0 = 1, B 0 = 1. By substituting these values in Eq.(3.24), we find Here P 2p(N ) denotes the Newtonian regime terms of perturbed second Bianchi identity. Inserting value of T from Eq.(3.26) in the above equation and rearranging, we have where P 5 (r) is given in Appendix. It is worth mentioning here that adiabatic index depends upon the electric field intensity, pressure components, energy density and scalar curvature terms in this limit. Thus, collapsing star would be unstable as long as inequality (3.28) holds.

Asymptotic Behavior
The expression for Γ takes following form when α → 0 This result represents the Einstein solution.

Post Newtonian Regime
Here, we analyze relativistic impressions upto O( m 0 r + Q 2 2r 2 ). In this approximation, we take Insertion of Eq.(3.30) and (3.31) in (3.24), Γ reads where P 4(P N ) ,p ⊥(P N ) and P 2(P N ) corresponds to PN regime terms of P 4 ,p ⊥ and P 2 respectively. However, W and X constitute the following expressions As far as instability problem concerned, system is unstable in PN limit for the above inequality. We can see that how curvature terms alongwith material variables affect the instability range. All terms appeared in the above inequality must maintained positivity to fulfill the dynamical instability condition and hence the following constraints

Asymptotic Behavior
As α → 0, adiabatic index Γ is unchanged whereas W becomes This represents the GR solution in PN regime.

Summary and Discussion
The purpose of the present work is to determine the electromagnetic field impressions on the instability of spherically symmetric collapsing compact object in f (R) framework. In order to achieve the goal, locally anisotropic matter experiencing dissipative collapse has been considered. We employ Jordan frame to work out instability problem and to modify EH action for modified gravity, we consider f (R) = R + αR n and Maxwell source so that attributes of f (R) model alongwith electromagnetic field can be investigated. Einstein field equations are modified accordingly. Dynamical equations are developed to study the evolution of non-static spherical star with the help of perturbation approach.
Model under consideration provides a viable alternate to dark energy, it satisfies the condition f ′′ (R) > 0 to carry out stellar stable configuration for matter dominated regime. Dissipation in terms of heat flow plays an important role in dynamics of collapse, especially electric charge and its distribution imply drastic effects on evolution and stellar structure.
Since solution of field equations are not ascertained yet, that is why to discuss dynamics of celestial bodies perturbation scheme is used. Perturbed form of second Bianchi describes the evolution of the collapsing system and further used to establish the instability range in terms of adiabatic index Γ. It is evident from the results that Γ has dependency on electric field intensity, radiative effects, density and pressure configuration. Inclusion of Maxwell source alongwith higher order curvature invariants imply that the self-gravitating system becomes more stable in the presence of electromagnetic field. Results are reduced to GR as α → 0.
Lastly, we compare our findings with the previous literature and found that our results reduces to the work already done for various constraints on n and electromagnetic effects. Comparison is elaborated in the following • For p r = p ⊥ , n = −1 and α = δ 4 our results reduce to isotropic pressure case [32].
• In the absence of Maxwell source results correspond to the results presented in [29].
• When we take n = 2 in our Model, the results supports the arguments in [33]. Also, it is clear that addition of Maxwell invariant describe more general expanding universe with a wider range of instability in the f (R) framework.
Static part of P 2 (r, t) is Perturbed form of P 1 and P 2 reads Expression for P 3 and P 4 respectively is