Shearfree Condition and dynamical Instability in $f(R,T)$ gravity

The implications of shearfree condition on instability range of anisotropic fluid in $f(R,T)$ are studied in this manuscript. A viable $f(R, T)$ model is chosen to arrive at stability criterion, where $R$ is Ricci scalar and $T$ is the trace of energy momentum tensor. The evolution of spherical star is explored by employing perturbation scheme on modified field equations and contracted Bianchi identities in $f(R, T)$. The effect of imposed shearfree condition on collapse equation and adiabatic index $\Gamma$ is studied in Newtonian and post-Newtonian regimes.


Introduction
In a recent work [1], we have studied the effect of anisotropic fluid on dynamical instability of spherically symmetric collapsing star in f (R, T ) theory. Herein, we plan to explore the instability range of anisotropic spherically symmetric stars, considering shearfree condition. The role of shear tensor in evolution of gravitating objects and consequences of shearfree condition has been studied extensively. Collins and Wainwright [2] studied the impact of shear on general relativistic cosmological and stellar models. Herrera et al. [3,4] worked out the homology and shearfree conditions for dissipative and radiative gravitational evolution.
Shearfree collapse accounting heat flow is discussed in [5], it is established that shear plays critical role in gravitational evolution and may lead to the formation of naked singularities [6]. The dynamical analysis of celestial objects and final outcome of stellar evolution have gained significant importance in astrophysical and gravitational theories. Thus, relevance of shear tensor in structure formation and its impressions on dynamical instability range of self gravitating body is well motivated direction of study.
Stars shine by consuming their nuclear fuel, continuous fuel consumption cause imbalance between inward acting gravitational pull and outward drawn pressure giving rise to collapse [7]. The outcome of gravitational evolution is size dependent as well as other physical aspects [8] such as isotopy, anisotropy, shear, radiation, dissipation etc. In comparison to the stars of mass around one solar masses, massive stars tend to lose nuclear fuel more rapidly and so more unstable. Pressure to density ratio naming adiabatic index, denoted by Γ is utile in estimation of stability/instability range of stars. Chandrasekhar [9] explored the instability range of spherical stars in terms of Γ [18].
Herrera and his contemporaries [10]- [17] contributed majorally in addressing the instability problem in general relativity (GR), accompanying various situations, i.e., isotopy, anisotropy, shearfree condition, radiation, dissipation, expansionfree condition, shearing expansionfree fluids. In order to achieve the more precise and generic description of universe, the dark energy components are incorporated by introducing modified theories of gravity. Modified theories are significant in advancements towards accelerated expansion of the universe and to present corrections to GR on large scales. The modifications are introduced in Einstein Hilbert (EH) action by inducing minimal or non-minimal coupling of matter and geometry.
Dynamical analysis of self gravitating sources in modified theories of gravity has been discussed extensively in recent years. The null dust non-static exact solutions in f (R) gravity are studied in [19], Cembranos et al. [20] studied the evolution of gravitating sources in the presence of dust fluid. Instability range of spherically and axially symmetric anisotropic stars has been established in context of f (R) gravity [21,22], concluding that devia-tions from spherical symmetry complicates the subsequent evolution.
Harko et al. [23] presented the f (R, T ) theory of gravity as another alternate to GR and generalization of f (R) theory representing non-minimal matter to geometry coupling. The EH action in f (R, T ) gravity includes arbitrary function of Ricci scalar R and trace of energy momentum tensor T to take into account the exotic matter. After the introduction of f (R, T ) gravity, its cosmological and thermodynamic implications were widely studied [24]- [27] including the energy conditions. Recently, we have studied the evolution of anisotropic gravitating source with zero expansion [28]. Herein, we are interested in exploration of shearfree condition implications on spherically symmetric gravitating source in f (R, T ) gravity.
The EH action in f (R, T ) is as follows [23] where L (m) denote matter Lagrangian, and g represents the metric tensor. Various choices of L (m) can be taken into account, each of which leads to a specific form of fluid. Many people worked out this problem in GR and modified theories of gravity, stability of general relativistic dissipative axially symmetric and spherically symmetric with shearfree condition has been established in [29,30]. Dynamical analysis of shearfree spherically symmetric sources in f (R) gravity is presented in [31]. The manuscript arrangement is: Section 2 comprises of modified dynamical equations in f (R, T ) gravity. Section 3 includes model under consideration, perturbation scheme and corresponding collapse equation alongwith shear-free condition in Newtonian and post Newtonian eras. Section 4 contains concluding remarks followed by an appendix.

Dynamical Equations in f (R, T )
In order to study the implications of shearfree condition on evolution of spherically symmetric anisotropic sources, modified field equations in f (R, T ) gravity are formulated by varying EH action (1.1) with the metric g uv . Here, we have taken L (m) = ρ, for this choice of L (m) modified field equations in f (R, T ) gravity takes following form uv is energy momentum tensor for usual matter taken to be locally anisotropic.
The three dimensional spherical boundary surface Σ is considered that constitutes two regions named as interior and exterior spacetimes. The line element for region inside the boundary Σ is The line element for region beyond Σ is [1] where ν is retarded time and M denote the total mass. The expression for anisotropic energy momentum tensor T (m) uv is given by where ρ is energy density, V u describes four-velocity of the fluid, χ u is radial four vector, p r and p ⊥ represent the radial and tangential pressure respectively. These physical quantities are linked as The shear tensor denoted by σ uv is defined as where a u is four acceleration and Θ is expansion scalar, given by Components of shear tensor are found by variation of Eq. (2.7) and these are used to find expression for shear scalar in the following form where dot and prime indicate time and radial derivatives respectively. From shearfree condition we arrive at vanishing shear scalar, i.e., σ = 0, implyinġ B B =Ċ C . Since we are dealing with fluid evolving under shearfree condition, so we shall make use of this condition while evaluating the components of field equations and also in conservation equations. The components of modified Einstein tensor are The dynamical equations extracted from the conservation laws are vital in the study of stellar evolution. The conservation of full field equations is considered to incorporate the non-vanishing divergence terms, Bianchi identities are on simplification of Eq.(2.14), we have dynamical equations as followṡ where Z 1 (r, t) and Z 2 (r, t) are provided in appendix as Eqs.(5.1) and (5.2) respectively. Deviations from equilibrium in conservation equations with the time transition leads to the stellar evolution, perturbation approach is devised to the estimate the instability range.

Perturbation Scheme and Shearfree Condition
We consider a particular f (R, T ) model of the form where α and λ can be any positive constants. Perturbation approach is utilized to estimate the instability range of spherical star with shear free condition. This scheme is utile in determination of more generic analytical constraints on collapse equation, rather to establish dynamical analysis of special cases numerically. Also, field equations are highly nonlinear differential equations, in such scenario application of perturbation lead to beneficial observations. It is assumed that initially all quantities are independent of time and with the passage of time perturbed form depends on both time and radial coordinates. Taking 0 < ε ≪ 1, the physical quantities and their perturbed form can be arranged as where Z 1p and Z 2p are given in appendix, for the sake of simplicity we put Y in place of 1 + 2αR 0 and λ 1 = λ + 1, assuming that D 1 = D 2 = D and e 1 = e 2 = e. Above mentioned perturbed dynamical equations and perturbed field equations shall be used to arrive at perturbed physical quantities such asρ,p r andp ⊥ . The expression forρ can be found from Eq.(3.31), as follows.
The Harrison-Wheeler type equation of state relateρ andp r , given bȳ Puttingρ from Eq.(3.33) in Eq.(3.34), we found Perturbed form of field equation (2.13) yields expression forp ⊥ , that turns out to bē Matching conditions at boundary surface together with perturbed form of Eq.(2.13) can be written in the simplified form as follows provided that The valid solution of Eq.(3.38) turns out to be The terms of Z 4 must be constrained in a way that all terms maintain positivity. Impact of shear-free condition on dynamical instability of N and pN regimes is covered in following subsections.

Newtonian Regime
In order to establish instability range in Newtonian era, we set ρ 0 ≫ p r0 , ρ 0 ≫ p ⊥0 and A 0 = 1, B 0 = 1. Insertion of these assumptions and Eq.(3.40) in Eq.(3.37) leads to the instability condition, relating usual matter and dark source contribution as under The quantities Z 1p(N ) and Z 2p(N ) are terms of Z 1p and Z 2p belonging to Newtonian era. The gravitating source remains stable in Newtonian approximation until the inequality for Γ satisfies, for which following constraints must be accomplished.
The case when α → 0 and λ → 0 leads to GR corrections and results for f (R) can be retrieved by setting λ → 0.

Post Newtonian Regime
We assume A 0 = 1 − m 0 r and B 0 = 1 + m 0 r to evaluate stability condition in pN regime. On substitution of these assumptions in Eq. (3.37), we have following inequality for Γ to be fulfilled for stability range.
Z 1p(P N ) and Z 2p(P N ) are terms of Z 1p and Z 2p that lie in post-Newtonian era. The above inequality (3.42) holds for positive definite terms and describe the stability range of subsequent evolution. The positivity of each term appearing in (3.42) leads to following restrictions

Concluding Remarks
In this manuscript, we carried out a study on implications of shear-free condition on stability of spherically symmetric anisotropic stars in f (R, T ).
Our In f (R, T ) gravity, the non-minimal matter geometry coupling include the terms of trace T in EH action that is beneficent in the description of quantum effects or so-called exotic matter. The components of modified field equations together with the implementation of shear-free condition are developed in section 2. Further conservation laws are considered to arrive at dynamical equations by means of Bianchi identities. These equations are utilized to estimate the variations in gravitating system with the passage of time.
The complexities of more generic analytical field equations are dealt by using linear perturbation of physical quantities. Perturbation scheme induce significant ease in the description of dynamical system, rather to present stability analysis by means of numerical simulations. The analytic approach we have employed here is more general and substantially important in explorations regarding structure formation. The perturbed shear-free condition together with the dynamical and field equations lead to the evolution equation, relating Γ with the usual and dark source terms. It is found that the induction of trace of energy momentum tenor in action (1.1) contributes positive addition to Γ, that slows down subsequent evolution considerably.
The outcome of gravitational evolution is size dependent as well as other physical aspects such as isotopy, anisotropy, shear, radiation, dissipation etc. The instability range for N and pN approximations is considered that impose some restrictions on the physical variables. It is observed that the terms appearing in Γ are less constrained for both the regimes (N and pN) in comparison to the anisotropic sources [1]. Thus, shear-free condition benefits in more stable anisotropic configurations. Corrections to GR and f (R) establishments can be made by setting α → 0, λ → 0 and λ → 0 respectively. The local isotropy of the model can be settled by assuming p r = p ⊥ = p. The extension of this work for shearing expansion free evolution of anisotropic spherical and cylindrical sources is in process.