Evolution of axially symmetric anisotropic sources in f(R, T) gravity

We discuss the dynamical analysis in f(R, T) gravity (where R is the Ricci scalar and T is the trace of the energy momentum tensor) for gravitating sources carrying axial symmetry. The self-gravitating system is taken to be anisotropic and the line element describes an axially symmetric geometry avoiding rotation about the symmetry axis and meridional motions (zero vorticity case). The modified field equations for axial symmetry in f(R, T) theory are formulated, together with the dynamical equations. Linearly perturbed dynamical equations lead to the evolution equation carrying the adiabatic index Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}, which defines the impact of a non-minimal matter to geometry coupling on the range of instability for Newtonian and post-Newtonian approximations.


Introduction
Recent developments in astrophysics and structure formation theories reveals that gravitating sources might deviate from most commonly studied spherical symmetry.Such deviations in realistic scenarios appear incidently, giving rise to the worth of non-spherical symmetries in gravitating objects.Herein, we intend to look into the implications of restricted class of axially symmetric sources (avoiding reflection and rotation) on gravitational evolution in context of f (R, T ) theory of gravity.Consideration of dynamic sources together with the angular momentum is a cumbersome task, however, observational data suggests that the lack of spherical symmetry prevails the more practical and worthwhile situations.A viable f (R, T ) model ( df dr ≥ 0, d 2 f dr 2 ≥ 0) with locally anisotropic matter distribution has been taken into account for the dynamical analysis.
The evolution of gravitating sources has been studied with a great deal of interest in recent past.Stars tend to collapse when outward drawn pressure decreases because of continuous fuel consumption, leading to imbalance in outward forces and inward acting gravitational pull.In such situation, gravitational force becomes the only governing force, massive stars burn nuclear fuel more rapidly and so more unstable as compare to the stars with relatively less mass.There are many other factors other than mass of the gravitating source that implicate intense modifications in range of stability/instability such as isotropy, anisotropy, shear, dissipation and radiation.Chanderashekar [1] shared valuable explorations to set instability range for spherically symmetric gravitating source in the form of adiabatic index Γ comprising pressure to density ratio with time transition.
Hillebrandt and Steinmetz [2] presented the instability criterion for anisotropic matter configuration of gravitating objects.Herrera et al. [3]- [7] contributed majorally in establishment of instability range of general relativistic fluids for different cases (isotropic, anisotropic, dissipative collapse etc), remarked that pressure anisotropy largely participates in setting dynamical instability.Moreover, they also worked out the imprints of axially and reflection symmetric static and dynamic sources by a general framework and some analytic models [8].Axially symmetric shearing geodesic and shearfree dissipative fluids are discussed in [9,10], where shearing geodesic case represents the zero radiation production.
General Relativity (GR) is a self-consistent theory, it is adequate for the explanation of many gravitational phenomenons up to cosmological scales.The scheme of GR appears to disagree with progressing observational data such as large scale structures ranging from galaxies to galaxy clustering, IA-type Supernovae, cosmic microwave background [11]- [14] etc.Alternatively, it can be said that GR is not the only definite gravitational theory that is suitable for all scales.Many attempts have been made to validate gravitational theories on large scales and coup with the cosmic acceleration [15]- [24], by introducing modified theories of gravity [25]- [30] for e.g.f (R), f (G), Brans-Dicke theory, f (R, T ) and so on.
Since the introduction of f (R, T ) theory in 2011 [31], people [32]- [34] worked on energy conditions along with its cosmological and thermodynamic implications.The f (R, T ) theory represents generalization of f (R) theory carrying non-minimal matter to geometry coupling.Extensive work has been done on instability range of spherically symmetric stars in GR as well as in modified theories of gravity.Literature on dynamical analysis of axially symmetric sources can be witnessed in GR.However, being a heavier task to handle modified dynamical equations in modified theories, very few attempts have been made to explore axial symmetry.
The purpose of this manuscript is to work out the instability problem for axially symmetric (in absence of reflection and radiation) anisotropic sources in context of f (R, T ) gravity.The reason of avoiding reflection and rotation terms in axially symmetry is only to somehow reduce the complications in analysis.The modified EH action in f (R, T ) admits arbitrary function of R and T to account the exotic matter.The action in f (R, T ) is given by [31] where L (m) represent matter Lagrangian and g represents the metric.Several choices of L (m) can be considered, each of which stands for a specific form of fluid.
The article arrangement is: The matter configuration and components of field equations together with the dynamical equations are furnished in section 2. Section 3 covers the information about the f (R, T ) model and perturbed conservation equations leading to the collapse equation.Section 4 contains range of stability of N and pN limits in the form of adiabatic index.The last section consists of concluding remarks followed by an appendix.

Interior Spacetime and Dynamical Equations
The general line element for axially symmetric compact objects constituting five independent metric coefficients is given by where the metric functions A, B, C, G, H have dependence on time, radial and axial coordinates (t, r, θ).Herein, we have ignored the meridional motions and rotation about the symmetry axis.Absence of dtdθ and dtdφ terms lead to the restricted character i.e. vorticity free case.The modified equations are highly non-linear in nature, so it is a tough task to handle such equations with non diagonal entries in the metric tensor, that is why we have taken zero vorticity case to somehow manage dynamical analysis by analytic approach.
The reduced form of general axial symmetry with three independent metric functions is [35] (2.3) Taking L (m) = −ρ, 8πG = 1 and varying the EH action (3.28) with respect to the metric tensor g uv leads to the following form for modified field equations uv is the energy momentum tensor for usual matter.The matter configuration is considered to be locally anisotropic [28], given by where ρ is the energy density and P xx , P yy , P zz and P xy are respective stresses causing pressure anisotropy, provided that P xy = P yx and P xx = P yy = P zz .K u and L u represents the four vectors in radial and axial directions, respectively and V u is for four-velocity, these quantities are linked as The components of modified (effective) Einstein tensor are ) ) ) ) Here dot, prime and θ indicate the derivatives w.r.t t, r and θ coordinates respectively.The expression for Ricci scalar is In order to explore stellar evolution, one needs to arrive at dynamical equations that can be obtained by employing contracted Bianchi identities.Conservation laws play significant part in establishment of instability range by more generic analytic approach, the dynamical equations in our case are T 2v ;v (rB) = 0, (2.17) on simplification, we have The notations of 0, 1 and 2 indicates t, r and θ.Terms belonging to matter or effective part of the dynamical equations can be viewed separately by inserting components of Einstein tensor given in Eqs.(2.7)-(2.13).The dynamics of gravitating axial system can be explored with the help of perturbation scheme, that is useful in estimating the change in system with the passage of time.

f (R, T ) Model and Perturbation Approach
Selection of the model under observation is a crucial constituent of the analysis.Since we are dealing the system analytically, thus model selected shall bring fruitful mechanism for some particular form of f (R, T ).We found that the f (R, T ) form suitable for dynamical analysis is constrained to f (R, T ) = f (R) + λT , where λ is positive constant and f (R) is an arbitrary function of Ricci scalar.The origin of such restriction comes from the fact that non-linear terms of trace in f (R, T ) complicates the formation of modified field equations that can not be handled analytically.Such f (R, T ) models bearing non-linear terms of trace of energy momentum can be dealt by using numerical techniques leading to more specified outcomes, whereas, the findings of analytic approach yield more generic.The viable f (R, T ) model we have chosen is where any positive value can be assigned to α and λ.

N and pN Approximation
This section constitutes the terms belonging to N and pN limits with instability criterion in the form of adiabatic index.Making use of Eqs.(3.39) and (3.38) in Eq. (3.35) leads to the evolution equation.The N and pN approximations for considered system are discussed in following subsections.

Newtonian Approximation
To approximate instability/stability range in Newtonian regime, we let A 0 = 1, B 0 = 1, ρ 0 ≫ p i0 ; i = xx, yy, xy, yy and Schwarzschild coordinates C 0 = r, evolution equation along with these assumptions yield ) where Z 2 N p corresponds to the N-approximation terms of Z 2p , and The inequality for Γ contains both material functions and effective part entries, system remains stable as long as the inequality (4.40) holds.The terms appearing in expression for Γ are presumed in a way that all terms maintain positivity, this requirement impose some restrictions on physical parameters.The constraints in N-approximation are Violation of these constraints imply instability in the sources and thus lead to gravitational collapse.

Post Newtonian Approximation
In pN approximation, we assume A 0 = 1− m 0 r and B 0 = 1+ m 0 r , corresponding inequality for range of stability is where Likewise Newtonian limit metric coefficients and effective part terms can be constrained to maintain stability of self gravitating system.System is stable unless above mentioned inequality holds, system collapses when ordering relation (4.41) breaksdown.One can deduce results of GR approximations by choosing vanishing values of λ and α.

Summary and Discussion
Observational Signatures supports the argument that gravitating sources might deviate from spherical symmetry incidently.Thus non-spherical symmetries facilitate in examining the realistic situations such as large scale structures, weak lensing, CMB etc. Motivating from the significance of nonspherical symmetries, we intend to explore impact of axially symmetric gravitating source in context of f (R, T ) gravity.More particularly we are dealing with restricted axial symmetry by ignoring meridional motions and rotation about symmetry axis.The consequence of restricted character of spacetime leads to vorticity-free case, because absence of dtdθ and dtdφ terms indicates that vorticity of gravitating source vanishes for at rest observer.The metric under consideration is axially symmetric with three independent metric functions.
Implications of axial symmetry on gravitating system has been studied extensively in GR and modified theories of gravity.The alternative gravity theory we have chosen to establish instability range is f (R, T ) gravity, because dynamical instability of axially symmetric sources in f (R, T ) framework has not been ascertained yet.The model under study f (R, T ) = R + αR 2 + λT is viable for positive values of α and λ.The modified field equations are obtained by varying action (3.28) for anisotropic matter distribution.The components of field equations (2.7)-(2.13)are used to arrive at conservation equations (2.18)-(2.20).These equations are of fundamental importance in establishment of instability range analytically.
The field equations are non-linear in nature, its a difficult task to evaluate their general solution.To count with this issue, we consider linear perturbation of usual matter and dark source terms.The perturbed physical quantities such as energy density and anisotropic pressure stresses are extracted from linearly perturbed components of field equations, that are further inserted in perturbed Bianchi identities to arrive at collapse equation carrying both material and dark source ingredients.An ordinary differential equation is formed from perturbed Ricci scalar, whose solution together with evolution equation provides adiabatic index.
Adiabatic index defines range of instability for N and pN approximations inducing some constraints on physical quantities that are provided in previous section.Corrections to GR and f (R) gravity can be determined by setting α → 0, λ → 0 and λ → 0 respectively.

Appendix
Following equations contain linearly perturbed terms of conservation equations and Ricci scalar respectively.