Dynamics of Axial Symmetric System in Self-Interacting Brans-Dicke Gravity

This paper investigates dynamics of axial reflection symmetric model in self-interacting Brans-Dicke gravity for anisotropic fluid. We formulate hydrodynamical equations and discuss oscillations using time-dependent perturbation for both spin as well as spin-independent cases. The expressions of frequency, total energy density and equation of motion of oscillating model are obtained. We study instability of oscillating models in weak approximations. It is found that the oscillations and stability of the model depend upon the dark energy source along with anisotropy and reflection effects. We conclude that the axial reflection system remains stable for stiffness parameter $\Gamma=1$, collapses for $\Gamma>1$ and becomes unstable for $0<\Gamma<1$.


Introduction
Dark energy and stellar evolution are interesting issues of modern cosmology as well as gravitational physics. Different astronomical surveys (such as Sloan Digital Sky Survey, Wilkinson Microwave Anisotropy Probe, Supernova type Ia, large scale-structure, weak lensing and galactic cluster emission of X-rays etc.) reveal accelerated expansion of the universe [1]. It is assumed that a mysterious form of energy termed as dark energy is responsible for this accelerated expansion of the universe. The resolution of this mystery leads to various modified theories of gravity by modifying the Einstein-Hilbert action. In this context, the scalar-tensor theory is one of the most fascinated idea which has provided solutions of various cosmic problems such as early and late behavior of the universe, inflation, coincidence problem and cosmic acceleration [2].
The most explored and useful example of scalar-tensor gravitational framework is the Brans-Dicke theory of gravity. This is a natural generalization of general relativity constructed by the coupling of tensor field R and a massless scalar field φ. It also contains a constant tuneable parameter ω BD which can be tuned according to suitable observations. The concept of this theory is based upon the weak equivalence principle, Mach's principle and Dirac's large number hypothesis [3]. The basic idea of this theory is that the inertial mass of an object is not an intrinsic property of the object itself but is generated by the gravitational effect of all the other matter in the universe. For cosmic inflation, this theory is generalized with self-interacting scalar field by the inclusion of scalar potential function V (φ) [4] known as self-interacting Brans-Dicke (SBD) gravity. This has attracted a community of researchers for the viable discussion of cosmic problems in scalar-tensor framework [5,6].
The study of formation and evolution of stars, galaxies and cluster of galaxies has important implications in cosmology and gravitational physics. Many observational and experimental surveys such as Sloan Digital Sky Survey, University of Washington N-Body shop, the Virgo consortium, Leiden observatory and the Hubble telescope indicate stellar structures to resolve cosmic issues like dark matter, dark energy and completeness of big-bang theory. It is conventional that stellar models are mostly rotating and anisotropic in nature. Anisotropy plays a significant role in different dynamical phases of stellar evolutions [7].
The pattern of uniform as well as differential rotations of various evolving celestial bodies are investigated through analysis of stability and oscillations of axial configurations in weak approximations. Arutyunyan et al. [8] used Newtonian (N) and post-Newtonian (pN) regimes to explore the structure of rotating celestial object. Chandrasekhar and Friedman [9] described perturbation theory of axial symmetric models to discuss instability ranges of uniformly rotating stars. Clifford [10] explored oscillations and stability of differentially rotating axial symmetric system. Sharif and Bhatti [12] discussed reflection symmetric axial non-static models and found that instability ranges depend upon the stiffness parameter and also the spinning models are more stable.
Many people [6,13] investigated stellar evolutions in the modified theories. Since the evolution of such models passes through different dynamical stages, this study can lead to correct theory of gravity or it may reveal some modifications hidden in the structure formation of the universe. In this paper, we explore dynamics of non-static axial reflection model in the framework of SBD gravity and study stellar evolution under Mach's principle. The paper is organized in the following format. In the next section, we review SBD gravity and axial system with reflection symmetry as well as anisotropic fluid. Section 3 describes dynamical picture of evolving axial system such as hydrodynamics, oscillations and instability regimes. Final section summarizes the results.

Self-Interacting Brans-Dicke Theory
The SBD theory is represented by the following action [4] where L m shows matter distribution and 8πG 0 = c = 1. Varying the above action with respect to g µν and φ, we obtain Here G µν represents the Einstein tensor, T m µν φ indicates the contribution of matter in the presence of scalar field, T m = g µν T µν and is the d'Alembertian operator. The energy contribution due to scalar field is described by which is energy momentum tensor associated with Machian terms that describes the interaction of scalar field with the geometry of the distant matter distributions in the universe and the effects of its potentials upon them.
Equations (2) and (3) represent SBD field equations and SBD wave equation, respectively. The right hand side of Eq. (2) indicates that both terms are sources of geometry (gravitation). There also exists static field in axial symmetric SBD model which has φ = φ(t 0 ) = constant with respect to cosmic time t 0 and generalizes Einstein equations with an effective cosmological constant V (φ 0 ) [11]. These static field configurations lead to the dynamics of non-static axial system.
In order to discuss dynamics of non-static axial symmetric configurations, we consider non-static axially symmetric spacetime characterized with reflection [12,14] having matter contribution in the form of locally anisotropic fluid given by Here where ρ is the energy density, p shows the isotropic pressure, Π µν represents anisotropic stress tensor, Π I = Π II = Π kχ are the anisotropic scalars and h µν expresses projection tensor. The four velocity u µ , unit four-vectors k µ , s µ and χ µ are calculated as with ∆ = r 2 A 2 B 2 + L 2 and satisfy the following relations The non-zero components of the energy-momentum tensor due to scalar field can be represented as Here v i + w i represents diagonal and x j + y j shows non-diagonal components of stress tensor (4) in which w i and y j indicate axial reflection effects due to scalar field.

Dynamics
In this section, we carry out dynamical analysis of axial reflection symmetric system. For this purpose, we derive the hydrodynamical equations and discuss oscillations as well as instability ranges of the perturbed axial system.

Hydrodynamics
The dynamical equations representing hydrodynamics of axially symmetric system can be obtained with the help of Bianchi identity G µν ;ν = 0. This identity along with Eqs. (2), (4) and (6) provide the following equations for µ = 0, 1, 2, Here dot, prime and superscript θ indicate derivatives with respect to time, r and θ, respectively. The subscript (mφ) implies energy-momentum terms of matter distribution per scalar field ( T µν(m) φ ) which corresponds to contributions of matter dynamics in the presence of scalar field. The terms E 0 (t, r, θ), E 1 (t, r, θ) and E 2 (t, r, θ) represent energy contributions due to scalar field and their values are given in Eqs.(A1)-(A3). Equations (10)- (12) describe hydrodynamical equations of axial reflection symmetric fluid in SBD gravity.

Oscillations
Now we discuss oscillations of axial system through perturbation approach. We consider that initially the system is in hydrostatic equilibrium and after that all metric functions along with dynamical variables are perturbed with time dependence perturbation T (t) = e iwt and the system starts oscillating with frequency w. The metric tensor as well as the scalar field and scalar potential have the same time dependence, while the dynamical variables bear the same time dependence as follows Here 0 < ǫ ≪ 1 and the subscript zero indicates static distribution while terms having bar represent perturbed terms [12,15]. Using Eqs.(13)-(23), the perturbed configuration of 02-components of the field equations (2) can be represented as where values of l, m and n are given in (A4)-(A6). Since e iwt = 0, this implies that yielding frequency of the oscillating axial reflection system. This shows that the frequency of oscillations depend upon the DE source (scalar field), anisotropic effects and reflection configuration. The total density of oscillating system can be obtained from the perturbed form of the first law of conservation (10) as follows Here F (mφ) shows contribution of matter with scalar field,Ē 0(a) andĒ 0(b) represent scalar field distributions whose values are given in Eqs.(A7)-(A9). The terms with subscript (a) represents scalar field coupled to frequency while subscript (b) shows scalar field without frequency. The perturbed form of Eq. (11) provides the equation of motion of the oscillating system given by The superscript p indicates perturbed form,Ē 1 represents perturbed configurations of scalar field terms given in Eq.(A10).

Spin-Independent Oscillations
The local spinning of anisotropic system is calculated through the vorticity tensor. For the axial symmetric spacetime with reflection, the vorticity tensor can be expressed in terms of k µ and χ µ as is the vorticity scalar. This shows that the spin-independent motion occurs whenever whereK =K(t, θ) is an arbitrary function of integration. This implies that LK = A 2 , consequently, if L = 0 we have A = 0 and the existence of nonstatic configuration of axial spacetime is disturbed. Therefore, for Ω = 0, we take LK = A 2 , L = 0.
Thus, oscillations convert into spin-independent form whenever, LK = A 2 and L = 0. This implies that spin-independent oscillations depends upon reflection contribution. Equations (24)-(26) along with LK = A 2 , L = 0 provide frequency, total density and equation of motion of spin-independent oscillations of axisymmetric distribution.

Stability Analysis
Here, we discuss stability of oscillating collapsing axial system (with reflection symmetry) in the presence of scalar field. We assume that the system is perturbed adiabatically and satisfies the equation of state [16] where the equation of state parameter (Γ) represents a constant adiabatic index which calculates stiffness or rigidity in the fluid. This equation with Eq.(25) provides the perturbed part of anisotropic stresses as follows Using these relations in the equation of motion (26), we obtain the collapse equation (hydrostatic equation) of oscillating axial reflection system The quantity δ (m(BD)) shows pressure gradient forces and anti-gravitational forces (due to matter as well as scalar field distributions) coupled to adiabatic index whereas λ (m(BD)) +Ē 1 gives gravitational forces (forces opposite to pressure gradients forces) mediated by matter as well as scalar field contributions. The values of these terms are given in (A11) and (A12). The adiabatic index Γ is taken to be positive in order to balance the hydrostatic configurations between pressure gradient as well as gravitational force.

Newtonian Approximations
In order to evaluate stability criteria in the N limits, we use approximation A 0 = 1, B 0 = 1, C 0 = r, L 0 = r, ∆ 0 = r 2 , φ = φ 0 and V (φ) = V 0 . By using these limits in Eq. (28), it follows that where the values of δ (m(BD))N , λ (m(BD))N andĒ 1(N ) are given in (A13)-(A15). This provides a hydrostatic condition which implies that the system collapses whenever For Γ > 0, we need to take |(λ (m(BD))N +Ē 1(N ) )| and |δ (m(BD))N |. Thus the system remains unstable (collapses) as long as the inequality (30) holds. This implies that the instability ranges in N limits can be calculated through the stiffness of fluid (adiabatic index) which depends upon the configurations of pressure gradient forces as well as anti-gravitational forces coupled to adiabatic index and gravitational forces. These factors in turn depend upon the energy density, anisotropies, reflection effects and scalar field contributions. We can summarize the results as follows: • If the gravitational forces |(λ (m(BD))N +Ē 1(N ) )| are balanced by antigravitational and pressure gradient forces (δ (m(BD))N ) then (29) implies that Γ = 1 and the system is in complete hydrostatic equilibrium (remains stable).
• If the anti-gravitational and pressure gradient distribution related to stiffness parameter are greater than gravitational contribution, then according to (29), the system becomes unstable (but not collapses) for 0 < Γ < 1.
• Equation (29) implies that if gravitational effects are greater than that of the anti-gravitational and pressure gradient effects coupled to adiabatic index, the system collapses leading to instability for Γ > 1.
In the case of spin-independent oscillations, the inequality (30) with LK = A 2 and L = 0 provides the criteria for an unstable spin-independent oscillations. The numerical instability ranges (0 < Γ < 1 and Γ > 1) remain the same as calculated for spin-dependent oscillations.

Post-Newtonian Approximation
In pN limits, we use approximations upto order of m 0 r 1 (discarding terms having higher order of m 0 r 1 ) as follows: 17], ϕ represents local deviations of scalar field from φ 0 . The axial system becomes unstable in pN limits if the adiabatic index satisfies the following inequality the values of δ (m(BD))pN and (λ (m(BD))pN +Ē 1(pN ) ) are given in (A16)-(A18). Similar to N case, the instability criteria depends upon rigidity of the fluid and Γ = 1 provides stable configurations in pN regime while the system becomes unstable for other values of adiabatic index.

Concluding Remarks
According to recent observation, DE controls the dynamics of expansion of the present universe. General relativity is considered as a fundamental theory for the description of various astrophysical processes. It is an excellent theory of gravity which has gained many achievements but said to break down at Planck length. Its proposed DE candidate "cosmological constant" is not considered to be compatible with the calculated vacuum energy of quantum fields. This non-normalizable behavior of general relativity induces the concept of alternative theories of gravity (alternative candidates of DE) [18]. These theories are constructed by incorporating extra degrees of freedom in the Einstein-Hilbert Lagrangian density either in geometrical (gravitational) or matter part. Some of these DE models are Chaplygin gas, tachyon fields, quintessence, k-essence and modified gravities such as f (R) gravity, f (T ) theory, Gauss-Bonnet gravity, f (R, T ) gravity and scalar-tensor theories. Scalar-tensor theory of gravity is an alternative step to unify theoretically gravity and quantum mechanics at high energies by introducing scalar field as an extra degree of freedom in the Einstein-Hilbert Lagrangian density. Brans-Dicke gravity is the first proper scalar-tensor gravity assumed to be prototype of alternative theory of Einstein gravity. The principal features of this theory are the compatibility with Dirac's hypothesis and Mach's principle, i.e., this theory believes on dynamical gravitational coupling (dynamical gravitational constant) by means of dynamical scalar field φ (extra force field) which allows distributions of distant matter to affect the dynamics at a point. The basic idea of this theory is that the inertial mass of an object is not an intrinsic property of the object itself but is generated by the gravitational effect of all the other matter in the universe. In this way, this theory has generalized the Einstein gravity to Machian one (compatible with Mach principle) and has provided convenient solutions of many cosmic problems especially accelerated expansion of the universe.
In general relativity, the effects of stellar rotations cannot be neglected to fully investigate the formation of stars and black holes. During evolution, the self-gravitating fluid passes through many phases of dynamical activities that remain in hydrostatic equilibrium for a short span. In this paper, we have studied dynamical stability of non-static stellar model with axial reflection symmetric anisotropic fluid distribution under the influence of dynamical gravitational coupling through SBD gravity. We have generalized the dynamical analysis of general relativity by incorporating Mach's principle to explore effects of DE upon the cosmic evolution When a system is departed from its initial static phase, it becomes perturbed and starts oscillating. We have explored oscillations of the axial reflection configuration under time-frequency dependent perturbation. It is shown that frequency of the rotating oscillations depends upon anisotropic effects, reflection symmetric and DE contribution represented by scalar field. The perturbed form of conservation laws yields the total energy density and equation of motion that depend upon the behavior of anisotropic effects as well as frequency. We have also investigated spin-independent oscillation and found that reflection configuration is the factor which controls spin of the axial system.
In order to obtain viable models of the rotating system, we have studied different instability ranges with the help of collapse equation. It is found that the stable configurations of spin-dependent oscillations depend upon the stiffness of the fluid which in turn depends upon anisotropic effects, reflection parameter as well as distribution of scalar field. The instability of spin-independent oscillating system depends upon the rigidity of the fluids due to anisotropy, reflection effects with constraints LK = A 2 , L = 0 as well as SBD gravity contribution. We would like to mention here that dynamics of axial rotating system in general relativity depends upon the anisotropic as well as reflection effects but here the results are modified by the inclusion of an extra field (scalar field) as a DE candidate. Thus we can conclude that in the present accelerating universe, DE not only controls the expansion among celestial objects but it also affects stellar evolution.
According to Mach's principle, the dynamics of any evolving body in the universe is not an intrinsic property, but the surrounding distant matter also has its effect. In this way, it involves all the surrounding stellar structures in the analysis. It would be intersecting to explore collapsing phenomenon and its consequences on the stellar objects according to Mach's principle to enhance the study of stellar evolution in the presence of DE.