Stability of Oscillating Gaseous Masses in Massive Brans-Dicke Gravity

# Stability of Oscillating Gaseous Masses in Massive Brans-Dicke Gravity

## Abstract

This paper explores the instability of gaseous masses for the radial oscillations in post-Newtonian correction of massive Brans-Dicke gravity. For this purpose, we derive linearized perturbed equation of motion through Lagrangian radial perturbation which leads to the condition of marginal stability. We discuss radius of instability of different polytropic structures in terms of the Schwarzschild radius. It is concluded that our results provide a wide range of difference with those in general relativity and Brans-Dicke gravity.

Keywords: Brans-Dicke Theory; Hydrodynamics; Instability; Newtonian and post-Newtonian regimes.
PACS: 04.25.Nx; 04.40.Dg; 04.50.Kd.

## 1 Introduction

The study of evolution and formation of stellar structures has been issue of great interest in gravitational physics and cosmology. In this context, the phenomenon of dynamical stability of celestial objects has important implications in the analysis. It is believed that different stability ranges for stellar bodies lead to different phases of evolution or structure formation of the astronomical models. In general relativity (GR), Chandrasekhar [1, 2, 3] was the first who described a mechanism to explain dynamical instability of stellar structure in weak field approximation (at post-Newtonian (pN) limits). He used equation of state involving adiabatic index and concluded that the fluid remains unstable for . Herrera et al. [4] investigated dynamical evolution of self-gravitating fluids in different configurations (anisotropic fluid, adiabatic, non adiabatic as well as shearing viscous fluid). Sharif and his collaborators [5] also explored characteristics of different celestial fluid configurations in weak regimes through stability analysis.

The mystery of accelerating expansion of the universe has taken a remarkable attention in the last decade. In this context, the mechanism of modified theories of gravity has become a fascinated candidate. Modified theory of gravity means theory of gravity followed by modified Einstein-Hilbert action. The viability of these theories is an issue of great importance. For this reason, these theories are tested on different gravitational scales such as strong as well as weak field gravitational regime [6]. In this regard, the evolution and formation of celestial structure are considered to be the most suitable test-beds for modified theories. It is believed that modification of GR introduces some new astrophysical insights which can explain hidden parts of the universe. In this context, a large number of researchers have discussed modified astrophysical analysis [7]. Nutku [8] studied modified fluid hydrodynamics that affects the results of Chandraskhar. Recently, we have discussed modified dynamics of self-gravitating system in both weak and strong fields [9].

Brans-Dicke (BD) gravity (natural generalization of GR) [10] is one of the most explored examples of modified theory which is considered as a solution of many cosmic issues. This theory modifies the Einstein-Hilbert action according the Dirac hypothesis, i.e., it allows dynamical gravitational coupling (converts Newtonian gravitational constant into dynamical one) by means of dynamical massless scalar field . In this gravity, gravitational effects are described by coupling a massless scalar field with the curvature part (Ricci scalar). One of the main features of this theory is that it contains a constant tuneable parameter which is a coupling constant and can adjust required results. This theory provides suitable solutions of various cosmic problems but remains unable to probe ”graceful exist“ problem of old inflationary cosmology. The inflationary phenomenon described by BD gravity shows unacceptably large microwave background perturbations (by collisions between big bubbles) which can be controlled with the help of specific values of coupling parameter [11]. But these defined ranges of parameter are in conflict with observational limits [12].

In order to solve this problem, a massive scalar field is introduced in the framework of BD gravity [13] via a potential function . This new gravity is known as massive BD (MBD) gravity or self-interacting BD gravity. Moreover, BD gravity investigates all strong field issues (cosmological issues) for negative and small values of [14] but satisfies all weak field tests (related to solar system) for large and positive values of [15]. The MBD gravity provides a consistency with weak field gravitational test, i.e., explains cosmic acceleration for positive and large values of [16]. There has been a large body of literature which describes dynamics of MBD gravity in many cosmic issues [17, 18]. Olmo [19] calculated pN limits of MBD equations but he converted only lowest-order (order of limits of solutions in terms of potential functions to explore gravity as a special case of scalar-tensor gravity. Recently, we have explored hydrodynamics of different celestial configurations in complete pN correction of MBD gravity that modify the results of GR and BD gravity [20].

In this paper, we investigate gaseous system in MBD gravity and compare the results with GR and BD gravity. For this purpose, we explore stability of gaseous masses for radial oscillations in weak field approximation of MBD gravity. The paper is organized as follows. The next section represents complete pN approximation of MBD theory in terms of potential as well as super-potential functions and the dynamical equations. Section 3 explores instability of gaseous systems for radial oscillations by means of Lagrangian perturbation and variational principle. In section 4, we evaluate instability conditions of different polytropes in MBD theory. Finally, section 5 summarizes the results.

## 2 Massive Brans-Dicke Gravity and Dynamical Equations

The action of MBD gravity with () [16] is given by

 S=12κ2∫d4x√−g[ϕR−ωBDϕ∇αϕ∇αϕ−V(ϕ)]+Lm, (1)

where represents matter distribution depending upon metric. By varying the above action with respect to and , we obtain MBD equations as follows

 Gαβ = κ2ϕTαβ+[ϕ,α;β−gαβ□ϕ]+ωBDϕ[ϕ,αϕ,β−12gαβϕ,μϕ,μ]−V(ϕ)2gαβ, (2) □ϕ = κ2T3+2ωBD+13+2ωBD[ϕdV(ϕ)dϕ−2V(ϕ)], (3)

where shows the energy-momentum tensor, and represents the d’Alembertian operator. Equations (2) and (3) indicate MBD field equations as well as evolution equation for the scalar field, respectively. We assume matter distribution as a perfect fluid which can be compatible with pN regime

 Tαβ=[ρc2(1+πc2)+p]uαuβ−pgαβ, (4)

where indicate density, thermodynamics density, pressure and four velocity, respectively.

### 2.1 Post-Newtonian Approximation

The weak-field limits of any relativistic theory explain the order of deviations of the local system from its isotropic and homogenous background. The parameterized pN approximations are widely used as weak field approximated solutions that are obtained by using the following Taylor expansion of the metric functions [21]

 gαβ ≈ ηαβ+hαβ,

with

 h00≈h(2)00+h(4)00,h0i≈h(3)0i,hij≈h(2)ij.

Here shows the Minkowski metric (describing isotropic and homogenous background of ), indicates deviation of from background values , and the superscripts and describe approximation of order as well as . In this approximation scheme, the field equations are solved formally and the metric functions are expressed as a sequences of pN functions of source variables (source of metric function like matter) coupled to coefficients (pN parameter). These coefficients are based upon the matching conditions between the local system and cosmological models or on other constants of the theory. The pN functions are basically metric potentials which are chosen under reasonable assumption of Poisson’s equations and gauge conditions to have unique solutions according to pN order of correction [21].

In order to discuss stability of gaseous system in MBD gravity and check the compatibility of our results with the analysis of GR [1, 2], we approximate the system in pN limits. For this purpose, we use complete pN approximations (upto order of ) of MBD gravity. The parameterized pN limits of MBD solutions has been evaluated by using the following expansion of metric and dynamical scalar field [19, 20]

 gαβ ≈ ηαβ+hαβ, ϕ ≈ ϕ0(t0)+φ(2)(t,x)+φ(4)(t,x), V(ϕ) ≈ V0+φdV0dϕ0+φ2d2V0d2ϕ0+....

Here represents time of isotropic and homogenous background of local system. The term shows unperturbed or initial value of scalar field in isotropic and homogenous background of local system which vary very slowly with respect to . This implies that the cosmological considerations would allow a slow evolution of on cosmological timescales. Since these timescales are much larger than the solar system timescales, so its evolution may be ignored for physical setup in weak-field and it is considered as constant. The term shows the potential function of scalar field at and is the local deviation of scalar field from .

The parameterized pN approximations of MBD solutions are given by [20]

 gij ≈ (−1−2γBDUc2−ΛBDr23c2)δij, (5) g00 ≈ 1−2Uf(r)c2+12c4⎡⎣(−2U+ΛBDr23)2−(−2U−A(r)3+2ωBD+A(r))2⎤⎦ (6) − 2(Φ+ψ), g0i ≈ 1c3(4Uif(r)−12∂2χ∂t∂xi). (7)

Here ( is the Newtonian mass of the sun) is the effective gravitational potential determined by Poisson’s equation

 ∇2U=−4ΠρGeff, (8)

where indicates the effective gravitational constant (dynamical Newtonian gravitational constant) for massive scalar field defined by [18, 19]

 Geff=κ28Πϕ0f(r)=κ28Πϕ0(1+A(r)3+2ωBD), (9)

where

 A(r)={e−m0rm20>0cos(m0r)m20<0, m0=⎛⎜ ⎜ ⎜⎝ϕ0d2V0dϕ20−dV0dϕ03+2ωBD⎞⎟ ⎟ ⎟⎠1/2.

Here the term is the mass of the massive scalar field and “” represents scale of experiments and observations. It is actually a distance between two points in the local system and can be used to express radius of configuration (spherical or cylindrical) under consideration. The term represents the parameterized pN parameter given by

 γBD=3+2ωBD−A(r)3+2ωBD+A(r). (10)

The oscillatory solutions are unacceptable [23]. In this case, the inverse-square law modifies as

 M⊙r2→(1+cos(m0r)+(m0r)sin(m0r))M⊙r2, (11)

and for very light fields (showing long-range interactions), the arguments of cosine and sine are very small in solar system scales which provide and . These approximations lead to usual Newtonian limits upto an irrelevant redefinition of Newtonian Constant. This also yields for which is observationally unacceptable since . If the scalar interaction is short-range or mid-range, the Newtonian limits would dramatically be modified. In fact, the leading order term is then oscillating, , and is clearly incompatible with observations. That is why, we consider only the damped solutions .

The Yukawa-type correction in the Newtonian potential has not been observed over distances that range from meters to planetary scales. In addition, since the post-Newtonian parameter is observationally very close to unity, the mass function present in Eqs.(10) and (11) satisfy the constraint ( shows the scale of the observations or experiments testing the scalar field). For solar system scale observations, the relevant scale is the Astronomical Unit corresponding to a mass scale . Although this scale is small for particle physics considerations, but it is still much larger than the Hubble mass scale required for nontrivial cosmological evolution of [24]. Current solar system constraints [21, 23] of the parameter have been obtained under one of the following assumptions [19, 18]

• When the background value of is very small (negligible mass of the field) and , MBD system reduces to simple BD gravity (massive scalar field becomes massless scalar field) having

 Geff=κ28Πϕ04+2ωBD3+2ωBD,γBD=1+ωBD2+ωBD.

That is why the BD theory (massless scalar field) is consistent with solar system constraints of the Cassini mission for .

• For and , the observational constraints on are same as discussed for the case .

• For massive scalar field ( and ), the dynamics of the spatial part of is frozen on the solar system scale through potential function of scalar field and all values of are observationally acceptable [27]. It can be noticed that further limit reduces the value of to simple Newtonian gravitational constant and which is consistent with GR.

• For , all values of are observationally allowed.

The term indicates the cosmological term (where is a cosmological constant) which is based on the potential of the scalar field. In order to be consistent with observational data (ranging from the solar system to clusters of stellar structures), the contribution due to scalar density should be very small and the following constraint must be satisfied

 V0L2ϕ0<<1.

Here shows the length scale equal to or greater than the solar system. The term represents super-potential given by the following Poisson’s equations [20]

 ∇2~Φ = −4ΠGeffρσ,~Φ=Φ+2ψ, (12) ∇2ψ = −12ϕ0[V0(1+h(2)[ij]−φ(2)ϕ0)+φ(2)dV0dϕ0], (13)

here

 σ=1f(r)[π+2v2+h(2)[ij]−φ(2)ϕ0+3pρ],∇2Φ=−4ΠGρ~σ,~σ=f(r)σ. (14)

Similarly and are potential functions satisfying the following Poisson’s equations

 ∇2χ=h(2)00=1c2(−2U+ΛBDr23), (15) ∇2(Uif(r))=−4ΠGeffρvif(r), (16)

where [1]. The effect of is taken approximately constant and the effects of as well as are neglected. The solutions satisfy the following gauge condition

 hαμ,α−12hαα,μ−1c2ϕ0∂φ∂xμ=0.

All the assumptions and solutions are consistent with BD gravity in the limits [8] and the system reduces to GR with [1].

### 2.2 Hydrodynamics

According to pN approximation of MBD theory, the equation of continuity and equation of motion (generalized Euler equation of Newtonian hydrodynamics) are obtained using

 Tαβ;β=0. (17)

From Eqs.(6)-(17), the equation of continuity is given by [1, 8, 20]

 ∂~ρ∂t+∂∂xi(~ρvi)=0,

where

 ~ρ=ρ(1+1c2(12v2−ΛBDr23+9+6ωBD−e−m0r3+2ωBD+e−m0rU)). (18)

This shows that the mass function indicated by density remains conserved. The spatial components of Eq.(17) provide the equation of motion is given by [20]

 ∂ηvi∂t+∂ηvivj∂xj+∂∂xi[(1+2γBDU+ΛBDr23)p]+2ρc2ddt[(2γBDU +ΛBDr23)vi]−4ρc2ddt[Uif(r)]−ρc2[f(r)σ∂∂xi(Uf(r))+∂~Φ∂xi] −4ρc2vj∂∂xi(Ujf(r))−ρc2∂∂xi(Uf(r))+ρ2c2ddt(Ui−Uα;iα) −ρ2c2Wi+ρ2c2Zi(BD)=0, (19)

where represents material derivative and

 ∂3χ∂2t∂xi = ddt(Ui−Uα;iα), η = ρ(1+1c2(v2+2U−2ΛBDr23+π+pρ)),

the potential functions and are given in Appendix A.

## 3 Dynamical Stability of Gaseous Masses

To discuss stability of gaseous masses in the presence of massive scalar field, we use Chandrasekhar technique [2] which has been also used to explain stability of gaseous system in BD gravity [8]. For this, we assume that initially the spherically symmetric distribution of matter field is in complete hydrostatic equilibrium. Using Eq.(19), the hydrostatic condition is given by

 [(1+2γBDUc2+ΛBDr23c2)]∂p∂xi=ρc2[f(r)σ∂∂xi(Uf(r))+∂~Φ∂xi]−ρc2 ×∂∂xi(Uf(r))−p∂∂xi[(1+2γBDUc2+ΛBDr23c2)]−ρ2c2Zhi(BD), (20)

where represents hydrostatic case of and its value is mentioned in Appendix A. In hydrostatic equilibrium, the values of and density term are free from velocity term ().

### 3.1 Lagrangian Perturbation and Oscillations

In order to discuss stability of oscillating MBD fluid, we consider that the fluid is flowing according to Lagrangian description. In Lagrangian description of fluid flow, the spatial reference system is comoving with the fluid. The position of the particle (depending upon spatial coordinates) is not an independent variable and the material derivative reduces to simple partial derivative of time at specific constant position [25]. We assume that the system is initially in hydrostatic configuration. Then after certain time, the equilibrium configuration of the system is slightly perturbed such that the spherically symmetric distribution remains unchanged. The perturbed state is obtained by the following Lagrangian displacement [2, 8]

 ¯ξeiαt,

where is a displacement vector defined by representing position vector of Lagrangian particles from their initial position at time ). The term shows the characteristic frequency of oscillations. In order to determine frequency of the oscillations, we evaluate linearized Lagrangian form of Eq.(19) (which governs small oscillations about the equilibrium) by using lagrangian perturbation and Eq.(20) as follows

 α2{ηξi+2ρc2((2γBDUc2+ΛBD3c2r2)ξi−2Uif(r)) +ρc2(Ui−Uα;iα)}=−∂∂xi[(1+2γBDUc2+ΛBDr23c2)Δp+2pγBDΔU] +ρc2[f(r)σ∂∂xi(ΔUf(r))−f(r)Δσ∂∂xi(Uf(r))] −∂∂xiΔ~Φ−Δρρ∂∂xi[(1+2γBDUc2+ΛBDr23c2)p] +ρc2∂∂xi(ΔUf(r))+ρc2ξiΔZi(BD). (21)

Here is converted into and becomes . The terms and denote the Lagrangian changes in the respective quantities. The value of is replaced by in the definition of as well as in and is expressed in appendix A.

Now we express the Lagrangian changes of various dynamical quantities in terms of . Under Lagrangian perturbation, the equation of continuity becomes

 Δ~ρ=−~ρ∇.¯ξ. (22)

Equations (18) and (22) imply

 Δ~ρ = Δρ(1+1c2(ΛBDr23+9+6ωBD−e−m0r3+2ωBD+e−m0rU))+ρc2 × 9+6ωBD−e−m0r3+2ωBD+e−m0rΔU=−ρ(1+1c2(−ΛBDr23 + 9+6ωBD−e−m0r3+2ωBD+e−m0rU))∇.¯ξ.

From the above equation, the explicit expressions of Lagrangian change in density can be evaluated in terms of as [2, 8]

 Δρ=−ρ(∇.¯ξ+9+6ωBD−e−m0r3+2ωBD+e−m0rΔU), (23)

where only linear terms of and are considered. The definition of adiabatic index and the relation

 dπ=pρ2dρ, (24)

yield

 Δp=γpρΔρ,ρΔπ=pρΔρ. (25)

Equations (23) and (25) give

 Δp = −γp(∇.¯ξ+9+6ωBD−e−m0r3+2ωBD+e−m0rΔU), (26) ρΔπ = −p(∇.¯ξ+9+6ωBD−e−m0r3+2ωBD+e−m0rΔU). (27)

From Eqs.(14) and (23)-(27), it follows that

 Δσ = 1f(r)(e−m0r3+2ωBD+e−m0rΔU (28) − pρ(3γ−2)(∇.¯ξ+1c29+6ωBD−e−m0r3+2ωBD+e−m0rΔU)),

where only linear terms of are considered. In order to obtain explicit expressions of and in terms of , we use relation between Eulerian and Lagrangian changes given by [2, 8]

 ΔU=δU+¯ξ.∇U,Δ~Φ=δ~Φ+¯ξ.∇~Φ. (29)

Here and represent Eulerian changes in the respective quantities that can be calculated from Eqs.(8) and (12) as follows

 ∇2δU=−4ΠGeffδρ,∇2δ~Φ=−4ΠGeffδ(ρσ).

Integration of the above equation gives [2, 8]

 δU = ∫vG(eff)ρ(~x)ξi(~x% )∂∂xi1|x−~x|d~x (30) − 6c2∫v3+2ωBD3+2ωBD+e−m0rG(eff)ρ(~x)ΔU(~x)x−~xd~x, δ~Φ = ∫vG(eff)ρ(~x)σ(~x)ξi(~x)∂∂xi1|% x−~x|d~x (31) − ∫vG(eff)ρ(~x)Δ~Φ(~x)x−~xd~x.

Equations (29)-(31) provide

 ΔU = ¯ξ.∇U+∫vG(eff)ρ(~x)ξi(~x)∂∂xi1|x−~x|d~x (32) − 6c2∫v3+2ωBD3+2ωBD+e−m0rG(eff)ρ(~x)ΔU(~x)x−~xd~x, Δ~Φ = ¯ξ.∇~Φ+∫vG(eff)ρ(~%x)σ(~x)ξi(~x)∂∂xi1|x−~x|d~x (33) − ∫vG(eff)ρ(~x)Δ~Φ(~x)x−~xd~x.

With the help of Eqs.(23)-(33), Eq.(21) can be expressed explicitly in terms of .

### 3.2 The Variational Principle

The stability criteria of oscillating body depends upon the behavior of frequency. For , the system becomes marginally stable, i.e., the model will expand and contract with homologous property. Therefore, in order to discuss the behavior of frequency, we use variational principle with the help of Eq.(21). For this purpose, we assume that on the boundary and at the origin , each quantity is nonsingular [2, 3]. In this way, Eq.(21) along with boundary conditions represent a self-adjoint characteristic value problem for . Thus a variational base is obtained by converting Eq.(21) into and then integrating over the configuration of fluid by contracting with [2]. The resulting equation becomes

 Qα2 = ∫v(∇.¯ξ)p[(1+2γBDUc2+ΛBDr23c2)(9+6ωBD−e−m0r3+2ωBD+e−m0rΔU (34) + γ∇.¯ξ)+2pγBDΔU]dx+ρc2∫v[f(r)σξi∂∂xi(ΔUf(r))−f(r)Δσ × ξi∂∂xi(Uf(r))+ξi∂~Φ∂xi]dx+ρc2∫v(∇.¯ξ+9+6ωBD−e−m0r3+2ωBD+e−m0rΔU) × f(r)ξi∂∂xi((1+2γBDU+ΛBDr23)p)dx+ρc2∫vξi∂∂xi × (ΔUf(r))dx−2ρc2∫vξi∂∂xi(e−m0r3+2ωBD+e−m0rΔU)dx + 2c2∫v(γp(∇.¯ξ+9+6ωBD−e−m0r3+2ωBD+e−m0rΔU) × (ξi∂∂xi(3+2ωBD+2e−m0r3+2ωBD+e−m0r+γBD)U))dx + 2pc2∫v(ξi∂∂xi(3+2ωBD+2e−m0r3+2ωBD+e−m0r)ΔU)dx + 2∫v[ξi∂∂xi(γBDΔU)+ΔUΛBDξi∂∂xir23]dx].

The left hand of this equation is

 Qα2 = α2{∫vη|¯ξ|2dx+ΛBDr23|¯ξ|2dx+∫v∫vG(eff)c2ρ(x)ρ(~x) × |¯ξ(x)−¯ξ(~x)||x−~x|dxd~x[2γBD−4(3+2ωBD)3+2ωBD+e−m0r] − ∫v∫vG(eff)2ρ(x)ρ(~x)[¯ξ(x).(x−~x)][¯ξ(~x).(x−~x)]|x−~x|3dxd~x},

where represents positive-definite quantity.

### 3.3 The Onset of Instability for the Radial Oscillations in the Post-Newtonian Approximation

Here, we discuss the criteria for the onset of dynamical instability in pN limits of MBD gravity. For this purpose, we consider radial oscillations having density as well as pressure distribution in the equilibrium conditions. According to definitions of vector spherical harmonics in radial oscillations, the Lagrangian displacement turns out to be [2, 8]

 ξr=r~η,ξ⊥=0,ξθ=0, (36)

where is an unknown function. The radial components of and can be obtained from Eqs.(28), (29) and (36) as follows

 Δσ = 1f(r)[e−m0r3+2ωBD+e−m0rΔU (37) − pρ(3γ−2)(ddr(r3~η))]+O(c−2), ΔU = δU+r~ηdUdr,Δ~Φ=δ~Φ+r~ηd~Φdr. (38)

Here the values of and for radial oscillations are given by [2]

 δU = 4ΠG(eff)[∫Rrρ(s)s~ηds−3c2(1r∫r0ρ(s)ΔU(s)s2ds (39) + ∫Rrρ(s)ΔU(s)sds)