# Spherically Symmetric Fluid Cosmological  Model with Anisotropic Stress Tensor in  General Relativity

Authors: D. D. Pawar; V. R. Patil; S. N. Bayaskar
Journal: ISRN Mathematical Physics
 (2012)

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## Abstract

This paper deals with the cosmological models for the static spherically symmetric spacetime for perfect fluid with anisotropic stress energy tensor in general relativity by introducing the generating functionsg(r) and w(r) and also discussing their physical and geometric properties.


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## Body

## 1. Introduction

The study of anisotropic fluid sphere and gravitational collapse problem is important in relativistic astrophysics. Ruderman [1] investigated relativistic stellar model and shows that the stellar matter may be anisotropic at very high density ranges. Anisotropy in fluid pressure could be introduced by the existences of solid core by the presence of type 3A superfluid. Rago [2] presented the procedure to obtain the solution of the field equations by using two arbitrary functions in Einstein general relativity where two arbitrary functions are introduced: the anisotropic function that measures the degree of anisotropy and a generating function. These functions determine the relevant physical variables as well as metric coefficients. Kandalkar and Khadekar [3] have obtained analytical solutions for anisotropic matter distribution in the context of bimetric theory of gravitation.  The dynamical equations governing the gravitational nonadiabatic collapse of a shear-free spherical distribution of anisotropic matter in the presence of charge have been studied by Tikekar and Patel [4]. According to Ruderman [1] and Canuto [5], the pressure in the various gravitational collapse of spherical distribution consisting of super dense matter distribution, may not be isotropic throughout for such stars; the core region may be anisotropic. Gair [6] obtained the spherical universes with anisotropic pressure. Thomas and Ratanpal [7] studied various aspects of gravitational collapse by using analytical and numerical methods by considering the gravitational collapse for spherical distributions, consisting of superdense matter distribution. In the last few years there has been increasing interest in the interior solutions of Einstein field equations corresponding to fluid distributions with anisotropic pressures Letelier [8], Maharaj and Maartens [9], Bondi [10], Coley and Tupper [11], and Singh et al. [12]. The matter distribution is adequately described by perfect fluid due to the large-scale distribution of galaxies in our universe. Hence a relativistic treatment of the problem requires the consideration of material distribution other than the perfect fluid.In this paper, we have obtained cosmological models for static spherically symmetric spacetime with anisotropic stress energy tensor by introducing two generating functionsg(r) and w(r)  and also discussed their physical properties.


## 2. Field Equations

Consider the static spherically symmetric space-time:(2.1)ds2=eγdt2-eλdr2-r2(dθ2+sin2θdϕ2),
where λ and γ being the function of  r alone.The energy momentum tensor for perfect fluid with anisotropic stress energy with heat flux is given by(2.2)Tij=(p+ρ)uiuj-pgij+πij+qiuj+qjui,
where ρ,p,qi,andui denote the matter density, fluid pressure, heat conduction vector orthogonal to ui,  and components of unit time-like flow vector field of matter, respectively, And the anisotropic stress energy tensor πij is given by
(2.3)πij=3S[cicj-13(uiuj-gij)],
where S=S(r) symbolizes the magnitude of the anisotropic stress tensor and the radial vector ci  is obtained as
(2.4)ci=(-e-λ/2,0,0,0).
By choosing the commoving system,  uiuj=1 yields
(2.5)ui=(0,0,0,e-γ/2).
The energy momentum tensor (2.2) with (2.3) has the following nonvanishing components:
(2.6)T11=-(p+2S3),T22=T33=-(p-S3),T44=ρ.
The pressure along radial direction,
(2.7)pr=(p+2S3),
is different from the pressure along the tangential direction,(2.8)p⊥=(p-S3).
Using (2.7) and (2.8), the magnitude of anisotropic stress tensor is(2.9)S=(pr-p⊥3).
The Einstein field equations for space-time (2.1) with (2.6) yield
(2.10)e-λ(γ′r+1r2)-1r2=8πpr,(2.11)e-λ(γ′′2-λ′γ′4+γ′24-(λ′-γ′)2r)=8πp⊥,(2.12)e-λ(-λ′r+1r2)-1r2=-8πρ,
where the prime over the letters indicates the derivative with respect to r.The consequence of conservation of energy momentum tensorTi;jj=0  leads to
(2.13)dprdr=-(pr+ρ)γ′2+2r(p⊥-pr).
From (2.12),
(2.14)e-λ=1-2m(r)r,
where m(r)=∫4πρr2dr = mass function.Again, from (2.13) we obtain
(2.15)γ′=4r(p⊥-pr)(pr+ρ)-2pr′(pr+ρ).
Using (2.14), (2.15), and then (2.10) yields
(2.16)(8πprr2+1)=[1-2m(r)r][4(p⊥-pr)(pr+ρ)-2rpr′(pr+ρ)+1].
Now, we define generating function g(r) and also introduce the anisotropic function w(r), respectively, as
(2.17)g(r)=1-2m(r)/r(8πprr2+1),(2.18)w(r)=4(pr-p⊥)(pr+ρ)g(r).
From (2.17) and (2.18), we can obtain ρ,pr,p⊥,S, and the metric potentials eλ,eγ are as follows.Using (2.17) and (2.18) in (2.16),
(2.19)(1-g+w)8π(pr+ρ)=-16πrpr′g.
Differentiating (2.14) yields
(2.20)e-λλ′=2m′(r)r-2m(r)r2.
Adding  8πpr on both sides of (2.12) and using (2.20) and (2.17), we obtain
(2.21)8π(pr+ρ)=2m′(r)r2-2m(r)r3+8πprr2+1r2(1-g).
Differentiating (2.17) yields
(2.22)2m′(r)r=-(8πprr2+1)g′-16πprrg-8πpr′r2g+2m(r)r2.
On simplifying (2.21) and (2.22), we get
(2.23)8πpr′+(1-3g-rg′)(1-g+w)rg(1+g-w)8πpr+(1-g-rg′)(1-g+w)r3g(1+g-w)=0.
Equation (2.23) is linear differential equation in pr. We obtain its solution as
(2.24)8πpr=e-∫B(r)dr[α0+∫C(r)[e∫B(r)dr]dr],
where α0 is constant of integration and B(r),  C(r) are, (2.25)B(r)=(1-3g-rg′)(1-g+w)rg(1+g-w),C(r)=(1-g-rg′)(1-g+w)r3g(1+g-w).
Equation (2.14) yields
(2.26)m′(r)r=4πρr.
Putting this value in (2.22) and using (2.17), we obtain
(2.27)8πρ=(1-g)1r2-8π(3pr+rpr′)g-(8πpr+1r2)rg′,
which is the expression for effective density ρ.Equation (2.18) yields,
(2.28)p⊥=pr-w4g(pr+ρ).
From (2.14) and (2.17), we have
(2.29)e-λ=(8πprr2+1)g.
From (2.10) and (2.29),
(2.30)eγ=A2re∫1/rgdr,
where A is constant of integration.By using (2.28) and (2.29) the space-time (2.1) becomes
(2.31)ds2=A2re∫(1/rg)drdt2-[1-2m(r)r]-1dr2-r2(dθ2+sin2θdϕ2).
The cosmological model (2.31) is physically meaningful with (2.9), (2.24), (2.25), and (2.27).Here we consider the following three cases.Case 1.
We define the generating function from (2.17) and (2.18) as
(2.32)g(r)=1-αr2,(2.33)w(r)=-αr2,
Whare   α is a constant such that  g-w=1≠0,  and this choice should lead to a physically reasonable model  since the function g(r)~1 as r→0 that implies the Minkowskian space via (2.29),  then the (2.25) yields,
(2.34)B(r)=0,C(r)=0.
Equation (2.24) and hence (2.7) yield
(2.35)pr=α08π⟹p=α08π-2S3.
If the constantα0=0,  then  pr=0.
Hence from (2.25),
(2.36)ρ=3α8π.
Also from (2.27) and (2.35) we obtain
(2.37)S=18π3[α0-3α2r24(1-αr2)].
Using (2.32) and (2.33), (2.29) and (2.30) give
(2.38)e-λ=(1-αr2)(α0r2+1),eγ=[A2c1+A2reℏ],
where ℏ=α/2tan-1αr.
Using (2.38), the cosmological model for the space-time (2.1) is,
(2.39)ds2=[A2c1+A2reℏ]dt2-[(1-αr2)(α0r2+1)]-1dr2-r2(dθ2+sin2θdϕ2).Case 2.
We choose the generating function as(2.40)g(r)=β,w(r)=constant.
From (2.25),
(2.41)B(r)=Dr,C(r)=(1-β+w)(1-β)r3β(1+β-w),
where D=((1-3β)(1-β+w))/(β(1+β-w)).
From (2.40) we obtain
(2.42)pr=18π[α0rD+Vr2].
From (2.7) we get
(2.43)p=18π[α0rD+Vr2]-2S3,
where V=((1-β+w)(1-β))/(β(1+β-w)(D-2)).
From (2.25),
(2.44)8πρ=(1-β-βV)1r2+α0β(D-3)r-D,(2.45)S=1323(wπβ){(V+1)(1-β)r2+α0(1+βD-3β)rD}.
The metric potentials in (2.29) and (2.30) become
(2.46)e-λ=(βα0r-D+2+βV+β),(2.47)eγ=A2r(1/β-1).
The space-time (2.1) can be written as
(2.48)ds2=[A2r(1/β-1)]dt2-[(βα0r-D+2+βV+β)]-1dr2-r2(dθ2+sin2θdϕ2).Case 3.
Whenw(r)=0, then (2.18) gives
(2.49)pr=p⊥=p⟹S=0.
On (2.10) and (2.11), we have
(2.50)e-λ(γ′′2-λ′γ′4+γ′24-λ′γ′2r-1r2)+1r2=0.
With (2.13), (2.49), and (2.50), we obtain
(2.51)dpdr=-(p+ρ)γ′2,⟹dp(p+ρ)=-dγ′2.
On integrating, we get
(2.52)8π(p+ρ)=c2e-γ/2,
where c2=8πc1, c1 is the constant of integration.
Subtracting (2.12) from (2.10), we get
(2.53)8π(p+ρ)=e-λ(γ′+λ′r).
Equations From (2.52) and (2.53) yield
(2.54)e-λ(γ′+λ′r)=c2e-γ/2.
On differentiating and simplifying (2.12), we get
(2.55)e-λ=1-r2R2,
where  1/R2=8πρ/3.
Using (2.54) and (2.55) we have
(2.56)eγ/2=A′-B(1-r2R2),
where A′=c3=(c2/2)R2  and  B=c4R, c3andc4is the constant of integration.
Thus the space-time (2.1) becomes
(2.57)ds2=(1-r2R2)-1dt2-[A′-B(1-r2R2)]2dr2-r2(dθ2+sin2θdϕ2).
Equation (2.57) perfectly matches with Schwarzschild interior solution with
(2.58)8πp={3B(1-r2/R2)-A′}R2{A′-B(1-r2/R2)},8πρ={c2R2+A′-3B(1-r2/R2)-A′}R2{A′-B(1-r2/R2)}.


## 3. Discussion

The cosmological model (2.31) is physically meaningful with radial pressure (pr), tangential pressure (p⊥), and energy density (ρ) being given by (2.24), (2.28), and (2.27) respectively. The model has initial singularity at r=0.Here we discuss the following three cases.In Case1, we consider generating function and anisotropic function as defined in (2.32) and (2.33) such that g-w=1≠0 and the radial pressure (pr) and energy density (ρ) become constant.In Case2, as r→0 and the pressure (p), energy density (ρ),  and stress tensor S all are infinite, the model starts with big bang. As r→∞, p=ρ=0, the model (2.48) represents a vacuum model.While in Case3,  we consider w(r)=0⇒S=0this gives; pr=p⊥ which implies that the cosmological model (2.57) is isotropic with pressure (p) density (ρ) given by (2.58), and our result perfectly matches with Schwarzschild interior solution.


## 4. Conclusion

We have investigated the spherically symmetric cosmological model for perfect fluid with anisotropic stress tensor in general relativity. Here we discuss the three different cases in which the last case forw(r)=0 matches with the Schwarzschild interior solution.



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*Source: 965164-2012-08-09.xml*