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 functions 𝑔(𝑟) and 𝑤(𝑟) and also discussing their physical and geometric properties.

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 functions 𝑔(𝑟) and 𝑤(𝑟)  and also discussed their physical properties.

2. Field Equations

Consider the static spherically symmetric space-time: 𝑑𝑠2=𝑒𝛾𝑑𝑡2−𝑒𝜆𝑑𝑟2−𝑟2𝑑𝜃2+sin2𝜃𝑑𝜙2,(2.1) 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 𝑇𝑖𝑗=(𝑝+𝜌)𝑢𝑖𝑢𝑗−𝑝𝑔𝑖𝑗+𝜋𝑖𝑗+ğ‘žğ‘–ğ‘¢ğ‘—+ğ‘žğ‘—ğ‘¢ğ‘–,(2.2) where 𝜌,𝑝,ğ‘žğ‘–,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 𝜋𝑖𝑗=√𝑐3𝑆𝑖𝑐𝑗−13𝑢𝑖𝑢𝑗−𝑔𝑖𝑗,(2.3) where 𝑆=𝑆(𝑟) symbolizes the magnitude of the anisotropic stress tensor and the radial vector 𝑐𝑖 is obtained as 𝑐𝑖=−𝑒−𝜆/2.,0,0,0(2.4) By choosing the commoving system, 𝑢𝑖𝑢𝑗=1 yields 𝑢𝑖=0,0,0,𝑒−𝛾/2.(2.5) The energy momentum tensor (2.2) with (2.3) has the following nonvanishing components: 𝑇11=−𝑝+2𝑆√3,𝑇22=𝑇33𝑆=−𝑝−√3,𝑇44=𝜌.(2.6) The pressure along radial direction, 𝑝𝑟=𝑝+2𝑆√3,(2.7) is different from the pressure along the tangential direction,𝑝⟂=𝑆𝑝−√3.(2.8) Using (2.7) and (2.8), the magnitude of anisotropic stress tensor is𝑝𝑆=𝑟−𝑝⟂√3.(2.9) The Einstein field equations for space-time (2.1) with (2.6) yield 𝑒−𝜆𝛾′𝑟+1𝑟2−1𝑟2=8𝜋𝑝𝑟𝑒,(2.10)âˆ’ğœ†î‚µğ›¾î…žî…ž2−𝜆′𝛾′4+𝛾′24−(𝜆′−𝛾′)2𝑟=8𝜋𝑝⟂,𝑒(2.11)−𝜆−𝜆′𝑟+1𝑟2−1𝑟2=−8𝜋𝜌,(2.12) where the prime over the letters indicates the derivative with respect to 𝑟.

The consequence of conservation of energy momentum tensor 𝑇𝑗𝑖;𝑗=0 leads to 𝑑𝑝𝑟𝑝𝑑𝑟=−𝑟𝛾+ğœŒî…ž2+2𝑟𝑝⟂−𝑝𝑟.(2.13) From (2.12), 𝑒−𝜆=1−2𝑚(𝑟)𝑟,(2.14) where ∫𝑚(𝑟)=4𝜋𝜌𝑟2𝑑𝑟 = mass function.

Again, from (2.13) we obtain ğ›¾î…ž=4𝑟𝑝⟂−𝑝𝑟𝑝𝑟−+𝜌2ğ‘î…žğ‘Ÿî€·ğ‘ğ‘Ÿî€¸+𝜌.(2.15) Using (2.14), (2.15), and then (2.10) yields 8𝜋𝑝𝑟𝑟2=+11−2𝑚(𝑟)𝑟4𝑝⟂−𝑝𝑟𝑝𝑟−+𝜌2ğ‘Ÿğ‘î…žğ‘Ÿî€·ğ‘ğ‘Ÿî€¸îƒ­+𝜌+1.(2.16) Now, we define generating function 𝑔(𝑟) and also introduce the anisotropic function 𝑤(𝑟), respectively, as 𝑔(𝑟)=1−2𝑚(𝑟)/𝑟8𝜋𝑝𝑟𝑟2,4𝑝+1(2.17)𝑤(𝑟)=𝑟−𝑝⟂𝑝𝑟+𝜌𝑔(𝑟).(2.18) 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), 𝑝(1−𝑔+𝑤)8𝜋𝑟+𝜌=−16ğœ‹ğ‘Ÿğ‘î…žğ‘Ÿğ‘”.(2.19) Differentiating (2.14) yields ğ‘’âˆ’ğœ†ğœ†î…ž=2ğ‘šî…ž(𝑟)𝑟−2𝑚(𝑟)𝑟2.(2.20) Adding 8𝜋𝑝𝑟 on both sides of (2.12) and using (2.20) and (2.17), we obtain 𝑝8𝜋𝑟=+𝜌2𝑚′(𝑟)𝑟2−2𝑚(𝑟)𝑟3+8𝜋𝑝𝑟𝑟2+1𝑟2(1−𝑔).(2.21) Differentiating (2.17) yields 2𝑚′(𝑟)𝑟=−8𝜋𝑝𝑟𝑟2𝑔+1−16𝜋𝑝𝑟𝑟𝑔−8ğœ‹ğ‘î…žğ‘Ÿğ‘Ÿ2𝑔+2𝑚(𝑟)𝑟2.(2.22) On simplifying (2.21) and (2.22), we get 8ğœ‹ğ‘î…žğ‘Ÿ+(1−3𝑔−𝑟𝑔′)(1−𝑔+𝑤)𝑟𝑔(1+𝑔−𝑤)8𝜋𝑝𝑟+(1−𝑔−𝑟𝑔′)(1−𝑔+𝑤)𝑟3𝑔(1+𝑔−𝑤)=0.(2.23) Equation (2.23) is linear differential equation in 𝑝𝑟. We obtain its solution as 8𝜋𝑝𝑟=𝑒−∫𝐵(𝑟)𝑑𝑟𝛼0+𝑒𝐶(𝑟)∫𝐵(𝑟)𝑑𝑟𝑑𝑟,(2.24) where 𝛼0 is constant of integration and 𝐵(𝑟),  𝐶(𝑟) are, 𝐵(𝑟)=(1−3𝑔−𝑟𝑔′)(1−𝑔+𝑤)𝑟𝑔(1+𝑔−𝑤),𝐶(𝑟)=(1−𝑔−𝑟𝑔′)(1−𝑔+𝑤)𝑟3𝑔.(1+𝑔−𝑤)(2.25) Equation (2.14) yields 𝑚′(𝑟)𝑟=4𝜋𝜌𝑟.(2.26) Putting this value in (2.22) and using (2.17), we obtain 18𝜋𝜌=(1−𝑔)𝑟2−8𝜋3𝑝𝑟+ğ‘Ÿğ‘î…žğ‘Ÿî€¸î‚€ğ‘”âˆ’8𝜋𝑝𝑟+1𝑟2î‚ğ‘Ÿğ‘”î…ž,(2.27) which is the expression for effective density 𝜌.

Equation (2.18) yields, 𝑝⟂=𝑝𝑟−𝑤𝑝4𝑔𝑟.+𝜌(2.28) From (2.14) and (2.17), we have 𝑒−𝜆=8𝜋𝑝𝑟𝑟2+1𝑔.(2.29) From (2.10) and (2.29), 𝑒𝛾=𝐴2𝑟𝑒∫1/𝑟𝑔𝑑𝑟,(2.30) where 𝐴 is constant of integration.

By using (2.28) and (2.29) the space-time (2.1) becomes 𝑑𝑠2=𝐴2𝑟𝑒∫(1/𝑟𝑔)𝑑𝑟𝑑𝑡2−1−2𝑚(𝑟)𝑟−1𝑑𝑟2−𝑟2𝑑𝜃2+sin2𝜃𝑑𝜙2.(2.31) 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 𝑔(𝑟)=1−𝛼𝑟2,(2.32)𝑤(𝑟)=−𝛼𝑟2,(2.33) 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, 𝐵(𝑟)=0,𝐶(𝑟)=0.(2.34) Equation (2.24) and hence (2.7) yield 𝑝𝑟=𝛼0𝛼8𝜋⟹𝑝=0−8𝜋2𝑆√3.(2.35) If the constant𝛼0=0, then 𝑝𝑟=0.
Hence from (2.25), 𝜌=3𝛼.8𝜋(2.36) Also from (2.27) and (2.35) we obtain 1𝑆=√8𝜋3𝛼0−3𝛼2𝑟241−𝛼𝑟2.(2.37) Using (2.32) and (2.33), (2.29) and (2.30) give 𝑒−𝜆=1−𝛼𝑟2𝛼0𝑟2,𝑒+1𝛾=𝐴2𝑐1+𝐴2𝑟𝑒ℏ,(2.38) where √ℏ=𝛼/2tan−1√𝛼𝑟.
Using (2.38), the cosmological model for the space-time (2.1) is, 𝑑𝑠2=𝐴2𝑐1+𝐴2𝑟𝑒ℏ𝑑𝑡2−1−𝛼𝑟2𝛼0𝑟2+1−1𝑑𝑟2−𝑟2𝑑𝜃2+sin2𝜃𝑑𝜙2.(2.39)

Case 2. We choose the generating function as 𝑔(𝑟)=𝛽,𝑤(𝑟)=constant.(2.40) From (2.25), 𝐷𝐵(𝑟)=𝑟,𝐶(𝑟)=(1−𝛽+𝑤)(1−𝛽)𝑟3,𝛽(1+𝛽−𝑤)(2.41) where 𝐷=((1−3𝛽)(1−𝛽+𝑤))/(𝛽(1+𝛽−𝑤)).
From (2.40) we obtain 𝑝𝑟=1𝛼8𝜋0𝑟𝐷+𝑉𝑟2.(2.42) From (2.7) we get 1𝑝=𝛼8𝜋0𝑟𝐷+𝑉𝑟2−2𝑆√3,(2.43) where 𝑉=((1−𝛽+𝑤)(1−𝛽))/(𝛽(1+𝛽−𝑤)(𝐷−2)).
From (2.25), 18𝜋𝜌=(1−𝛽−𝛽𝑉)𝑟2+𝛼0𝛽(𝐷−3)𝑟−𝐷,1(2.44)𝑆=√323𝑤(𝜋𝛽𝑉+1)(1−𝛽)𝑟2+𝛼0(1+𝛽𝐷−3𝛽)𝑟𝐷.(2.45) The metric potentials in (2.29) and (2.30) become 𝑒−𝜆=𝛽𝛼0𝑟−𝐷+2,𝑒+𝛽𝑉+𝛽(2.46)𝛾=𝐴2𝑟(1/𝛽−1).(2.47) The space-time (2.1) can be written as 𝑑𝑠2=𝐴2𝑟(1/𝛽−1)𝑑𝑡2−𝛽𝛼0𝑟−𝐷+2+𝛽𝑉+𝛽−1𝑑𝑟2−𝑟2𝑑𝜃2+sin2𝜃𝑑𝜙2.(2.48)

Case 3. When 𝑤(𝑟)=0, then (2.18) gives 𝑝𝑟=𝑝⟂=𝑝⟹𝑆=0.(2.49) On (2.10) and (2.11), we have ğ‘’âˆ’ğœ†î‚µğ›¾î…žî…ž2−𝜆′𝛾′4+𝛾′24−𝜆′𝛾′−12𝑟𝑟2+1𝑟2=0.(2.50) With (2.13), (2.49), and (2.50), we obtain 𝑑𝑝𝛾𝑑𝑟=−(𝑝+𝜌)2,⟹𝑑𝑝(𝑝+𝜌)=âˆ’ğ‘‘ğ›¾î…ž2.(2.51) On integrating, we get 8𝜋(𝑝+𝜌)=𝑐2𝑒−𝛾/2,(2.52) where 𝑐2=8𝜋𝑐1, 𝑐1 is the constant of integration.
Subtracting (2.12) from (2.10), we get 8𝜋(𝑝+𝜌)=𝑒−𝜆𝛾′+𝜆′𝑟.(2.53) Equations From (2.52) and (2.53) yield 𝑒−𝜆𝛾′+𝜆′𝑟=𝑐2𝑒−𝛾/2.(2.54) On differentiating and simplifying (2.12), we get 𝑒−𝜆𝑟=1−2𝑅2,(2.55) where 1/𝑅2=8𝜋𝜌/3.
Using (2.54) and (2.55) we have 𝑒𝛾/2=𝐴′−𝐵𝑟1−2𝑅2,(2.56) where 𝐴′=𝑐3=(𝑐2/2)𝑅2 and 𝐵=𝑐4𝑅, 𝑐3and𝑐4is the constant of integration.
Thus the space-time (2.1) becomes 𝑑𝑠2=𝑟1−2𝑅2−1𝑑𝑡2âˆ’âŽ¡âŽ¢âŽ¢âŽ£ğ´î…žîƒŽâˆ’ğµî‚µğ‘Ÿ1−2𝑅2⎤⎥⎥⎦2𝑑𝑟2−𝑟2𝑑𝜃2+sin2𝜃𝑑𝜙2.(2.57) Equation (2.57) perfectly matches with Schwarzschild interior solution with 8𝜋𝑝=3𝐵1−𝑟2/𝑅2−𝐴′𝑅2𝐴′−𝐵1−𝑟2/𝑅2,𝑐8𝜋𝜌=2𝑅2+𝐴′−3𝐵1−𝑟2/𝑅2−𝐴′𝑅2𝐴′−𝐵1−𝑟2/𝑅2.(2.58)

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 Case 1, 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 Case 2, as 𝑟→0 and the pressure (𝑝), energy density (𝜌), and stress tensor 𝑆 all are infinite, the model starts with big bang. As ğ‘Ÿâ†’âˆž, 𝑝=𝜌=0, the model (2.48) represents a vacuum model.

While in Case 3, 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 for 𝑤(𝑟)=0 matches with the Schwarzschild interior solution.


The authors are grateful to the referee for his valuable comments and suggestions.