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ISRN Mathematical Physics
VolumeΒ 2012Β (2012), Article IDΒ 965164, 10 pages
http://dx.doi.org/10.5402/2012/965164
Research Article

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

1Department of Mathematics, Government Vidarbha Institute of Science and Humanities, Amravati 444404, India
2Department of Mathematics, Arts, Science and Commerce College, Chikhaldara 444807, India
3Department of Mathematics, Adarsha Science, J. B. Art and Birla Commerce College, Dhamangaon Rly 444709, India

Received 23 April 2012; Accepted 1 July 2012

Academic Editors: M.Β Rasetti and W.-H.Β Steeb

Copyright Β© 2012 D. D. Pawar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 functions 𝑔(π‘Ÿ) and 𝑀(π‘Ÿ) 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(π‘Ÿ) and 𝑀(π‘Ÿ)  and also discussed their physical properties.

2. Field Equations

Consider the static spherically symmetric space-time: 𝑑𝑠2=𝑒𝛾𝑑𝑑2βˆ’π‘’πœ†π‘‘π‘Ÿ2βˆ’π‘Ÿ2ξ€·π‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€Έ,(2.1) where πœ† and 𝛾 being the function of π‘Ÿ alone.

The energy momentum tensor for perfect fluid with anisotropic stress energy with heat flux is given by 𝑇𝑖𝑗=(𝑝+𝜌)π‘’π‘–π‘’π‘—βˆ’π‘π‘”π‘–π‘—+πœ‹π‘–π‘—+π‘žπ‘–π‘’π‘—+π‘žπ‘—π‘’π‘–,(2.2) where 𝜌,𝑝,π‘žπ‘–,and𝑒𝑖 denote the matter density, fluid pressure, heat conduction vector orthogonal to 𝑒𝑖, and components of unit time-like flow vector field of matter, respectively, And the anisotropic stress energy tensor πœ‹π‘–π‘— is given by πœ‹π‘–π‘—=βˆšξ‚ƒπ‘3π‘†π‘–π‘π‘—βˆ’13ξ€·π‘’π‘–π‘’π‘—βˆ’π‘”π‘–π‘—ξ€Έξ‚„,(2.3) where 𝑆=𝑆(π‘Ÿ) symbolizes the magnitude of the anisotropic stress tensor and the radial vector 𝑐𝑖 is obtained as 𝑐𝑖=ξ€·βˆ’π‘’βˆ’πœ†/2ξ€Έ.,0,0,0(2.4) By choosing the commoving system, 𝑒𝑖𝑒𝑗=1 yields 𝑒𝑖=ξ€·0,0,0,π‘’βˆ’π›Ύ/2ξ€Έ.(2.5) The energy momentum tensor (2.2) with (2.3) has the following nonvanishing components: 𝑇11=βˆ’π‘+2π‘†βˆš3ξƒͺ,𝑇22=𝑇33𝑆=βˆ’π‘βˆ’βˆš3ξƒͺ,𝑇44=𝜌.(2.6) The pressure along radial direction, π‘π‘Ÿ=𝑝+2π‘†βˆš3ξƒͺ,(2.7) is different from the pressure along the tangential direction,π‘βŸ‚=ξƒ©π‘†π‘βˆ’βˆš3ξƒͺ.(2.8) Using (2.7) and (2.8), the magnitude of anisotropic stress tensor is𝑝𝑆=π‘Ÿβˆ’π‘βŸ‚βˆš3ξƒͺ.(2.9) The Einstein field equations for space-time (2.1) with (2.6) yield π‘’βˆ’πœ†ξ‚΅π›Ύβ€²π‘Ÿ+1π‘Ÿ2ξ‚Άβˆ’1π‘Ÿ2=8πœ‹π‘π‘Ÿπ‘’,(2.10)βˆ’πœ†ξ‚΅π›Ύξ…žξ…ž2βˆ’πœ†β€²π›Ύβ€²4+𝛾′24βˆ’(πœ†β€²βˆ’π›Ύβ€²)ξ‚Ά2π‘Ÿ=8πœ‹π‘βŸ‚,𝑒(2.11)βˆ’πœ†ξ‚€βˆ’πœ†β€²π‘Ÿ+1π‘Ÿ2ξ‚βˆ’1π‘Ÿ2=βˆ’8πœ‹πœŒ,(2.12) where the prime over the letters indicates the derivative with respect to π‘Ÿ.

The consequence of conservation of energy momentum tensor 𝑇𝑗𝑖;𝑗=0 leads to π‘‘π‘π‘Ÿξ€·π‘π‘‘π‘Ÿ=βˆ’π‘Ÿξ€Έπ›Ύ+πœŒξ…ž2+2π‘Ÿξ€·π‘βŸ‚βˆ’π‘π‘Ÿξ€Έ.(2.13) From (2.12), π‘’βˆ’πœ†=1βˆ’2π‘š(π‘Ÿ)π‘Ÿ,(2.14) where βˆ«π‘š(π‘Ÿ)=4πœ‹πœŒπ‘Ÿ2π‘‘π‘Ÿ = mass function.

Again, from (2.13) we obtain π›Ύξ…ž=4π‘Ÿξ€·π‘βŸ‚βˆ’π‘π‘Ÿξ€Έξ€·π‘π‘Ÿξ€Έβˆ’+𝜌2π‘ξ…žπ‘Ÿξ€·π‘π‘Ÿξ€Έ+𝜌.(2.15) Using (2.14), (2.15), and then (2.10) yields ξ€·8πœ‹π‘π‘Ÿπ‘Ÿ2ξ€Έ=ξ‚Έ+11βˆ’2π‘š(π‘Ÿ)π‘Ÿξ‚Ήξƒ¬4ξ€·π‘βŸ‚βˆ’π‘π‘Ÿξ€Έξ€·π‘π‘Ÿξ€Έβˆ’+𝜌2π‘Ÿπ‘ξ…žπ‘Ÿξ€·π‘π‘Ÿξ€Έξƒ­+𝜌+1.(2.16) Now, we define generating function 𝑔(π‘Ÿ) and also introduce the anisotropic function 𝑀(π‘Ÿ), respectively, as 𝑔(π‘Ÿ)=1βˆ’2π‘š(π‘Ÿ)/π‘Ÿξ€·8πœ‹π‘π‘Ÿπ‘Ÿ2ξ€Έ,4𝑝+1(2.17)𝑀(π‘Ÿ)=π‘Ÿβˆ’π‘βŸ‚ξ€Έξ€·π‘π‘Ÿξ€Έ+πœŒπ‘”(π‘Ÿ).(2.18) From (2.17) and (2.18), we can obtain 𝜌,π‘π‘Ÿ,π‘βŸ‚,𝑆, and the metric potentials π‘’πœ†,𝑒𝛾 are as follows.

Using (2.17) and (2.18) in (2.16), 𝑝(1βˆ’π‘”+𝑀)8πœ‹π‘Ÿξ€Έ+𝜌=βˆ’16πœ‹π‘Ÿπ‘ξ…žπ‘Ÿπ‘”.(2.19) Differentiating (2.14) yields π‘’βˆ’πœ†πœ†ξ…ž=2π‘šξ…ž(π‘Ÿ)π‘Ÿβˆ’2π‘š(π‘Ÿ)π‘Ÿ2.(2.20) Adding 8πœ‹π‘π‘Ÿ on both sides of (2.12) and using (2.20) and (2.17), we obtain 𝑝8πœ‹π‘Ÿξ€Έ=+𝜌2π‘šβ€²(π‘Ÿ)π‘Ÿ2βˆ’2π‘š(π‘Ÿ)π‘Ÿ3+8πœ‹π‘π‘Ÿπ‘Ÿ2+1π‘Ÿ2(1βˆ’π‘”).(2.21) Differentiating (2.17) yields 2π‘šβ€²(π‘Ÿ)π‘Ÿξ€·=βˆ’8πœ‹π‘π‘Ÿπ‘Ÿ2𝑔+1ξ…žβˆ’16πœ‹π‘π‘Ÿπ‘Ÿπ‘”βˆ’8πœ‹π‘ξ…žπ‘Ÿπ‘Ÿ2𝑔+2π‘š(π‘Ÿ)π‘Ÿ2.(2.22) On simplifying (2.21) and (2.22), we get 8πœ‹π‘ξ…žπ‘Ÿ+(1βˆ’3π‘”βˆ’π‘Ÿπ‘”β€²)(1βˆ’π‘”+𝑀)π‘Ÿπ‘”(1+π‘”βˆ’π‘€)8πœ‹π‘π‘Ÿ+(1βˆ’π‘”βˆ’π‘Ÿπ‘”β€²)(1βˆ’π‘”+𝑀)π‘Ÿ3𝑔(1+π‘”βˆ’π‘€)=0.(2.23) Equation (2.23) is linear differential equation in π‘π‘Ÿ. We obtain its solution as 8πœ‹π‘π‘Ÿ=π‘’βˆ’βˆ«π΅(π‘Ÿ)π‘‘π‘Ÿξ‚Έπ›Ό0+ξ€œξ‚ƒπ‘’πΆ(π‘Ÿ)∫𝐡(π‘Ÿ)π‘‘π‘Ÿξ‚„ξ‚Ήπ‘‘π‘Ÿ,(2.24) where 𝛼0 is constant of integration and 𝐡(π‘Ÿ),  𝐢(π‘Ÿ) are, 𝐡(π‘Ÿ)=(1βˆ’3π‘”βˆ’π‘Ÿπ‘”β€²)(1βˆ’π‘”+𝑀)π‘Ÿπ‘”(1+π‘”βˆ’π‘€),𝐢(π‘Ÿ)=(1βˆ’π‘”βˆ’π‘Ÿπ‘”β€²)(1βˆ’π‘”+𝑀)π‘Ÿ3𝑔.(1+π‘”βˆ’π‘€)(2.25) Equation (2.14) yields π‘šβ€²(π‘Ÿ)π‘Ÿ=4πœ‹πœŒπ‘Ÿ.(2.26) Putting this value in (2.22) and using (2.17), we obtain 18πœ‹πœŒ=(1βˆ’π‘”)π‘Ÿ2ξ€·βˆ’8πœ‹3π‘π‘Ÿ+π‘Ÿπ‘ξ…žπ‘Ÿξ€Έξ‚€π‘”βˆ’8πœ‹π‘π‘Ÿ+1π‘Ÿ2ξ‚π‘Ÿπ‘”ξ…ž,(2.27) which is the expression for effective density 𝜌.

Equation (2.18) yields, π‘βŸ‚=π‘π‘Ÿβˆ’π‘€ξ€·π‘4π‘”π‘Ÿξ€Έ.+𝜌(2.28) From (2.14) and (2.17), we have π‘’βˆ’πœ†=ξ€·8πœ‹π‘π‘Ÿπ‘Ÿ2ξ€Έ+1𝑔.(2.29) From (2.10) and (2.29), 𝑒𝛾=𝐴2π‘Ÿπ‘’βˆ«1/π‘Ÿπ‘”π‘‘π‘Ÿ,(2.30) where 𝐴 is constant of integration.

By using (2.28) and (2.29) the space-time (2.1) becomes 𝑑𝑠2=𝐴2π‘Ÿπ‘’βˆ«(1/π‘Ÿπ‘”)π‘‘π‘Ÿπ‘‘π‘‘2βˆ’ξ‚Έ1βˆ’2π‘š(π‘Ÿ)π‘Ÿξ‚Ήβˆ’1π‘‘π‘Ÿ2βˆ’π‘Ÿ2ξ€·π‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€Έ.(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(π‘Ÿ)=1βˆ’π›Όπ‘Ÿ2,(2.32)𝑀(π‘Ÿ)=βˆ’π›Όπ‘Ÿ2,(2.33) Whare 𝛼 is a constant such that π‘”βˆ’π‘€=1β‰ 0, and this choice should lead to a physically reasonable model since the function 𝑔(π‘Ÿ)∼1 as π‘Ÿβ†’0 that implies the Minkowskian space via (2.29), then the (2.25) yields, 𝐡(π‘Ÿ)=0,𝐢(π‘Ÿ)=0.(2.34) Equation (2.24) and hence (2.7) yield π‘π‘Ÿ=𝛼0𝛼8πœ‹βŸΉπ‘=0βˆ’8πœ‹2π‘†βˆš3.(2.35) If the constant𝛼0=0, then π‘π‘Ÿ=0.
Hence from (2.25), 𝜌=3𝛼.8πœ‹(2.36) Also from (2.27) and (2.35) we obtain 1𝑆=√8πœ‹3𝛼0βˆ’3𝛼2π‘Ÿ24ξ€·1βˆ’π›Όπ‘Ÿ2ξ€Έξƒ­.(2.37) Using (2.32) and (2.33), (2.29) and (2.30) give π‘’βˆ’πœ†=ξ€·1βˆ’π›Όπ‘Ÿ2𝛼0π‘Ÿ2ξ€Έ,𝑒+1𝛾=𝐴2𝑐1+𝐴2π‘Ÿπ‘’β„ξ‚Ή,(2.38) where βˆšβ„=𝛼/2tanβˆ’1βˆšπ›Όπ‘Ÿ.
Using (2.38), the cosmological model for the space-time (2.1) is, 𝑑𝑠2=𝐴2𝑐1+𝐴2π‘Ÿπ‘’β„ξ‚Ήπ‘‘π‘‘2βˆ’ξ€Ίξ€·1βˆ’π›Όπ‘Ÿ2𝛼0π‘Ÿ2+1ξ€Έξ€»βˆ’1π‘‘π‘Ÿ2βˆ’π‘Ÿ2ξ€·π‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€Έ.(2.39)

Case 2. We choose the generating function as 𝑔(π‘Ÿ)=𝛽,𝑀(π‘Ÿ)=constant.(2.40) From (2.25), 𝐷𝐡(π‘Ÿ)=π‘Ÿ,𝐢(π‘Ÿ)=(1βˆ’π›½+𝑀)(1βˆ’π›½)π‘Ÿ3,𝛽(1+π›½βˆ’π‘€)(2.41) where 𝐷=((1βˆ’3𝛽)(1βˆ’π›½+𝑀))/(𝛽(1+π›½βˆ’π‘€)).
From (2.40) we obtain π‘π‘Ÿ=1𝛼8πœ‹0π‘Ÿπ·+π‘‰π‘Ÿ2ξ‚„.(2.42) From (2.7) we get 1𝑝=𝛼8πœ‹0π‘Ÿπ·+π‘‰π‘Ÿ2ξ‚„βˆ’2π‘†βˆš3,(2.43) where 𝑉=((1βˆ’π›½+𝑀)(1βˆ’π›½))/(𝛽(1+π›½βˆ’π‘€)(π·βˆ’2)).
From (2.25), 18πœ‹πœŒ=(1βˆ’π›½βˆ’π›½π‘‰)π‘Ÿ2+𝛼0𝛽(π·βˆ’3)π‘Ÿβˆ’π·,1(2.44)𝑆=√323𝑀(πœ‹π›½ξ‚Άξ‚»π‘‰+1)(1βˆ’π›½)π‘Ÿ2+𝛼0(1+π›½π·βˆ’3𝛽)π‘Ÿπ·ξ‚Ό.(2.45) The metric potentials in (2.29) and (2.30) become π‘’βˆ’πœ†=𝛽𝛼0π‘Ÿβˆ’π·+2ξ€Έ,𝑒+𝛽𝑉+𝛽(2.46)𝛾=𝐴2π‘Ÿ(1/π›½βˆ’1).(2.47) The space-time (2.1) can be written as 𝑑𝑠2=𝐴2π‘Ÿ(1/π›½βˆ’1)𝑑𝑑2βˆ’ξ€Ίξ€·π›½π›Ό0π‘Ÿβˆ’π·+2+𝛽𝑉+π›½ξ€Έξ€»βˆ’1π‘‘π‘Ÿ2βˆ’π‘Ÿ2ξ€·π‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€Έ.(2.48)

Case 3. When 𝑀(π‘Ÿ)=0, then (2.18) gives π‘π‘Ÿ=π‘βŸ‚=π‘βŸΉπ‘†=0.(2.49) On (2.10) and (2.11), we have π‘’βˆ’πœ†ξ‚΅π›Ύξ…žξ…ž2βˆ’πœ†β€²π›Ύβ€²4+𝛾′24βˆ’πœ†β€²π›Ύβ€²βˆ’12π‘Ÿπ‘Ÿ2ξ‚Ά+1π‘Ÿ2=0.(2.50) With (2.13), (2.49), and (2.50), we obtain π‘‘π‘π›Ύπ‘‘π‘Ÿ=βˆ’(𝑝+𝜌)ξ…ž2,βŸΉπ‘‘π‘(𝑝+𝜌)=βˆ’π‘‘π›Ύξ…ž2.(2.51) On integrating, we get 8πœ‹(𝑝+𝜌)=𝑐2π‘’βˆ’π›Ύ/2,(2.52) where 𝑐2=8πœ‹π‘1, 𝑐1 is the constant of integration.
Subtracting (2.12) from (2.10), we get 8πœ‹(𝑝+𝜌)=π‘’βˆ’πœ†ξ‚΅π›Ύβ€²+πœ†β€²π‘Ÿξ‚Ά.(2.53) Equations From (2.52) and (2.53) yield π‘’βˆ’πœ†ξ‚΅π›Ύβ€²+πœ†β€²π‘Ÿξ‚Ά=𝑐2π‘’βˆ’π›Ύ/2.(2.54) On differentiating and simplifying (2.12), we get π‘’βˆ’πœ†π‘Ÿ=1βˆ’2𝑅2,(2.55) where 1/𝑅2=8πœ‹πœŒ/3.
Using (2.54) and (2.55) we have 𝑒𝛾/2ξƒŽ=π΄β€²βˆ’π΅ξ‚΅π‘Ÿ1βˆ’2𝑅2ξ‚Ά,(2.56) where 𝐴′=𝑐3=(𝑐2/2)𝑅2 and 𝐡=𝑐4𝑅, 𝑐3and𝑐4is the constant of integration.
Thus the space-time (2.1) becomes 𝑑𝑠2=ξ‚΅π‘Ÿ1βˆ’2𝑅2ξ‚Άβˆ’1𝑑𝑑2βˆ’βŽ‘βŽ’βŽ’βŽ£π΄ξ…žξƒŽβˆ’π΅ξ‚΅π‘Ÿ1βˆ’2𝑅2ξ‚ΆβŽ€βŽ₯βŽ₯⎦2π‘‘π‘Ÿ2βˆ’π‘Ÿ2ξ€·π‘‘πœƒ2+sin2πœƒπ‘‘πœ™2ξ€Έ.(2.57) Equation (2.57) perfectly matches with Schwarzschild interior solution with 8πœ‹π‘=3𝐡1βˆ’π‘Ÿ2/𝑅2ξ€Έξ‚Όβˆ’π΄β€²π‘…2ξ‚»ξ”π΄β€²βˆ’π΅ξ€·1βˆ’π‘Ÿ2/𝑅2ξ€Έξ‚Ό,𝑐8πœ‹πœŒ=2𝑅2+π΄β€²βˆ’3𝐡1βˆ’π‘Ÿ2/𝑅2ξ€Έξ‚Όβˆ’π΄β€²π‘…2ξ‚»ξ”π΄β€²βˆ’π΅ξ€·1βˆ’π‘Ÿ2/𝑅2ξ€Έξ‚Ό.(2.58)

3. Discussion

The cosmological model (2.31) is physically meaningful with radial pressure (π‘π‘Ÿ), tangential pressure (π‘βŸ‚), and energy density (𝜌) being given by (2.24), (2.28), and (2.27) respectively. The model has initial singularity at π‘Ÿ=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 π‘”βˆ’π‘€=1β‰ 0 and the radial pressure (π‘π‘Ÿ) and energy density (𝜌) become constant.

In Case 2, as π‘Ÿβ†’0 and the pressure (𝑝), energy density (𝜌), and stress tensor 𝑆 all are infinite, the model starts with big bang. As π‘Ÿβ†’βˆž, 𝑝=𝜌=0, the model (2.48) represents a vacuum model.

While in Case 3, we consider 𝑀(π‘Ÿ)=0⇒𝑆=0this gives; π‘π‘Ÿ=π‘βŸ‚ 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 𝑀(π‘Ÿ)=0 matches with the Schwarzschild interior solution.

Acknowledgement

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

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