Some Investigation LRS Bianchi Type- I Model in General Relativity

DOI : 10.17577/IJERTV3IS041077

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Some Investigation LRS Bianchi Type- I Model in General Relativity

S. D. Deo

Head, Department of Mathematics

  1. S. Science & Arts College, Bhadrawati, Dist.Chandrapur-

    442902(M.S) India

    1. S. Punwatkar

      Department of Mathematics Datta Meghe Institute of Engineering, Technology & Research, Sawangi (Meghe), Wardha (M.S), India

      1. M. Patil

        Head, Department of Mathematics Shri Shivaji Science College, Amravati-444603 (M.S) India

        Abstract-Locally Rotationally symmetric (LRS) Bianchi type-I cosmological model is studied in the context of general theory of relativity with the matter cosmic cloud string and electromagnetic field respectively. Further some physical and kinematical properties are discussed.

        Keyword- LRS Bianchi Model, cosmic cloud string, electromagnetic field, general relativity.

        MSC2010 classification- 83C05, 83C15. PACS- 98.80.-k, 95.30.-k, 98.80.Cq, 04.20.-q.


          Space-times admitting a three parameter group of automorphisms are important in discussing the cosmological models. The case where the group is simply transitive over the 3-dimensional, constant time subspace is particularly useful for two reasons. First, Bianchi has shown that there are only nine distinct sets of structure constants for groups of this type. Therefore, we can use algebra to classify the homogeneous Space -times. The second reason for the importance of Bianchi type Space – times is the simplicity of the field equations.

          When we study the Bianchi type models, we observe that

          It is well known that the exact solutions of general theory of relativity for homogeneous space times belong to either Bianchi types or KantowskiSachs. For simplification and description of the large scale behavior of the actual universe, locally rotationally symmetric (Henceforth referred as to LRS) Bianchi-I space-time has been widely studied. In order to study problems like the formation of galaxies of the process of homogenization and isotropization of the universe, it is necessary to study problems relating to inhomogeneous and anisotropic space- time.Mazumder [1] has obtained cosmological solutions for LRS Bianchi-I space-time filled with a perfect fluid with arbitrary cosmic scale functions and studied kinematical properties of the particular form of the solution. Mohanti [2,3] also obtained some solutions for the same field equations by using solution generation technique with the matter perfect fluid. Banerjee [4] studied Bianchi type I cosmologies with cosmic strings. Numbers of researches [5-17] have studied Bianchi Type- I cosmological models in general relativity.

        2. Metric and Field Equation

          The LRS Bianchi type I line element is given by

          the models contain isotropic special cases and they permit arbitrarily small anisotropic levels at some instant of

          ds2 dt2 A2dt2 B2 (dy2 dz2 )


          cosmic times.

          Bianchi type cosmological models are important in the sense that these models are homogeneous and anisotropic, from which the process of isotropization of the universe is studied through the passage of time. Moreover, from the theoretical point of view, anisotropic universe has a greater

          Where, A and B are the scale factor (metric potential) and function of the cosmic time t only (non-static case).

          The Einstein field equation in the general relativity is given by

          i i


          generality than isotropic models. The simplicity of the field equations made Bianchi space time useful in constructing models of spatially homogeneous and anisotropic

          R j



          Rg j 8 kT j



          Hence, these models are to be known as suitable models of

          Where, R j

          is known as Ricci Tensor


          our universe. Therefore, study of Bianchi type models create much more interest.

          Now, we discuss LRS Bianchi type – I cosmological space

          – times in general relativity. The present day observations indicates that the universe at large scale is homogeneous and isotropic, and we have accelerating phase of universe

          And R gij R is the Ricci Scalar



          T j is energy momentum tensor for the matter.

          Einsteins field equation (2.2) for line element (2.1) lead to

          A B


          2 A

          R11 A A 2 B


          Where, A = t ,

          A t2


          From equation (2.11) and (2.12) we get

          22 B B


          A B


          B 2

          • B


            =0 (2.14)

            Thus, cosmic cloud string does not exist.

            33 B B


            A B


            B 2

          • B


          Hence Vacuum solution are



          B 2


          2 B 0



          R44 A 2 B

          Case I: – For Massive cloud string


          A B A B


          A B AB


          Here we consider the energy momentum tensor for a cloud massive string is given by

          j j j

          B 2

          2 A B 0


          Ti viv

          • xi x


          B AB

          Where, is the rest energy density for a cloud of string with particles attached along the extension.

          Solving the equation (2.15) to (2.17) we obtain

          Thus, p


          A e(kt k1 )

          e(kt k2 )


          Where, p is particle energy and is the tension density of the string.

          B k k3


          vi is the four vector representing the velocity of cloud of particles and xi is the four vector representing the direction anisotropy, i.e. z- direction.

          Where, ki s are constants of integration.

          The corresponding vacuum cosmological model can be written as

          Where vi

          and xi

          satisfy condition

          (ktk2 )

          2 2(ktk ) e2 2

          ds dt e 1 dx k

          (dy2 dz2 )



          k 3

          vv j 1 , x x j 1 and v xi 0


          i i i

          The field equation (2.2) together with (2.7) for the line element (2.1) subsequently lead to the following system of equation

          Case II: – For electromagnetic field

          Here, the energy momentum tensor for electromagnetic field is defined as

          B 2


          E j F F jr 1 F F ab g j




          2 0



          i ir

          4 ab i



          Where, E j is electromagnetic energy tensor and

          j is



          A B

          electromagnetic field tensor.


          A B AB


          In comoving coordinate system, the magnetic field is taken



          A B

          along Z-axis so that the non-vanishing components of

          8 k A B AB


          electromagnetic field tensor Fij is F12 .

          The first set of Maxwells equation

          B 2


          2 A B 8 k AB


          Fij,k Fjk ,i Fki, j 0

          Lead to


          F12 = constant [Here F14 0 F24 F34 ]


          The non-vanishing component of shear tensor ij



          Now from equation (2.21) we obtain the components of the electromagnetic field

          ij ui; j u j;i



          ij uiu j ) are obtained as

          1 2 3 4 2

          AA A2

          E1 E2 E3 E4 2A2 B2


          11 3

          Uing comoving coordinate system, the field equations for the matric (2.1) can be written as

          BB B2 22 33 3

          B 2

          B 2



          2 4 k

          B A2 B2



          Thus shear scale is obtained as



          A B 2

          1 A 2

          B 2

          4 k

          A B AB A2 B2



          2 A2

          2 2





          A B 2

          4 k

          A B AB A2 B2


          1. CONCLUSION

            We have investigated non-static LRS Bianchi type I

            B 2

            2 A B


            2 2

            4 k


            Cosmological models with the matter cosmic cloud string and electromagnetic field respectively and further observed


            AB A B

            that in this model cosmic cloud string as well as electromagnetic field does not exist and vacuum solutions

            From equation (2.26) and (2.27) we have

            F12 0


            have been obtained.

          2. REFERENCES

Here we observed that, in anisotropic bianchi type I cosmological model electromagnetic field does not exist. Thus we have a new set of vacuum field equations which are same as the case I equation (2.15) to (2.17). We obtain the same vacuum cosmological model defined as in equation (2.20).

3. Physical and Kinematical Properties The spatial volume for the model (2.1) is given by

  1. Mazumder A., Solutions of LRS Bianchi-I Space- Time Filled Witha Perfect Fluid. Gen.Rela.Grav, 26,307-310, 1994.

  2. Mohanti G., LRS Bianchi Type I Cosmological Models with Perfect Fluid.

    Bulg.J.Phys, 28,185-192, 2001.

  3. Mohanti G., Sahoo R.C., Sahoo P.K., Some Geometrical Aspects of Bianchi Type -I Space Time.

    Theo.Appl.Mechan, 27, 79-86,2002.

  4. Banerjee A., Sanyal A.K., Chakrabarty S., String Cosmology In Bianchi I Space Time.

    Pramana-The J. Phys., 34, 1-11, 1990.

  5. Bali R.,Bianchi Type I magnetized strings Cosmological Model In General Relativity.

    Astrophys. and space-Science, 302, 201, 2006.

    V 3 AB2


    Where, V ( AB2 )3


    as the average scale factor.

  6. Bali R., PareekU.K., Pradhan A., Bianchi Type I Massive String Magnetized Baratropic Perfect Fluid Cosmological Model. Chin.Phys. Lett. 24, 2455, 2007.

  7. Bali R., UpadhyayaR.D., LRS Bianchi Type I Strings Dust Magnetized Cosmological Models.

    Astrophys. Space Sci., 283, 97, 2003.

    So that the Hubble parameter in anisotropic model can be defined as

  8. Pradhan A., Kumar A., LRS Bianchi I Cosmological Universe Models With Varying Cosmological Term .

    Int. J. Mod. Phys. D Vol.10, No. 3, 291-298, (IUCAA-24/2000),


    1 A





  9. Pradhan A., Kumar A., LRS Bianchi I Cosmological Models with

    V 3 A B

    The physical quantities of the expansion scalar is defined as

    Perfect Fluid. SUJST, Vol. XIII, Sec.B, 44-47, 2001.

  10. Pradhan A., Chakrabarty I., LRS Bianchi I Models With Time Varying Gravitational And Cosmological Constants. Gravitation & Cosmology, Vol.7, No.1 (25), 55-57, 2001.

  11. Pradhan A., TiwariK.L., Beesham A. LRS Bianchi I Cosmological Model In The Presence Of Zero-Mass Scalar Fields. Ind. J.Pure.Appl.math, Vol. 32, No. 6, 789-795, 2001.






  12. Pradhan A., Vishwakarma A.K., A New Class of LRS Bianchi Type-I Cosmological Models with Variable G and . SUJST, Vol.

    V A B

    XII, Sec.B, 42-48., 2000.

  13. Pradhan A., Vishwakarma A.K., A New Class of LRS Bianchi Type-I Cosmological Models with Perfect Fluid.

    Ind. J.Pure.Appl.math, 33. (8), 1239-1250, 2002.

  14. Kumar S., Sing, C. P., Exact Bianchi Type I Cosmological Model In A Scalar Tensor Theory. Int.J.Theo.Phys.47, 1722-1730, 2008.

  15. Singh C.P., Suresh Kumar. , Viscous Fluid Cosmology in Bianchi type I Space Time. Int.J.Theo.Phys.48, 925-936, 2009.

  16. Deo S. D., Singh S. P., Bianchi type I Wet dark inverse in Bimetric relativity. Mathematical theory and Modeling Vol3 (a), 30-32, 2013.

  17. Deo S. D., Singh, S. P.,Bianchi type I Maxwell field coupled with perfect fluid in Bimetric relativity. American J. Sci. and Application Vol 1, No.1, 67-72, 2013.

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