Static Spherically Symmetric Cosmological Space- Time in Bimetric Relativity

Static Spherically Symmetric Cosmological Space- Time in Bimetric Relativity

S. D. Deo

Head, Department of Mathematics

N. S. Science & Arts College, Bhadrawati, Dist.Chandrapur-442902(M.S) India

Abstract- Spherically symmetric static Kantowski – Sachs space time is studied in the context of Rosens (1940) bimetric relativity with the source of matter perfect fluid and scalar massive meson field respectively. It is observed that, Kantowski – Sachs cosmological model representing perfect

2. FIELD EQUATIONS

The field equations of bimetric relativity derived from variation principles are K j N j 1 Ng j 8T j

fluid does not exist where as massive meson scalar field exists

when the space time is Spherically symmetric static model and

M 0 .Moreover ,when M=0 , the scalar meson field reduces

(2.1)

i i 2 i i

to zero mass scalar field .

Key Words- Spherically symmetric, perfect fluid, scalar massive meson field, Bimetric relativity

1. INTRODUCTION

   Where Nij 1 ghj ghi

   2

   1

   N N , g 2

   (2.3)

   (2.2)

   In general theory relativity, the spherical symmetry has its own importance by virtue of its simplicity. Many noteworthy spherically symmetric space times (for examples -Schwarzschild solutions (exterior and interior),

   g = det.gij , = det. ij (2.4)

   Avertical bar () denotes a covariant differentiation with respect to .

   the Robertson Walker model of the expanding universe, j ij

   etc.) are playing a vital role in general theory of relativity.

   Space time symmetries are features of space time that can be described as exhibiting some form of symmetry .The role of symmetry in physics is important in simplifying solutions to many problems .Space time symmetries finding ample applications in the study of exact solutions of Einsteins field equations of general relativity.

   Deo(1) studied Spherically symmetric Kantowski – Sachs space time in the context of Rosens (1973) bimetric relativity with the source of matter cosmic strings and domain walls respectively. Observed that, in this theory spherically symmetric Kantowski – Sachs space time does not accommodate cosmic strings as well as domain walls. Hence, one can obtain the vacuum solutions. And it is interesting to note that resulting space time

   Ti is the energy momentum tensor for the matter.

   1. PERFECT FLUID

      Here we consider the spherically symmetric Kantowski –

      Sachs space time in the form

      ds2 = dt2 – 2dr2 k2(d2 + Sin2 d2) (3.1)

      Where and k are functions of r only.

      The background metric corresponding to the metric (3.1) is taken as

      d2 = dt2 dr2 – d2 – Sin2 d2 (3.2)

      For this metric the only non-vanishing Christoffel symbols are

      represents the Robertson Walker flat metric.

      2 sin cos , 3

      3

      cot

      Here we have studied the static spherically symmetric 33

      space times in the context of Rosens (2-4) bimetric theory of relativity with the source of matter perfect fluid and

      32 23

      (3.3)

      scalar massive meson field respectively.

      The energy momentum tensor for cosmic perfect fluid is given by

      i i i

      T j pv v j pg j

      (3.4)

      Together with

      v v4 1

      4

      Here viis the four velocity vector of the fluid distribution having p and as the proper pressure and energy density of the fluid, respectively.

      Using co-moving coordinate system the field equations (2.1) for the metric (3.1) and (3.2) corresponding to the energy momentum tensor (3.4) in bimetric relativity can be written explicitly as

   2. MASSIVE SCALAR MESON FIELD

Here we consider the region of the space time with massive meson scalar field .The energy momentum tensor for massive scalar field is given by

2 k 16 kp

T V V 1 g V V ,m M 2V 2 (4.1)

k

(3.5)

ij ,i , j 2 ij ,m

Together with

g V

16 kp

(3.6)

ij

;ij

M 2V 0

(4.2)

k

Where M is the mass of the parameter of the scalar meson field V. And the suffix comma and the semicolon

2

16 k (3.7) Using the equations

after a field variable represent ordinary and covariant

k

(3.5) and (3.6) we get

differentiations with respect to rand gij, respectively.

Using the equations (3.1), (3.2) and (4.1) then (2.1)

k

k

(3.8)

yield-

2 k 8 k(V 2 M 2V 2 )

Using the equation (3.8) with (3.5) and (3.7) we get

k

3p 0

(3.9)

(4.3)

In view of reality conditions ,i.e.

0, p 0 the

8 k V 2 M 2V 2

equation (3.9) implies that

0 p

(3.10)

k

(4.4)

Thus in bimetric relativity the spherically symmetric Kantowski-Sachs cosmological perfect fluid model does not exist and hence only vacuum model exists.

Using the equations (3.10) and (3.5)-(3.7) we have

2

k

8 k V 2 M 2V 2

(4.5)

k

0

k

(3.11)

Using equations (4.3) and (4.4) we get

k

(4.6)

Which give us

k

0

(3.12)

kqhr

(4.7)

1

Using the equations (3.11) and (3.12) we get n em1r

(3.13)

Where h 0,q are constants of integration. Using equations (4.6) in (4.3) and (4.5), we have

k

k n em2r

(3.14)

k 2 2 2

2

Where m1, m2, n1, n2 are constants of integration.

8 V

• M V

  (4.8)

  Using the equations (3.13) and (3.14) in (3.1), then

  ds2 dt2 e2m1r dr2 e2m2r d 2 sin2 d2

  (3.15)

  3 k

  k

  8 V 2 M 2V 2

  (4.9)

  For which m1=m2=m, then

  ds2 dt2 e2mr dr2 d 2 sin2 d2

  This turns to be

  2V 2 M 2V 2 0

  (4.10)

  Equation (4.10) gives two solutions for V as

  (3.16)

  V exp M r

  V exp M r

  2

  This vacuum model represents Robertson Walker flat static metric.

  (4.12)

  (4.11)

  2

  Klein-Gordan equation (4.2) for the metric (3.1) is

  V 2 k V M 2V 0

  (4.13)

  k

  Using the values of and V from the equations (4.7) and (4.11) and similarly from (4.7) and (4.12), we get the values of and k.

  CONCLUSION

  It is shown that in Rosens bimetric relativity, the static spherically symmetric cosmological model representing perfect fluid does not exist where as massive meson scalar field exists when the space time is (3.1)and M 0

  .Moreover ,when M=0 , the scalar meson field reduces to zero mass scalar field .

  ACKNOWLEDGEMENTS

  The author wishes to acknowledge the UGC for sanctioning research project and financial support.

  REFERENCES

  1. S. D. Deo, International Journal of Applied Compuational Science and Mathematics

2. (1)2012, 23-28.

3. N. Rosen, Gen. Rela. Grav. 4 (1973) 435. 4. N. Rosen, Ann. Phys.84 (1974)455.

5. N. Rosen, Gen. Rela. Grav. 12 (1980) 493.