NON-EXISTENCE OF COSMIC STRINGS IN BIMETRIC THEORY OFGRAVITATION

D.R.K. REDDYDepartment of Applied Mathematics, Andhra University, Visakhapatnam – 530 003, India

(Received 5 March 2003; accepted 16 July 2003)

Abstract. Bianchi type cosmological models are considered in Bimetric theory of gravitation pro-posed by Rosen (1973) in the context of cosmic strings. It is interesting to note that cosmic stringsdo not occur in Bianchi type cosmologies.

1. Introduction

In recent years there has been lot of interest in the study of cosmic strings. Gaugetheories with spontaneous symmetry breaking in elementary particle physics haveinitiated study of cosmic strings. After the big bang the universe may have exper-ienced phase transitions (Linde, 1979) which can produce vacuum domain struc-tures such as domain walls, strings and monopoles (Kibbles, 1976). Of all thesecosmological structures cosmic strings are interesting since they may act as grav-itational lenses (Vilenkin, 1981) and may give rise to density perturbations leadingto the formation of galaxies (Vilenkin, 1981; Stachel, 1980). Banerjee et al. (1990),Shriram and Singh (1995), Krori et al. (1994), Mahanta and Mukherjee (2001) aresome of the authors who have extensively investigated several aspects of cosmicstrings in the context of general relativity.

A new theory of gravitation has been proposed by Rosen (1973) which is knownas bimetric theory at gravitation. It is based on a simple form of Lagrangian andhas a simpler mathematical structure than that of general theory of relativity. In thistheory, at each point of space-time, there exists two metric tensors; a Riemannianmetric tensor gij and the background flat space-time metric tensor γij . The tensorgij describes the geometry of a curved space-time and the gravitational fields. Herethe background metric tensor γij refers to inertial forces. This theory also satisfiescovariance and equivalence principles. It is pointed out that this theory agrees withgeneral theory of relativity upto the accuracy of observations made upto now.

The field equations of bimetric theory of gravitation proposed by Rosen (1973)are

Nij − 1

2Nδi

j = −8πkT ij

Astrophysics and Space Science 286: 397–400, 2003.© 2003 Kluwer Academic Publishers. Printed in the Netherlands.

398 D.R.K. REDDY

where

Nij = 1

2γ ab(ghighj |a)|b (1)

and

k =(g

r

)1/2

together with g = det(gij ) and γ = det(γij ). A vertical bar (|) stands for covariantdifferentiation with respect to γij and T i

j is the energy momentum tensor of matterfields.

Rosen (1973, 1975), Yilmaz (1975), Karade and Dhoble (1980), Karade (1980)and Israelit (1976, 1979, 1981) have studied several aspects of bimetric theoryof gravitation. In particular Reddy and Venkateswarlu (1989), Reddy and Ven-kateswara Rao (1998), Mohanty and Sahoo (2002) and Mohanty et al. (2002) haveestablished the non-existence of anisotropic spatially homogeneous Bianchi typecosmological models in bimetric theory when the source of gravitation is governedby either perfect fluid or mesonic perfect fluid.

In this paper, spatially homogeneous and anisotropic Bianchi type metrics inbimetric theory of gravitation in connection with cosmic strings are considered. Itis observed that Bianchi type string cosmological models do not exist in this theory.

2. Cosmic Strings and Their Non-Existence

The energy momentum tensor for system of cosmic strings is

Tij = ρuiuj − λxixj (2)

here ρ is the rest energy density of the system of strings with massive particlesattached to them, ρ = ρp + λ, ρp being the rest energy of particles attached to thestrings and λ the tension density of the system of strings. As pointed out by Letelier(1983), λ may be positive or negative, ui describes the system four-velocity and xi

represents a direction of anisotropy, i.e. the direction of strings.We have

uiui = −xixi = 1 and uixi = 0. (3)

We consider spatially homogeneous and anisotropic Bianchi type–I metric

ds2 = dt2 − A2dx2 − B2dy2 − C2dz2 (4)

where A, B, C are functions of t only. From Equations (4) and (3) we write

ui = ui = (1, 0, 0, 0) (5)

NON-EXISTENCE OF COSMIC STRINGS IN BIMETRIC THEORY OF GRAVITATION 399

and xi can be taken parallel to any of the directions ∂∂x

, ∂∂y

, ∂∂z

. We choose xi parallel

to ∂∂x

, so that

xi = (0, A−1, 0, 0) (6)

with the help of Equations (2)–(6), the field equations (1) of bimetric theory ofgravitation can be written as(

A4

A

)4

+(

B4

B

)4

+(

C4

C

)4

= −16πkρ

(A4

A

)4

−(

B4

B

)4

−(

C4

C

)4

= 16πkλ

(A4

A

)4

−(

B4

B

)4

−(

C4

C

)4

= 0 (7)

(A4

A

)4

+(

B4

B

)4

−(

C4

C

)4

= 0

where a suffix 4 indicates differentiation with respect to t .The set of field equations (7) admit an exact solution given by

A = exp(α) = exp{k1t}B = (c1t + c2) exp{k2t}C = exp{k2t} (8)

16πkρ = 16πkλ = c21/(c1t + c2)

2

where k1, k2, c1 and c2 are constants of integration. Now, using the fact that thesolution (8) should necessarily satisfy each of the field equations (8), it can beeasily seen that c1 = 0. Hence from solution (8) we obtain

ρ = λ = 0 (9)

which shows that cosmic strings do not occur in bimetric theory of gravitation inBianchi type-I space-time. Thus the situation here is quite different from generalrelativity. However in view of (9) solution (8) reduces to the Bianchi type-I vacuummodel presented by Reddy and Venkateswarlu (1989).

One can easily find that field equations for more general Bianchi type-III, V, VIo

and other metrics in bimetric theory are identical to the set of equations (7) (Reddyand Venkateswara Rao, 1998). Hence in each of the cases, by a similar argumentthat follows the set of equations (7), we conclude that Bianchi type cosmic stringsdo not exist in bimetric theory of gravitation. It may be noted, here, that Krori et

400 D.R.K. REDDY

al. (1994) have shown that in the context of general relativity cosmic strings do notoccur in Bianchi type-V cosmology only.

3. Conclusions

Cosmic strings are important in the early stages of evolution of the universe beforethe particle creation. The present day observations do not rule out the possibleexistence of large scale networks of strings in the early universe. However we haveshown, here, that cosmic strings, which have received considerable attention incosmology do not exist in the framework of Rosen’s (1973) bimetric theory ofgravitation.

References

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