Bianchi type i wet dark universe in bimetric relativity

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.9, 2013-Special issue, International Conference on Recent Trends in Applied Sciences with Engineering Applications
30
Bianchi Type-I Wet Dark Universe in Bimetric Relativity
S.D.Deo1
, S.P.Singh2
( 1) Head Department Of Mathematics, N.S. Science and Arts College, Bhadrawati , Dist.: Chandrapur(M.S.),
Pin- 442902 , India.
Email: S_deo01@yahoo.in
(2) Department of Mathematics, Anandibai Damodar Kale Degree college of Arts & Commerce,
Saibaba Nagar , Borivali (west), Mumbai-400092, Maharashtra, India.
Email: spsinghcollege@gmail.com
Abstract
In this paper , we have investigated the role of Wet Dark Fluid in Bianchi Type-I cosmological model within the
framework of bimetric theory of relativity proposed by Rosen N. Here we have used a new equation of state for
the matter dark energy component of the universe known as Wet Dark Fluid given by = ϒ( - ∗ )
which can describe a liquid for example water. It is further observed the non existence of Wet Dark Fluid in
bimetric relativity.
Keywords: Bianchi Type-I Universe , Wet Dark Fluid , Bimetric Relativity
AMS Subject Code- 83C05 ( general relativity)
1. INTRODUCTION:-
Einstein’s theory of general relativity is one of the most beautiful structures of theoretical physics which
describes the theory of gravitation in terms of geometry. In the last decades, several theories of gravitation have
been proposed as alternatives to Einstein’s theory of general relativity.The most popular among them is Rosen’s
(1973) bimetric theory of relativity.
Rosen(1) proposed the bimetric theory of relativity to remove some of the unsatisfactory features of the general
theory of relativity. In the bimetric theory , there exist two metric tensors at each point of space-time , which
describes the gravitation and the background metric which enters into the field equation and interacts
with , ℎ ℎ .
One can regard ℎ ℎ no matter were present. Accordingly , at each
space-time point ,one has two line elements:
=
and ! =
where ds is the interval between two neighbouring events as measured by means of a clock and a measuring rod.
The interval ! is an abstract or geometrical quantity not directly measurable .One can regard it as describing
the geometry that would exist if no matter were present.
When we study the Bianchi type models , we observed that the models contain isotropic special cases and they
permit arbitrarily small anisotropic levels at some instant of cosmic times. Hence these models are to be known
as suitable models of our universe. Therefore study of Bianchi type models creates much more interest.
One of the mysteries of cosmology is the nature of dark energy component of the universe. There is , definitely
no lack of cosmological constant, phantom energy , quintessence k-essence etc. To explain the acceleration of
the universe, modified Friedmann equation such as cardassion expansion and also what might be derived from
brane cosmology have been used. In this work , Wet Dark Fluid ( WDF) as a model for dark energy is used in
Bimetric Relativity. This model is generalized Chaplygin gas( GCG) where an equation of state is offered with
properties relevant for the dark energy problem.
Riess et al (2) , Perimuttar et al (3) and Sahani(4) have studied the nature of the dark component of the universe
as one of the deepest mysteries of cosmology. We are motivated to use the wet dark fluid (WDF) as a model for
dark energy which stems from an equation of state proposed by Tait(5) and Hayward(6) to treat water and
aqueous solutions.
The equation of state for Wet Dark Fluid is = ϒ( - ∗ ) (1.1)
Where the parameters ϒ and ∗ " # #
0 ≤ ≤ 1.
We have energy conservation equation as:
+ 3)* + + = 0 -1.2/
Using equation of state and 3H = 0
.
1
in the above equation , we get

31
=
2
342 ∗ +
5
6 -342/
(1.3)
ℎ ℎ 7 # ℎ # .
Wet Dark Fluid ( WDF) has two components : one behaves as cosmological constant and other as standard fluid
with equation of state p= Type equation here.
If we take c>0 then this fluid will not violet the strong energy condition p + ≥ 0 ;
* + + = - 1 + / − ∗
* + += - 1 + /
5
6 -342/
- 1.4/
Holman and Naidu (7) used the wet dark fluid as dark energy in the homogeneous isotropic FRW case . Singh
and Chaubey (8) studied Bianchi type-I universe with wet dark fluid. And also Deo S. D. (9) studied Bianchi
type-I universe with Wet Dark Energy in Bimetric Relativity. This work concludes that the Bianchi Type-I
cosmological model in Bimetric Relativity does not accommodate Wet Dark Energy .
2.METRIC AND SOLUTIONS OF FIELD EQUATIONS:-
We consider Bianchi type-I Metric in the form:
= I - − / + J d + K L (2.1)
ℎ I, J, K ℎ 7 of t only.
Mℎ background flat metric corresponding to equation (2.1) is
! = − + + d + L (2.2)
The field equations in bimetric theory of gravitation proposed by Rosen (1973) are:
N -
3
O = -8P"M (2.3)
ℎ N =
3 QR
( S
.
S ⃒Q
)⃒U
and k=-
V
2
/
W
X
together with g=det( ) and = det - )
Here the vertical bar (⃒) denotes the covariant differentiation with respect to
and M is the energy momentum tensor of the matter field.
Rosen’s field equation in bimetric theory for the metric (2.1) with the help of equations
(2.2)-(2.3) , can be written in the form:
Z
[
[
]
.
+ Z
^
^
]
.
= -16P"M3
3
(2.4)
2 Z
_
_
]
.
− Z
[
[
]
.
+ Z
^
^
]
.
= -16P"M (2.5)
2 Z
_
_
]
.
+ Z
[
[
]
.
− Z
^
^
]
.
= -16P"M`
`
(2.6)
Z
[
[
]
.
+ Z
^
^
]
.
= -16P"Ma
a
(2.7)
Here the overhead dot( . ) denotes differentiation with respect to t.
The energy momentum tensor by Singh and Chaubey(2008) for the Wet Dark Fluid source is given by,
M = * + + - O (2.8)
Here
ℎ is the matter density and is the flow vector of the fluid. The
flow of the matter is taken orthogonal to the hyper surfaces of the homogenity so that
. = 1 (2.9)
b # 7 , 7 c -2.8) &(2.9) ℎ # ,
M3
3
= M = M`
`
= - & Ma
a
= (2.10)
Now using equation (2.10), equations (2.4)-(2.7) becomes,

32
Z
[
[
]
.
+ Z
^
^
]
.
= 16P" (2.11)
− Z
[
[
]
.
+ Z
^
^
]
.
= 16P" (2.12)
2 Z
_
_
]
.
+ Z
[
[
]
.
− Z
^
^
]
.
= 16P" (2.13)
Z
[
[
]
.
+ Z
^
^
]
.
= -16P" (2.14)
d c -2.11) & (2.14) ,we get
+ = 0 ( 2.15)
e ℎ , ℎ ,
> 0
> 0 ℎ .
Mℎ # c -2.15/ 7 ℎ ,
=0 and = 0 (2.16)
ℎ ℎ ℎ ℎ ℎ # L. h i " e
( ) and wet dark fluid energy density( /, both are identically zero. Thus Bianchi type-I universe with
wet dark fluid in bimetric theory of relativity does not survive and hence only vacuum model is obtained.
For vacuum case, =0 = , the field equations (2.11)-(2.14),gives the solution of the form,
A=B=C= jk
(2.17)
Where n is the constant of integration.
Thus in view of equation (2.17), the metric (2.1) takes the form:
d = jk
( + d + L -d ) (2.18)
This can be transformed through a proper choice of coordinates to
d = l
( m + dn + L -dM ) (2.19)
3.CONCLUSION:-
Here we have constructed Bianchi Type-I cosmological model in Rosen’s bimetric theory of relativity with a
new equation of state for the dark energy component of the universe known as Wet Dark Fluid. Further, we
conclude the nil contribution of Wet Dark Fluid in Bianchi Type-I cosmological model in Rosen’s theory and
hence only vacuum model is obtained.
ACKNOWLEDGEMENT
The authors are thankful to Dr. R.D.Giri , Professor Emiratus , PGTD (Mathematics) ,RTM Nagpur
University,Nagpur (M.S.), India , for his constant inspiration and guidance. Also special thanks is given to
Principal, Dr. Yasmeen Bhatia of Anandibai Damodar Kale Degree college, Mumbai (M.S.), India.
REFERENCES:-
(1) Rosen , N: Gen. Rel.Grav.,4,435(1973).
(2)Riess , A.G. et al. 1009(1998).
(3)Perimutter , S.et al : Astrophys.J.,157,565(1998).
(4)Sahni,V.:arXiv:astro-ph/0403324(2004).
(5)Tait,P.G.: The Voyage of HMS Challenger( H.M.S.O. , London,1998).
(6)Hayward,A.T.J.,Brit.J.Appl.Phys.18,965,(1967).
(7)Holman,R. and Naidu,S., arXiv: Astro-phy/0408102(preprint)(2005).
(8)Singh,T. and Chaubey,R.: Pramana Journal of Physics,71,No.3(2008).
(9)Deo,S.D.: OIIRJ,II(1), 104-107(2012).

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