Non-vacuum solutions of five dimensional Bianchi type-I spacetime in f (R) theory of gravity

Research Inventy: International Journal of Engineering And Science
Vol.4, Issue 8 (August 2014), PP 74-82
Issn (e): 2278-4721, Issn (p):2319-6483, www.researchinventy.com
74
Non-vacuum solutions of five dimensional Bianchi type-I space-time
in f (R) theory of gravity
V. K. Jaiswal, R. A. Hiwarkar, Jyotsna Jumale and K D Thengane
J. L. Chaturvedi College of Engineering, Nagpur, email : vkjassi@gmail.com
Guru Nanak Institute of Engineering Technology, Nagpur, rhivarkar@gmail.com
R.S.Bidkar College, Hinganghat , Distt. Wardha, jyotsnajumale@yahoo.com
and N.S.Science & Arts College, Bhadrawati, thenganekd@gmail.com
ABSTRACT: The present paper is dealt with the study of the exact non-vacuum perfect fluid solutions of five
dimensional Bianchi-type-I space-time in the metric version of f ( R ) gravity considering stiff matter to obtain
energy density and pressure of the universe on the lines of M. Sharif and M. Farasat Shamir (2010). In
particular, we obtain two exact solutions which correspond to two models of the universe in higher five
dimensional Bianchi type-I space-time. The function f ( R ) are also evaluated for both the models. Finally the
physical properties of these models have been discussed.
KEYWORDS: Higher order theory of gravitation, f ( R ) theory of gravity, Five dimensional Bianchi type-I
Space-time.
I. INTRODUCTION
S. N. Pandey (2008) has developed a higher order theory of gravitation based on a Lagrangian density
consisting of a polynomial of scalar curvature R to obtain gravitational wave equations conformally flat.
Recently S. N. Pandey and B. K. Sinha (2009) have studied spherically symmetric metric in the field equations
of higher order theory of gravitation which is obtained by modifying Einstein’s field equations in general
relativity theory. Thus Einstein’s general relativity is nothing but the particular case of this higher order
modified theory i.e, f ( R ) theory of gravity.
Many authors have obtained spherical, cylindrical and plane symmetric solutions in f ( R ) gravity
[Hollenstein and Lobo (2008), Azadi et al.(2008), M. Sharif and M. Farasat Shamir (2009), etc.]. Sotiriou and
Faraoni (2010) presented some important aspects of f ( R ) theory of gravity in Metric, Palatini and Metric-
Affine formalisms.
Recently, in the paper [1], M. Sharif and M. Farasat Shamir (2010) have studied the non-vacuum
perfect fluid solutions of four dimensional Bianchi type-I space-time in the framework of metric f ( R ) gravity.
For this purpose they have considered stiff matter to find energy density and pressure of the universe and found
two exact solutions which correspond to two models of the universe and observed that the first solution yields a
singular model while the second solution gives a non-singular model. Finally the physical behavior of these
models has been discussed using some physical quantities. The function f ( R ) of the Ricci scalar is evaluated
also by them. It is noted that the work of M. Sharif and M. Farasat Shamir (2010) is in the framework of four
dimensional Bianchi type-I space-time. In the present paper we wish to investigate the similar situation in higher
five dimensional Bianchi type-I space-time.
In a past few years there have been many attempts to construct a unified field theory based on the idea
of multidimensional space-time. The idea that space-time should be extended from four to higher five dimension
was introduced by Kaluza and Klein (1921, 26) to unify gravity and electromagnetism. Several aspects of five
dimensional space-time have been studied in different theories by many authors [Wesson (1983, 84), Reddy
D.R.K. (1999),
Khadekar et al. (2001), Ghosh and Dadhich (2001), Adhao (1994), Thengane (2000), Ambatkar (20002),
Jumale(2006) etc.].
We observed that the four dimensional work of M. Sharif and M. Farasat Shamir (2010) regarding non-vacuum
perfect fluid solutions in the metric version of f ( R ) gravity can further be extended to the higher five
dimensional Bianchi type-I space-time and therefore, an attempt has been made in the present paper.

Non-vacuum solutions of five dimensional Bianchi type…
75
The paper is organized as follows : In section-2, we briefly give the five dimensional field equations in
metric f ( R ) gravity. Section-3 is used to find exact non-vacuum solutions of Bianchi type-I space-time in 5 V ,
section-4 is dealt with some physical quantities. Section-5 and 6 are dealt with five dimensional models of the
universe and in the last section-7, we summarize and conclude the results.
II. FIVE DIMENSIONAL FIELD EQUATIONS IN f ( R )
THEORY OF GRAVITY
The five dimensional field equations in f ( R )
theory of gravity are given by :
ij ij i j ij F R R  f ( R ) g    F ( R )  g
2
1
( ) □ ij F ( R )  kT , (i , j  1,2,3,4,5) (1)
where
dR
df R
F R
( )
( )  , □ i
i
  
, i  is the covariant derivative and ij T is the standard matter energy
momentum tensor.
Contracting the above field equations we have
( ) 4
2
5
F ( R ) R  f R  □ F ( R )  kT . (2)
Using this equation in (1), the field equations take the form
F R R F R kT [ F ( R ) R
5
1
( )    ( )       □  F ( R )  kT ] g . (3)
In this way we have eliminated f ( R ) from the field equations and therefore, the equation (3) helps us to solve
the field equations.
III. EXACT NON-VACUUM SOLUTIONS OF BIANCHI TYPE - I SPACE-TIME IN 5 V
In this section we find exact non vacuum solutions of five dimensional Bianchi type-I space time in f ( R )
theory of gravity. The line element of Bianchi type-I space-time in 5 V is given by
2 2 2 2 2 2 2 2 2 2
ds  dt  A (t )dx  B (t )( dy )  C (t )dz  D (t )du (4)
where A, B and C are cosmic scale factors. The corresponding Ricci scalar is
2[ ]
.. .. .. .. . . . . . . . . . . . .
CD
C D
BD
B D
BC
B C
AD
A D
AC
A C
AB
A B
D
D
C
C
B
B
A
A
R            , (5)
where dot means derivative with respect to t . The energy momentum tensor for perfect fluid gives
ij i j ij T  (  p )u u  pg (6)
satisfying the equation of state
p  w  , 0  w  1 , (7)
where  and p are energy density and pressure of the fluid while (1,0,0,0,0 ) 00 u g i  is the five-velocity
in co-moving co-ordinates. Since the metric (4) depends only on t , equation (3) yields a set of

76
differential equation for F ( t ), A, B , C ,  and p . Thus the subtraction of the 00-component and 11-
component gives
( ) 0
.. .. .. . . . . . . . . ..
           p 
F
k
F
F
AF
A F
AD
A D
AC
A C
AB
A B
D
D
C
C
B
B
. (8)
Similarly the subtraction of the 00-component and 22- component gives
( ) 0
.. .. .. . . . . . . . . ..
           p 
F
k
F
F
BF
B F
BD
B D
BC
B C
AB
A B
D
D
C
C
A
A
. (9)
The subtraction of the 00-component and 33- component gives
( ) 0
.. .. .. . . . . . . . . ..
           p 
F
k
F
F
CF
C F
CD
C D
BC
B C
AC
A C
D
D
B
B
A
A
, (10)
and the subtraction of the 00-component and 44- component gives
( ) 0
.. .. .. . . . . . . . . ..
           p 
F
k
F
F
DF
D F
CD
C D
BD
B D
AD
A D
C
C
B
B
A
A
. (11)
It is interesting to note that there are three independent differential equations in five dimensional case also.
The conservation equation, 0 ; 
ij
j T leads to
( )[ ] 0
. . . .
.
       
D
D
C
C
B
B
A
A
p . (12)
Thus we have four differential equations with six unknowns namely A, B , C , F ,  & p .
From equations (8) - (9), (9) -(10) and (8) - (10), we get respectively
( ) ( ) 0
.. .. . . . . .
       
F
F
D
D
C
C
B
B
A
A
B
B
A
A
, (13)
( )( ) 0
.. .. . . . . .
      
F
F
D
D
A
A
C
C
B
B
C
C
B
B
, (14)
( )( ) 0
.. .. . . . . .
     
F
F
B
B
A
A
D
D
C
C
D
D
C
C
. (15)
( )( ) 0
.. .. . . . . .
      
F
F
C
C
B
B
A
A
D
D
A
A
D
D
. (16)
These equations imply that

77
exp[ ]
1 1 4
 
a F
dt
d c
A
B
, (17)
exp[ ]
2 2 4
 
a F
dt
d c
B
C
, (18)
exp[ ]
3 3 4
 
a F
dt
d c
C
D
(19)
exp[ ]
4 4 4
 
a F
dt
d c
D
A
(20)
where 1 c , 2 c , 3 4 c , c and 1 d , 2 d , 3 d , 4 d are constants of integration which satisfy the relation
0 1 2 3 4 c  c  c  c  , 1 1 2 3 4 d d d d  . (21)
Using equation (17), (18), (19) and (20), we can write the metric functions explicitly as
exp[ ]
1 1  4 
a F
dt
A ap q , (22)
exp[ ]
2 2  4 
a F
dt
B ap q , (23)
exp[ ]
3 3  4 
a F
dt
C ap q , (24)
exp[ ]
4 4 4
 
a F
dt
D ap q , (25)
where 1 1 / 4
3
2
2
3
1 1 ( )
  
p  d d d , 1 1 / 4
3
2
2 1 2 ( )
 
p  d d d , 1 1 / 4
3
2
3 1 2 ( )

p  d d d , 3 1 / 4
3
2
4 1 2 p  ( d d d ) (26)
and
4
3 2 1 2 3
1
c c c
q
 
  ,
4
2 1 2 3
2
c c c
q
 
 ,
4
2 1 2 3
3
c c c
q
 
 ,
4
2 3 1 2 3
4
c c c
q
 
 (27)
i c and i d are constants of integration. Using power law relation between F and a we have
m
F  ka , (28)
where k is the constant of proportionality, m is any integer ( here taken as -3 ) and a is given by
n
a nlt k
1 /
1  (  ) , n  0
exp( ) 2 a  k lt , n  0 , (29)

78
where 1 k and 2 k are constants of integration. It is mentioned here that we have used n
H la

 ,
l  0 , n  0 to get the above equation. Thus we obtain two values of the average scale factor corresponding to
two different models of the universe.
IV. SOME IMPORTANT PHYSICAL QUANTITIES
In this section we define some important physical quantities
The average scale factor and the volume scale factors are defined respectively as under :
4
1
a  ( ABCD ) , V  a  ABCD
4 . (30)
The generalized mean Hubble parameter H is defined by
[ ]
4
1
1 2 3 4 H  H  H  H  H , (31)
where
A
A
H
.
1  ,
B
B
H
.
2  ,
C
C
H
.
3  ,
D
D
H
.
4  are the directional Hubble parameters in the directions
of x , y , z and u axis respectively. Using equations (30) and (31), we obtain
a
a
H H H H
V
V
H
.
1 2 3 4
.
[ ]
4
1
4
1
      . (32)
The mean anisotropy parameter A is given by
2
4
1
( )
4
1




i
i
H
H H
A . (33)
The expansion scalar  and shear scalar 2
 are defined as under
C
C
B
B
A
A
u
i
i
. . .
;
2
     , (34)
ij
ij    
2
1 2 (35)
where        ij j i i j ij u u g
4
1
( )
2
1
. (36)
In thermodynamics, the entropy of the universe is given by
Tds  d ( V )  pdV . (37)
V. FIVE DIMENSIONAL MODEL OF THE UNIVERSE WHEN n  0
For this model, F becomes n
F k nlt k
3 /
1 ( )

  and the corresponding metric coefficients A, B and C turn
out to be

79
]
( 1)
( )
( ) exp[
1
1 / 1 1
1 1


 

kl n
q nlt k
A p nlt k
n
n
n , n  1 (38)
]
( 1)
( )
( ) exp[
1
1 / 2 1
2 1


 

kl n
q nlt k
B p nlt k
n
n
n , n  1 (39)
]
( 1)
( )
( ) exp[
1
1 / 3 1
3 1


 

kl n
q nlt k
C p nlt k
n
n
n , n  1 (40)
]
( 1)
( )
( ) exp[
1
1 / 4 1
4 1


 

kl n
q nlt k
D p nlt k
n
n
n , n  1 (41)
The mean generalized Hubble parameter and the volume scale factor become
1 nlt k
l
H

 , n
V nlt k
4 /
1  (  ) . (42)
The mean anisotropy parameter A turns out to be
n
k nlt k
q q q q
A
2 /
1
2
2
4
2
3
2
2
2
1
2 (  )
  
 . (43)
The expansion  and shear scalar 2
 are given by
1
4
nlt k
l

  ,
n
k nlt k
q q q q
2 /
1
2
2
4
2
3
2
2
2
2 1
2 (  )
  
  . (44)
For stiff matter (  1) , were have p   . Thus the energy density and pressure of the universe become
n n
k nlt k
q q q q
nlt k
kl
kp k
5 /
1
2
4
2
3
2
2
2
1
3 / 2
1
2
( ) ( )
12
2 2

  



  

. (45)
The entropy of universe is given by
1 / 1
1
2
4
2
3
2
2
2
1
1 / 3
1
3
( )( )
2
3
[6 ( 2 5 )( )
1   
       
n n
q q q q nlt k
k
l
kl n nlt k
k
Tds . (46)
Also equation (12) leads to
2
V
c
  , (47)
where c is an integration constant. It is mentioned here that this value of  , when compared with the value
obtained in equation (41), gives a constraint
6 0
2
kc  kl  (48)

80
Which holds only when n  5 / 2 and 0
2
4
2
3
2
2
2
1 q  q  q  q  . The function of Ricci scalar, f ( R ) is
n n n
nlt k R kl n nlt k
k
f R
2 3 /
1
3 / 2
1 ( 2 5 )( )
5
12
( )
5
2
( )
  
     , (49)
where 2
1
2
1 4 (2 5)( )

R  R  l n  nlt  k . For a special case
2
1
n  , f ( R ) turns out to be
3 / 2 1
2
3 / 2 2
2
)
16
(
5
48
)
16
(
5
2
( )

   
n n
l
R
kl
l
k R
f R (50)
which gives f ( R ) in terms of R .
VI. FIVE DIMENSIONAL MODEL OF THE UNIVERSE WHEN n  0
Here the metric coefficient take the form
]
exp( )
exp( ) exp[
2
1
1 2
klk
q lt
A p k lt

  , (51)
]
exp( )
exp( ) exp[
2
2
2 2
klk
q lt
B p k lt

  , (52)
]
exp( )
exp( ) exp[
2
3
3 2
klk
q lt
C p k lt

  . (53)
]
exp( )
exp( ) exp[
2
4
4 2
klk
q lt
D p k lt

  . (54)
The mean generalized Hubble parameter will become
H  l (55)
while the volume scale factor turns out to be
exp( 4 )
4
2 V  k lt . (56)
The mean anisotropy parameter A becomes
] exp( 2 )
4
[
2
2
2 2
2
4
2
3
2
2
2
1 lt
l k k
q q q q
A 
  
 (57)
while the quantizes  and 2
 are given by
  4 l , ] exp( 2 )
2
[
2
2
2
2
4
2
3
2
2
2
2 1
lt
k k
q q q q

  
  . (58)
For stiff matter, the energy density and pressure turn out to be

81
[ ] exp( 5 )
12 exp( 3 )
2 2
5
2
2
4
2
3
2
2
2
1
3
2
2
lt
kk
q q q q
k
kl lt
k kp 
  

 
   . (59)
The corresponding entropy is
( ) exp( )]
2
3
[ 30 exp( )
1 2
4
2
3
2
2
2
1
2
3
2 q q q q lt
kk
l
kk l lt
k
Tds        . (60)
The constraint equation with the condition, 0
2
4
2
3
2
2
2
1 q  q  q  q  , is given by
12 exp( 5 ) 0
5 2
2 kc  kk l lt  . (61)
The function of Ricci scalar, f ( R ) , takes the form
exp( 3 )( 30 )
5
2
( )
2
3
2
lt R l
k
k
f R    . (62)
which reduces to ] ( 30 )
12
[
5
2
( )
3 / 5 2
2 8 / 3
R l
kC
l k
f R   (63)
using the constraint equation (62). This corresponds to the general function f ( R )
 
n
n f ( R ) a R , (64)
where n may take values from negative or positive.
VII. CONCLUDING REMARK
(i) In this paper we have studied the expansion of the universe in metric f ( R ) theory of gravity. Using the non-vacuum
field equations, we have obtained exact solutions of the Bianchi type-I space-time. These exact
solutions correspond to two models of the universe. For n  0 , we have obtained singular model of the
universe and for n  0 we found a non-singular model. For these solutions we have evaluated some important
cosmological physical quantities such as expansion scalar  , shear scalar 2
 and mean anisotropy parameter
A . The entropy of the universe is also found.
Our observations regarding the model of the universe are as under
(i) For n  0 , singular model of the universe
For this model, we have a singularity at
nl
k
t t s
1    .
The physical parameters 2
1 2 3 4 H , H , H , H H ,  and  are all infinite at this point for n  0 but volume
scale factor vanishes.
The mean anisotropic parameter A is also infinite at this point for 0  n  1 and it will vanishes
n  1 . The function of the Ricci scalar f ( R ) , energy density  , pressure p and T are also infinite while
the metric function A, B and C vanish at this point of singularity.

82
The model suggests that expansion scalar  and shear scalar 2
 decrease for n  0 with the time.
The mean anisotropic parameter also decreases for n  1 with the increase in time.
This indicate that after a large time the expansion will stop completely and universe will achieve isotropy.
The isotropy condition i.e, / 0
2
   as t   , is also satisfied. The entropy of the universe is
infinite for n  1 / 3 .
Thus we can conclude from these observations that the model starts its expansion from zero volume
with infinite energy density and pressure at s t  t and it continues to expand with time.
(ii) For n  0 , non-singular model of the universe
The physical parameters H , H , H , H , H , and A
2
1 2 3 4  are all finite for all finite values of t .
The mean generalized Hubble parameter H and expansion scalar  is constant while f ( R ) is also finite here.
The metric functions A, B and C do not vanish for this model.
The entropy of the universe is finite.
The energy density and pressure become infinite as t   which shows that the universe started its
evolution in an infinite past with a strong pressure and energy density.
The isotropy condition is also verified for this model.
The volume scale factor increases exponentially with time which indicates that the universe starts its expansion
with zero volume from infinite past.
We observed that all the four dimensional results of Sharif and Farasat (2010) can be obtained from our
investigations after the reduction of dimension.. Therefore it is pointed out that the solution of Sharif and Farasat
(2010) is a particular case of solution presented here.
We think that this new exact higher dimensional solution should bring some additional information and
therefore, they need to be further investigated.
VIII. ACKNOWLEDGEMENT
We are thankful to Professor S N Pandey from India for his constant inspiration.
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Non-vacuum solutions of five dimensional Bianchi type-I spacetime in f (R) theory of gravity

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Non-vacuum solutions of five dimensional Bianchi type-I spacetime in f (R) theory of gravity