Vol.2, No.5, 484-488 (2010) Natural Science
http://dx.doi.org/10.4236/ns.2010.25060
Higher dimensional bianchi type-V universe in
creation-field cosmology
Kishor S. Adhav*, Shivdas D. Katore, Abhijit S. Bansod, Prachi S. Gadodia
Department of Mathematics, Sant Gadge Baba Amravati University, Amravati, India;
*Corresponding Author:
Received 16 December 2009; revised 6 March 2010; accepted 15 March 2010.
ABSTRACT
We have studied the Hoyle-Narlikar C-field cos-
mology with Bianchi type-V non static space-
time in higher dimensions. Using methods of
Narlikar and Padmanabham [1], the solutions
have been studied when the creation field C is a
function of time t only as space time is non
static. The geometrical and physical aspects for
model are also studied.
Keywords: Bianchi Type-V Space-Time, Creation
Field Cosmology, Cosmological Model of Universe,
Higher Dimensions
1. INTRODUCTION
The study of higher dimensional physics is important
because of several prominent results obtained in the de-
velopment of the super-string theory. In the latest study
of super-strings and super-gravity theories, Weinberg [2]
studied the unification of the fundamental forces with
gravity, which reveals that the space-time should be dif-
ferent from four. Since the concept of higher dimensions
is not unphysical, the string theories are discussed in 10-
dimensions and 26-dimensions of space-time. Because
of this, many researchers are inspired to study the higher
dimensional to explore the hidden knowledge of the uni-
verse. Chodos and Detweller [3], Lorentz-Petzold [4],
Ibanez and Verdaguer [5], Gleiser and Diaz [6], Banerjee
and Bhui [7], Reddy and Venkateswara [8], Khadekar
and Gaikwad [9], Adhav et al. [10] have studied the
multi-dimensional cosmological models in general rela-
tivity and in other alternative theories of gravitation.
The three important observations in astronomy
namely the phenomenon of expanding universe, primor-
dial nucleon-synthesis and the observed isotropy of
cosmic microwave background radiation (CMBR) were
supposed to be successfully explained by big-bang cos-
mology based on Einstein’s field equations. However,
Smoot et al. [11] revealed that the earlier predictions of
the Friedman-Robertson-Walker type of models do not
always exactly meet our expectations. Some puzzling
results regarding the red shifts from the extra galactic
objects continue to contradict the theoretical explana-
tions given from the big bang type of the model. Also,
CMBR discovery did not prove it to be a out come of big
bang theory. Infact, Narlikar et al. [12] have proved the
possibility of non-relic interpretation of CMBR. To ex-
plain such phenomenon, many alternative theories have
been proposed from time to time. Hoyle [13], Bondi and
Gold [14] proposed steady state theory in which the uni-
verse does not have singular beginning nor an end on the
cosmic time scale. Moreover, they have shown that the
statistical properties of the large scale features of the
universe do not change. Further, the constancy of the
mass density has been accounted by continuous creation
of matter going on in contrast to the one time infinite
and explosive creation of matter at t = 0 as in the earlier
standard model. But the principle of conservation of
matter was violated in this formalism. To overcome this
difficulty Hoyle and Narlikar [15] adopted a field theo-
retic approach by introducing a mass less and charge less
scalar field C in the Einstein-Hilbert action to account
for the matter creation. In the C-field theory introduced
by Hoyle and Narlikar there is no big bag type of singu-
larity as in the steady state theory of Bondi and Gold
[14]. A solution of Einstein’s field equations admitting
radiation with negative energy mass less scalar creation
fields C was obtained by Narlikar and Padmanabhan [1].
The study of Hoyle and Narlikar theory [15-17] to the
space-time of dimensions more than four was carried out
by Chatterjee and Banerjee [18]. The solutions of Ein-
stein’s field equations in the presence of creation field
have been obtained for Bianchi type-V universe in four
dimensions by Singh and Chaubey [19].
Here, we have considered a spatially homogeneous
and anisotropic non static Bianchi type-V cosmological
model in Hoyle and Narlikar C-field cosmology with
five dimensions. Therefore, we have assumed that the
creation field C is a function of time t only i.e.

K. S. Adhav et al. / Natural Science 2 (2010) 484-488
485
485
  tCtxC ,.
This study is important because of the fact that the
resulting cosmological model is considered to be ame-
nable to the model obtained by Singh and Chaubey [19].
2. HOYLE AND NARLIKAR C-FIELD
COSMOLOGY
Introducing a mass less scalar field called as creation
field namely C-field, Einstein’s field equations are mo-
dified. Hoyle and Narlikar [15-17] field equations are

ij
c
ij
m
ijij TTRgR 
 8
2
1 (1)
where ij
mT is matter tensor of Einstein theory and ij cT
is matter tensor due to the C-field which is given by





 k
k
ijjiij
cCCgCCfT 2
1 (2)
where 0f and i
ix
C
C

.
Because of the negative value of
 0
0000 TT , the
C-field has negative energy density producing repulsive
gravitational field which causes the expansion of the
universe. Hence, the energy conservation equation re-
duces to
j
ji
j
ijc
j
ijmCfCTT ;;;  (3)
i.e. matter creation through non-zero left hand side is
possible while conserving the over all energy and mo-
mentum.
Above equation is similar to
0 j
i
ij C
ds
dx
mg (4)
which implies that the 4-momentum of the created parti-
cle is compensated by the 4-momentum of the C-field. In
order to maintain the balance, the C-field must have
negative energy. Further, the C-field satisfy the source
equation i
i
i
iJCf ;; and i
i
iv
ds
dx
J   , where

is homogeneous mass density.
3. METRIC AND FIELD EQUATIONS
The five-dimensional Bianchi-Type-V line element can
be written as
22
2
4
22
2
3
22
2
2
2
2
1
22 dueadzeadyeadxadtds mxmxmx  
(5)
where 1 a, 2 a, 3 a and 4 a are functions of t only and
m is constant.
Here the extra coordinate is taken to be space like.
The above space time is non static, hence, we have
assumed that creation field C is function of time t only
i.e.



 tCtxC

, and  ppppdiagT i
j
m ,,,,

(6)
We have assumed that velocity of light and gravita-
tional constant are equal to one unit.
Now, the Hoyle-Narlikar field Eq.1 for metric (5)
with the help of Eqs.2, 3, and 6 can be written as







2
2
1
2
43
43
42
42
32
32
41
41
31
31
21
21
2
1
8
6
Cf
a
m
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa

     

(7)







2
2
1
2
43
43
42
42
32
32
4
4
3
3
2
2
2
1
8
3
Cfp
a
m
aa
aa
aa
aa
aa
aa
a
a
a
a
a
a

     

(8)







2
2
1
2
43
43
41
41
31
31
4
4
3
3
1
1
2
1
8
3
Cfp
a
m
aa
aa
aa
aa
aa
aa
a
a
a
a
a
a

     

(9)







2
2
1
2
42
42
41
41
21
21
4
4
2
2
1
1
2
1
8
3
Cfp
a
m
aa
aa
aa
aa
aa
aa
a
a
a
a
a
a



(10)







2
2
1
2
32
32
31
31
21
21
3
3
2
2
1
1
2
1
8
3
Cfp
a
m
aa
aa
aa
aa
aa
aa
a
a
a
a
a
a

   

(11)
4
4
3
3
2
2
1
1
3a
a
a
a
a
a
a
a  (12)


























C
a
a
a
a
a
a
a
a
CCf
p
a
a
a
a
a
a
a
a









4
4
3
3
2
2
1
1
4
4
3
3
2
2
1
1

(13)
where dot )(

indicates the derivative with respect to t.
From Eq.12, we get
432
3
1aaaa  (14)
Assume that V is a function of time t defined by
4321 aaaaV

(15)
From Eqs.14 and 15, we get

K. S. Adhav et al. / Natural Science 2 (2010) 484-488
486
4
1
1Va  (16)
Above Eq.13 can be written in the form
 

VCV
dV
d
VCfpV
dV
d
  (17)
In order to obtain a unique solution, one has to specify
the rate of creation of matter-energy (at the expense of
the negative energy of the C-field). Without loss of gen-
erality, we assume that the rate of creation of matter en-
ergy density is proportional to the strength of the exist-
ing C-field energy-density. i.e. the rate of creation of
matter energy density per unit proper-volume is given by
  VgCpV
dV
d2222
   (18)
where

is proportionality constant and we have de-
fined
  VgVC 
.
Substituting it in Eq.17 , we get
  Vg
dV
d
VfgpV
dV
d
 (19)
Comparing right hand sides of Eqs.18 and 19, we get
   Vg
f
gV
dV
d
Vg 2
2

 (20)
Integrating , which gives













1
1
2
f
VAVg

(21)
where 1
A is arbitrary constant of integration.
We consider the equation of state of matter as

 p (22)
Substituting Eqs.21 and 22 in the Eq.18, we get













12
2
1
2
2
f
VAV
dV
d

 (23)
which further yields




















12
2
2
1
2
2
12
f
V
f
A




 (24)
From Eq.22, we get




















12
2
2
1
2
2
12
f
V
f
A
p




(25)
Subtracting Eq.8 from Eq.9, we get
0
4
4
3
3
2
2
1
1
2
2
1
1
2
2
1
1























a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
dt
d
(26)
Now, from Eqs.15 and 26, we get
0
2
2
1
1
2
2
1
1















V
V
a
a
a
a
a
a
a
a
dt
d

Integrating, which gives






V
dt
xd
a
a
11
2
1exp , 1
d=constant, 1 x=constant
(27)
Subtracting Eq.9 from Eq.10, we get
0
3
3
2
2
3
3
2
2















V
V
a
a
a
a
a
a
a
a
dt
d

Integrating, we get






V
dt
xd
a
a
22
3
2exp , 2
d=constant, 2 x=constant
(28)
Subtracting Eq.10 from Eq.11, we get
0
4
4
3
3
4
4
3
3















V
V
a
a
a
a
a
a
a
a
dt
d

which on integration gives






V
dt
xd
a
a
33
4
3exp , 3
d=constant, 3 x=constant
(29)
Subtracting Eq.8 from Eq.11, we get
0
4
4
1
1
4
4
1
1















V
V
a
a
a
a
a
a
a
a
dt
d

Integrating, we get






V
dt
xd
a
a
44
4
1exp , 4
d= constant, 4 x= constant
(30)
where 3214 dddd

, 3214 xxxx 

and 4321 aaaaV

.
Using Eqs.27, 28, 29 and 30, the values of


ta1,

 ta2,

 ta3 and

 ta4 can be written explicitly as







V
dt
XVDta 1
41
11 exp (31)







V
dt
XVDta 2
41
22 exp (32)







V
dt
XVDta 3
41
33 exp (33)

K. S. Adhav et al. / Natural Science 2 (2010) 484-488
487
487







V
dt
XVDta 4
41
44exp (34)
where the relations 1
4321 DDDD and


21 XX
0
43  XX are satisfied by

4,3,2,1iDi and
 4,3,2,1iX i. From Eqs.10 and 31, we get 1
1

D
and 0
1X.
Adding Eqs.8, 9, 10, 11 and 4 times Eq.7, we get
 p
a
m
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
a
a
a
a
a
a
a
a


















3
3212
2
2
1
2
43
43
42
42
41
41
13
13
32
32
21
21
4
4
3
3
2
2
1
1


(35)
From Eq.15, we have
















43
43
42
42
41
41
13
13
32
32
21
21
4
4
3
3
2
2
1
1
2aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
aa
a
a
a
a
a
a
a
a
V
V



(36)
From Eqs.35, 36 and 22, we get
   1
3
3212
2
1
2
a
m
V
V

(37)
Substituting Eq.16 in Eq.37, we get
   1
3
3212
21
2
V
m
V
V

(38)
Substituting Eq.24 in Eq.38, we get





















12
2
22
21
2
2
12
1
3
3212 f
V
f
A
V
m
V
V






(39)
which further gives

t
kVmV
f
fA
dV
f












1
2322
2
2
16
123
132 2




(40)
where 1 k is integration constant.
For 1

(Zeldovich fluid or Stiff fluid) and 0
1

k,
the above equation gives
44tmV  (41)
Substituting Eq.28 in Eq.21, we get























1414
1
22
ff tmAg

(42)
Also, from equation

 VgVC
, we get































34
2
3414
1
22
f
tmA
C
ff


(43)
Substituting Eq.41 in Eq.24, the homogeneous mass
density becomes























1818
2
1
22
2
1ff tfmA

 (44)
Using Eq.25 and 1


, pressure becomes























1818
2
1
22
2
1ff tfmAp

(45)
From Eqs.38 and 39, it is observed that for 2  f,
there is no singularity in density and pressure.
Using Eq.41 in Eqs.31, 32, 33 and 34, we get

 mtta

1 (46)






34
2
22
1
3
exp tm
X
mtDta (47)
 




34
3
33
1
3
exp tm
X
mtDta (48)






34
4
44
1
3
exp tm
X
mtDta (49)
4. PHYSICAL PROPERTIES
The expansion scalar

is given by
H4


t
4
 (50)
The mean anisotropy parameter is given by







4
1
4
1
i
i
H
H
A

68
2
4
2
3
2
2
4tm
XXX 
 (51)
The shear scalar 2  is given by
22
4
1
2
2
2
4
4
2
1AHHH
ii







 
88
2
4
2
3
2
2
2tm
XXX

(52)
The deceleration parameter q is given by
1
1






Hdt
d
q=0 (53)
where HHH ii



and H is the Hubble parameter.

K. S. Adhav et al. / Natural Science 2 (2010) 484-488
488
For large t, the expansion scalar and shear scalar tend
to zero. Further, if 2
 f, for large t, the model re-
duces to the vacuum case.
5. CONCLUSIONS
In this paper, we have considered the space-time geome-
try corresponding to Bianchi type-V in Hoyle-Narlikar
[15-17] creation field theory of gravitation with five di-
mensions. Bianchi type-V universe in creation-field cos-
mology has been investigated by Singh and Chaubey [19]
whose work has been extended and studied in five di-
mensions. An attempt has been made to retain Singh and
Chaubey’s [19] forms of the various quantities. We have
noted that all the results of Singh and Chaubey [19] can
be obtained from our results by assigning appropriate
values to the functions concerned.
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