J. Mod. Phys., 2010, 1, 190-195
doi:10.4236/jmp.2010.13028 Published Online August 2010 (http://www.scirp.org/journal/jmp)
Kasner Universe in Creation-Field
Cosmology
Kishor S. Adhav, Meena V. Dawande, Maya S. Desale, Ranjita B. Raut
Department of Mathematics, Sant Gadge Baba Amravati University, Amravati, India
Received April 18, 2010; revised July 9, 2010; accepted July 20, 2010
Abstract
We have studied the Hoyle-Narlikar C-field cosmology with Kasner [1,2] space-time. Using methods of Nar-
likar and Padmanabhan [3], the solutions have been studied when the creation field C is a function of time t
only. The geometrical and physical properties of the models, thus obtained, are also studied.
Keywords: Kasner Space-Time, Creation Field Cosmology, Cosmological Models of Universe.
1. Introduction
The three important observations in astronomy viz., the
phenomenon of expanding universe, primordial nucleon-
synthesis and the observed isotropy of cosmic micro-
wave background radiation (CMBR) were supposed to
be successfully explained by big-bang cosmology which
is based on Einstein’s field equations. However, Smoot
et al. [4] revealed that the earlier predictions of the
Friedman-Robertson-Walker types of models do not al-
ways exactly meet our expectations. Some puzzling re-
sults regarding the red shifts from the extra galactic ob-
jects continue to contradict the theoretical explanations
given from the big bang type of the models. Also,
CMBR discovery did not prove it to be an out come of
big bang theory. Infact, Narlikar et al. [5] 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 [6], Bondi and
Gold [7] 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 models. But, the principle of conservation of
matter was violated in this formalism. To overcome this
difficulty, Hoyle and Narlikar [8] adopted a field theo-
retic approach by introducing a massless and chargeless
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 singularity
as in the steady state theory of Bondi and Gold [7].
The solutions of Einstein’s field equations admitting
radiation with negative energy mass less scalar creation
field C was obtained by Narlikar and Padmanabhan [3].
The study of Hoyle and Narlikar theory [9,10] and [8] to
the space-time of dimensions more than four was carried
out by Chatterjee and Banerjee [11]. RajBali and Tikekar
[12] studied C-field cosmology with variable G in the flat
Friedmann-Robertson-Walker model. Whereas, C-field
cosmological models with variable G in FRW space-time
has been studied by RajBali and Kumawat [13]. Singh
and Chaubey [14] studied various Bianchi types models
and Kantowski-Sach model in creation field cosmology.
The way in which the Kasner [1,2] metric has played a
central role in the elucidation of the existence and struc-
ture of anisotropic cosmological models and their singu-
larities in general relativity motivates the authors to stu-
dy this problem. The Kasner metric is one of the more
widely studied metric. Its usefulness for the construction
of cosmological models and its utility for certain studies
of elementary particles have made it particularly attrac-
tive for exploitation. Because of its simplicity it has been
“rediscovered” many times and is itself very closely re-
lated to metrics given several years earlier by Weyl,
Levi-Civita and Wilson. The form in which Kasner pre-
sented it has been virtually forgotten in favor of the dy-
namic form of the synchronous Bianchi I metric. Here,
we have considered a spatially homogeneous and anisot-
ropic Kasner form of Bianchi type-I metric in Hoyle and
Narlikar C-field cosmology. We have assumed that the
creation field C is a function of time t only i.e.


,Cxt


Ct

.
2. Hoyle and Narlikar C-Field Cosmology
Introducing a massless scalar field called as creation

K. S. ADHAV ET AL. 191
field viz. C-field, Einstein’s field equations are modified.
Einstein’s field equations modified by Hoyle and Nar-
likar [8-10] are

18
2
mc
ijijij ij
RgR TT    (1)
where m
ij
T is matter tensor of Einstein theory and c
ij
T
is matter tensor due to the C-field which is given by
1
2
ck
iji jijk
TfCCgCC

 


(2)
where 0f and ii
C
C
x

 .
Because of the negative value of

0000 0TT

, 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
;; ;
mijciji j
j
jj
TTfCC  (3)
i.e. matter creation through non-zero left hand side is
possible while conserving the over all energy and mo-
mentum.
[Here semicolon (;) denotes covariant derivative].
Above equation is similar to
0
i
ij j
dx
mg C
ds 
(4)
Which implies that the 4-momentum of the created
particle 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 ;;
ii
ii
f
CJ and
i
ii
dx
J
v
ds

 ,
where

is homogeneous mass density.
3. Metric and Field Equations
We consider an anisotropic [Bianchi type-I] metric in
Kasner form as
3 12
2 22 2222 q qq
dsdt tdxtdytdz (5)
where 1
q,2 q and 3 qare three parameters that we shall
require to be constants.
We have assumed that creation field C is function of
time t only i.e.
 
,Cxt Ctand

,,,
mi
j
Tdiag ppp   (6)
We have also assumed that velocity of light and gravi-
tational constant are equal to one unit.
We first find the components of Ricci tensor Rij.
Assuming the metric (5), the non vanishing compo-
nents of Christoffel’s symbols are
21 0i q
ii i
qt

 , 0
ii
i
q
t

 , i=1, 2, 3 .
Hence, we calculate


22
123
1i
q
ii i
Rqqqqt

 , i=1, 2, 3.


222 2
001 23123
Rqqqqqqt



 

.
Let 123 Sqq q

 and 222
123
qqq   , we get


22
2RS St 


Now, the Hoyle-Narlikar field Equations (1) for metric
(5) with the help of Equations (2), (3) and (6) can be
written as


22 2
1
11
12 8
22
qSSStpfC 


 
 



 
 (7)


22 2
2
11
12 8
22
qSSStp fC



 
 



 
 (8)


22 2
3
11
12 8
22
qSSStp fC



 
 



 
 (9)


22 2
11
28
22
SSStfC



 
 



 
 (10)

SS
pfCCC
tt


 


 
 (11)
Now, we assume that

123
qq qS
Vt t


 (12)
Above Equation (12) can be written in the form
 
dd
VpfCV VCV
dV dV


 

(13)
In order to obtain a unique solution, one has to satisfy
the rate of creation of matter-energy (at the expense of
the negative energy of the C-field).
Without loss of generality, we assume that the rate of
creation of matter energy density is proportional to the
strength of the existing C-field energy-density per unit
proper-volume.
This is given by
 
22 22
dVpC gV
dV
  
 (14)
where

is proportionality constant .
Let us define that


CV gV
 .
Substituting it in (13), we get
 
dd
VpfgV Vg
dV dV
  (15)
Comparing right hand sides of (13) and (14), we get

K. S. ADHAV ET AL. 192
  
2
2
d
g
VgVgV
dV f

. (16)
On integrating which gives

2
1
1
f
gV AV






 (17)
where 1
A
is arbitrary constant of integration.
We consider the equation of state of matter as
p

 , 01


 (18)
Substituting Equations (17) and (18) in the equation
(14), we get

2
21
22 f
dVAV
dV







 (19)
which further yields
2
22 21
1
2
21
f
AV
f




 






(20)
Subtracting Equation (7) from Equation (8), we get

2
21 10qqS t

 (21)
Equation (21) can be written as
12 12 0
qq qq dS
dttt ttt

 


Now, using Equation (12), the above equation be-
comes
12 120
qq qq dV
dt ttttV

 



(22)
This on integration gives
11
21
dt
qxV
q
tde
t

,11 ;dconstx const (23)
Subtracting Equation (9) from Equation (8), we get
 
2
23 10qqS t

 (24)
Equation (24) can be written as
33 220
qq qq dS
dt ttttt




Now, using Equations (12) the above equation be-
comes
33
220
qq qq dV
dttttt V





.
This on integration gives
22
32
dt
qxV
q
tde
t

;22 ;dconstx const (25)
Subtracting Equation (9) from Equation (7), we get




2
13 10qqS t


 (26)
Equation (26) can be written as
33 110
qq qq
dS
dtttttt


 


Now, using Equations (12) the above equation be-
comes
33 110
qq qq dV
dtttttV


 



This on integration gives
13
33
dt
qxV
q
tde
t

 33 ;dconstx const

 (27)
where 312312
;dddxxx

.
Using Equations (23), (25) and (27), the values of
12
,
qq
ttand 3
q
t can be explicitly written as,
113
11
exp
qdt
tDV X
V



 (28)
213
22
exp
qdt
tDV X
V



 (29)






V
dt
XVDtq
3
31
3exp
3 (30)
where the relations 123 1DDD  and 123
0XX X
are satisfied by


1, 2, 3
i
Di and

1, 2, 3
i
Xi.
Adding Equations (7)-(9) and subtracting from three
times Equation (10), we get


2
112SS tp 


 (31)
From Equation (12) and (18), we get

12 1
V
V

 

(32)
Substituting Equation (21) in Equation (32), we get

2
22 21
1
2
12 1
21
f
A
VV
V
f




 








(33)
This further gives



2
2
12
2
12
12 1
2
f
f
f
f
VAf t
ff


















(34)
Substituting Equation (34) in Equation (17), we get



12
22
12 1
11
2
gt
fff

 







(35)
Also, from equation


CV gV
, we get

K. S. ADHAV ET AL. 193



12
22
12 1
1log
2
Ct
fff

 







(36)
Substituting Equation (34) in Equation (21), the ho-
mogeneous mass density becomes


2
22
2
1
12 1
f
t
f

 
 (37)
Using Equation (18), pressure becomes


2
22
2
1
12 1
f
pt
f

 
 (38)
From Equations (37) and (38), it is observed that:
When time t, we get, density and pressure tend-
ing to zero i.e. the model reduces to vacuum. Also from
Equation (31), we can verify that [p = 0 and ρ = 0 gives]
S = 1. Which is consistent with the Kasner’s condition
for vacuum i.e. S ≡123 qqq= 1.
When 2
f

, there is singularity in density and pres-
sure.
There is also singularity in density and pressure
for 1

.
From Equation (18), for 1


, we get p

 which
further gives [using Equation (31)] S = 1. In this case, we
can interpret this result as “an anisotropic Kasner type
universe can be considered to be filled with an ideal
(non-viscous) fluid which has equation of state p


[stiff matter: the velocity of sound coincides with the
speed of light].
Using Equation (34) in Equations (28)-(30) we get

2
22
13
13 1
12
exp 1
f
f
q
f
Xf
tDKt t
K












(39)

2
22
23
13 2
22
exp 1
f
f
q
f
Xf
tDKt t
K












(40)

2
22
33
13 3
32
exp 1
f
f
q
f
Xf
tDKt t
K












(41)
where



2
12
2
12
12 1
2
f
f
KAf ff
















and
1
D, 2 D, 3 D and 1
X
, 2
X
, 3
X
are constants of in-
tegration, satisfying the relations 123 1DDD

and
123
0XX X.
4. Physical Properties
We define

3 12
1
3
q qq
attt as the average scale factor so
that the Hubble’s parameter in our anisotropic models
may be defined as
3
1
1
3i
i
a
H
H
a


, where i
i
i
a
Ha 
are directional
Hubble’s factors in the direction of i
x
s respectively.
The expansion scalar

is given by
3
H


2
1f
t
f 




(42)
The mean anisotropy parameter is given by
2
3
1
1
3
i
i
H
AH






2
2
2
22 2
2
3
f
Xf
At
f K











 (43)
The shear scalar 2

is given by
3
2222
1
13
3
22
i
i
H
HAH 






2
22
2
2
f
f
Xt
K






 (44)
The deceleration parameter is given by
11 d
qdtH





2
3
2q
f

 , (45)
where 2222
12 3
X
XXX
If 2 f

then for large t, the model tends to iso-
tropic case.
Case I : 0


(Dust Universe) :
In this case, we obtain the values of various parame-
ters as

12
2
2
2
11
12
f
gt
f









12
2
2
2
1log
12
f
Ct
f









2
22
2
1
12
f
t
f

 ,

K. S. ADHAV ET AL. 194

2
22
13
13 1
11 2
1
exp 1
f
f
q
f
Xf
tDKt t
K

















2
22
23
13 2
21 2
1
exp 1
f
f
q
f
Xf
tDKt t
K

















2
22
33
13 3
31 2
1
exp 1
f
f
q
f
Xf
tDKt t
K
















where


2
12
2
12
12
2
f
f
KAf f















Here 1 D, 2 D, 3 D and 1
X
, 2
X
, 3
X
are con-
stants of integration, satisfying the relations 123 1DDD

and 123 0XX X .
In this case, the expansion scalar

is given by
2
1f
t f





,
The mean anisotropy parameter is given by
2
2
2
22 2
2
1
3
f
Xf
At
f
K












The shear scalar 2

is given by
2
22
2
2
1
2
f
f
Xt
K








The deceleration parameter is given by
2
3
2q
f

 ,
where 2222
12 3
X
XXX
If 2 f

, this model also tends to isotropy for large
t
.
Case II : 1
3
  (Disordered Radiation Universe)
In this case, we obtain the values of various parame-
ters as

12
2
2
3
11
12
f
gt
f

 






12
2
2
3
1log
12
f
Ct
f

 






2
22
2
31
24
f
t
f

 ,


2
22
2
1
24
f
pt
f




2
22
13
13 1
12 2
2
exp 1
f
f
q
f
Xf
tDKt t
K













2
22
23
13 2
22 2
2
exp 1
f
f
q
f
Xf
tDKt t
K













2
22
33
13 3
32 2
2
exp 1
f
f
q
f
Xf
tDKt t
K












where


2
12
2
21 2
12
f
f
KAff















.
Here 1 D,2 D,3 D and 1
X
,2
X
,3
X
are constants of
integration, satisfying the relations 1231DDD

and
123
0XXX

 .
In this case, the expansion scalar

is given by
2
1f
t f





.
The mean anisotropy parameter is given by
2
2
2
22
2
2
2
3
f
Xf
At
f K












The shear scalar 2

is given by
2
22
2
2
2
2
f
f
Xt
K








The deceleration parameter is given by
2
3
2qf


where 2222
12 3
X
XXX or2 f

, this model also
tends to isotropy for large
t
.
5. Discussion
1) In both cases for2 f

 , we get negative decelera-
tion parameter indicating that the universe is accelerating.
This observation is consistent with the present day ob-
servation.
2) The expansion scalar

starts with an infinite
value at t = 0, further gradually decreases & the expan-
sion halts when t = ∞.
3) For2 f

, we get lim 0
t


 . Therefore, the mod-
els are isotropic for large value of t.
4) Also, we have noticed that matter density is in-
versely proportional to square of time t. When t →0, we

K. S. ADHAV ET AL. 195
get ρ→∞ and when t →∞, we get ρ→0.
These are all physically valid results indicating that
there is a situation where Kasner type C-field cosmology
starts from infinite mass density.
5) In general, we have observed that the creation field
C is proportional to time t. That is, the creation of matter
increases as time increases.
6. References
[1] E. Kasner, “Geometrical Theorem on Einstein’s Cosmo-
logical Equations,” American Journal of Mathematics,
Vol. 43, No. 2, 1921, pp. 217-221.
[2] E. Kasner, “An Algebraic Solution of the Einstein Equa-
tions,” Transactions of the American Mathematical Soci-
ety, Vol. 27, No. 1, 1925, pp. 101-105.
[3] J. V. Narlikar and T. Padmanabhan, “Creation-Field Co-
smology: A Possible Solution to Singularity, Horizon,
and Flatness Problems,” Physical Review D, Vol. 32, No.
8, 1985, pp. 1928-1934.
[4] G. F. Smoot, et al., “Structure in the COBE Differential
Microwave Radiometer First-Year Maps,” Astrophysical
Journal, Vol. 396, No. 1, 1992, pp. 21-25.
[5] J. V. Narlikar, et al., “Inhomogeneities in the Microwave
Background Radiation Interpreted within the Framework
of the Quasi-Steady State Cosmology,” Astrphysical
Journal, Vol. 585, No. 3, 2003, pp. 1-11.
[6] F. Hoyle, “A New Model for the Expanding Universe,”
Monthly Notices of the Royal Astronomical Society, Vol.
108, No. 1748, 1948, pp. 372-382.
[7] H. Bondi and T. Gold, “The Steady-State Theory of the
Expanding Universe,” Monthly Notices of the Royal Astr-
onomical Society, Vol. 108, No. 3, 1948, pp. 252-270.
[8] F. Hoyle and J. V. Narlikar, “A Radical Departure from
the ‘Steady-State’ Concept in Cosmology,” Proeedings of
Royal Society (London) A, Vol. 290, No. 1421, 1966, pp.
162-176.
[9] F. Hoyle and J. V. Narlikar, “Mach’s Principle and the
Creation of Matter,” Proceedings of Royal Society (Lon-
don) A, Vol. 273, No. 1352, 1963, pp. 1-11.
[10] F. Hoyle and J. V. Narlikar “The C-Field as a Direct Par-
ticle Field,” Proceedings of Royal Society (London) A,
Vol. 282, No. 1389, 1964, pp. 178-183.
[11] S. Chatterjee and A. Banerjee, “C-Field Cosmology in
Higher Dimensions,” General Relativity Gravitation, Vol.
36, No. 2, 2004, pp. 303-313.
[12] R. Bali and R. S. Tikekar, “C-Field Cosmology with
Variable G in the Flat Friedmann-Robertson-Walker
Model,” Chinese Physics Letters, Vol. 24, No. 11, 2007,
pp. 3290-3292.
[13] R. Bali and M. Kumawat, “C-Field Cosmological Models
with Variable G In FRW Space-Time,” International
Journal of Theoretical Physics, Vol. 48, No. 3, 2009,pp.
3410-3415
[14] T. Singh and R. Chaubey, “Bianchi type-I, III, V, VI and
Kantowski-Sachs Universes in Creation-Field Cosmol-
ogy,” Astrophysical Space Scence, Vol. 321, No. 1, 2009,
pp. 5-18.