J
. Mod. Ph
y
s
.
doi:10.4236/jm
p
Copyright ©
2
M
athemati
c
Abstract
In this pape
r
the space-ti
m
equations fo
r
studied.
Keywords:
S
1. Introdu
c
Several new t
h
which are c
o
theory of gra
v
tion, scalar te
n
[1], Nordved
t
Saez and Bal
l
In the theory
ists a variable
which admits
of Canuto et
fied theory b
y
an additional
g
tion principle
electromagne
t
cluded that a
n
scale-invaria
n
It is found
t
agrees with
g
servations m
a
[10] and Can
u
the scale inv
a
[11,12] form
u
the interactio
n
free manner.
Mohanty a
n
of Bianchi ty
p
p
endent gaug
e
p
erfect fluid.
diating mode
l
type VIII sp
a
non- static pl
a
.
, 2010, 1, 185
p
.2010.13027
P
2
010 SciRes.
Scal
e
Biv
c
s Group, Birl
a
E-m
a
Re
c
r
, we have st
u
m
e described
r
this space-
t
S
cale Invaria
n
c
tion
h
eories of gra
v
o
nsidered to
v
itation. In a
l
n
sor theories
p
t
[2], Wagone
r
l
ester [6] are
m
proposed by
B
gravitational
a variable G,
al. [7]. Dirac
y
introducing
t
g
auge functio
n
was proposed
t
ic field equat
i
n
arbitrary gau
g
n
t theories.
t
hat the scale
i
g
eneral relativ
i
a
de up now. D
i
u
to et al. [7] h
a
a
riant theory
o
u
lation is so f
a
n
s between
m
n
d Mishra [1
3
p
e VIII and I
X
e
function an
d
In that paper
,
l
of the univ
e
a
ce-time. Mis
h
a
ne symmetric
-189
P
ublished Onlin
e
e
Inva
r
Eins
t
udutta Mis
h
a
Institute of
T
a
il: {bivudutt
a
c
eived June 1
9
u
died the pe
r
by Einstein-
R
t
ime with ga
u
n
t, Space-Ti
m
v
itation have
b
be alternativ
e
l
ternative the
o
p
roposed by
B
r
[3], Ross [4
]
m
ost importa
n
B
rans and Dic
k
parameter G.
A
is the scale
c
[8,9] rebuilt
t
t
he notion of
t
n

. A scale
from which
g
i
ons can be d
e
g
e function is
i
nvariant theo
r
i
ty up to the
a
i
rac [8,9], Ho
y
a
ve studied se
o
f gravitation
.
a
r the best on
e
m
atter and gra
v
3
] have studie
d
X
space-times
d
a matter fiel
d
,
they have c
o
e
rse for the
fe
h
ra [14] has
Zeldovich fl
u
e
August 2010 (
h
r
iant T
h
t
ein-R
o
h
ra, Pradyu
m
T
echnology an
d
a
, addepallir}
@
9
, 2010; revise
r
fect fluid di
s
R
osen metri
c
u
ge function
m
e, Perfect F
l
b
een formulat
e
e
to Einstein
o
ries of gravi
t
a
B
rans and Dic
k
]
, Dunn [5] a
n
n
t among the
m
k
e [1] there e
x
A
nother theor
y
c
ovariant theo
r
t
he Weyl’s u
n
t
wo metrics a
n
invariant vari
a
g
ravitational a
n
e
rived. It is co
n
necessary in
a
r
y of gravitati
o
a
ccuracy of o
b
y
le and Narlik
a
veral aspects
o
.
But Wesson
e
to describe
a
v
itation in sca
d
the feasibili
t
with a time d
e
d
in the form
o
o
nstructed a
r
a
fe
asible Bianc
h
constructed t
h
u
id model in t
h
h
ttp://www.scir
p
h
eory o
f
o
sen Sp
a
m
n Kumar
S
d
Science, Hy
d
@
gmail.com, s
d July 19, 201
s
tribution in
t
c
with a time
are solved a
n
l
uid
e
d
’s
a
-
k
e
n
d
m
.
x
-
y
,
r
y
n
i-
n
d
a
-
n
d
n
-
a
ll
o
n
b
-
a
r
o
f
’s
a
ll
l
e
t
y
e
-
o
f
a
-
h
i
h
e
h
is
theory
Mishr
a
dovich
Rao
metric
It is f
o
theory
tein-R
o
taken
a
space-
t
cosmo
l
2. Fie
Wesso
n
gravita
t
,
1
i
x
i

space-
t
tric te
n
invaria
n
Wesso
n
are:
2
ij
G

with
Her
e
p
.org/journal
/
j
m
f
Grav
i
a
ce-Ti
m
S
ahoo, Adde
p
d
erabad Cam
p
e
0; accepted J
u
t
he scale inv
a
dependent
g
n
d some ph
y
with a time
d
a
[15] has con
s
fluid model i
n
et al. [16,17
]
scalar meson
o
und from th
e
of gravitation
o
sen space-ti
m
a
n attempt to
t
ime in the sc
a
l
ogical model
h
ld Equatio
n
n
[11,12] for
m
t
ion using
a
1
,2,3,4 are c
t
ime and the t
e
n
sor .
ij
gThis
n
t in nature.
n
[11,12] for t
h
;,,
2
4
iji j






G
e
,
ij
G is the co
n
m
p)
i
tation
i
m
e
p
alli Ramu
p
us, Andhra P
r
e
diffmail.com
u
ly 26, 2010
a
riant theory
g
auge functio
y
sical proper
t
d
ependent gau
g
s
tructed static
n
scale invaria
n
]
have discus
s
fields and Br
a
e
literature t
h
has not been
m
e. Hence, i
n
study the c
y
a
le invariant t
h
h
as been pres
e
n
s
m
ulated a sc
a
a
gauge fun
c
oordinates in
e
nsor field is i
theory is bot
h
The field eq
u
h
e combined
s
,,
2
ab ab
g







1
2
ij ij
G
RR
g

n
ventional Ei
n
i
n
r
adesh, Pilani,
of gravitatio
n
n. The cosm
o
t
ies of the m
o
g
e function.
R
plane symm
e
n
t theory.
s
ed cylindrica
l
a
ns-Dicke scal
a
h
at the scale
i
studied so far
n
this paper,
w
y
lindrically s
y
h
eory of gravi
t
e
nted.
a
le invariant t
h
c
tion

i
x
 ,
the fou
r
-di
m
dentified wit
h
h
coordinate
a
u
ations formu
s
calar and ten
s
;
2
ab ab i
j
gg






ij
g
n
stein tensor i
n
JMP
India
n
, when
o
logical
o
del are
R
ecently,
e
tric Zel-
l
ly sym-
a
r fields.
i
nvariant
in Eins-
w
e have
y
mmetric
t
ation. A
h
eory of
where,
m
ensional
h
the
m
e-
a
nd scale
lated by
s
or fields
j
ij
T  
(1)
(2)
n
volving

B. MISHRA ET AL. 186
ij
g
. Semicolon and comma respectively denote covariant
differentiation with respect to ij
g
and partial differen-
tiation with respect to coordinates. The cosmological
term Λij
g
of Einstein theory is transformed to
2
0
Λij
g

in scale invariant theory with a dimensionless
constant 0 Λ. ij T is the energy momentum tensor of the
matter field and 4
8. G
c

 
The line element for Einstein- Rosen metric with a
gauge function

ct   is.
222
WE
ds ds   (3)
with

222222 22222ABB B
E
dsecdtdrr ededz 


(4)
where A and B are functions of t only, and c is the veloc-
ity of light. Here we intend to build cosmological models
in this space-time with a perfect fluid having the energy
momentum tensor of the form

2m
ijmmijm ij
Tp cUUpg   (5)
together with 1
ij
ij
gUU
where i Uis the four-velocity vector of the fluid; m p
and m p are the proper isotropic pressure and energy
density of the matter respectively.
The non – vanishing components of conventional
Einstein’s tensor (2) for the metric (4) can be obtained as
2
11 4
2
1
GB
c
 (6)

14 4
1
GA
r
 (7)
2
2244 4
2
1
GAB
c


(8)
2
334444 4
2
12GABB
c


(9)
2
44 4
GB

 (10)
Here afterwards the suffix 4 after a field variable de-
notes exact differentiation with respect to time t only.
Using the comoving coordinate frame where 4
ii
U


,
the non-vanishing components of the field Equation (1)
for the metric (3) can be written in the following explicit
form:

11
2
22 2222 44 44
44 0
22
122
AB AB
m
G
peA Bce
c
 

 
 





(11)
14 0G (12)
i.e. 1
A
k

, where 1 kis an integrating constant.

22
2
22 2222 44 44
40
22
122
AB AB
m
G
peB ce
c
 

 









(13)

33
2
22 2222
44 44
40
22
122
AB AB
m
G
peB ce
c
 












(14)

44
2
42 2222 2 44
40
2
32
AB AB
m
G
ceB ce

 
 









(15)
Equation (12) reduces the above set of Equations
(11)-(15) as

1 1
11 22
2
22 22 22 44 44
40
22
122
kB kB
m
GG
peB ce
c
 

 









(16)

1
1
22
33
2
22 22 44 44
40
22
122
kB
m
kB
Gpe
Bce
c

 




 


 




(17)

1
1
22 4
44
2
22 22 44
40
2
32
kB
m
kB
Gce
Bce

 
 


 







(18)
Now, Equation (1) and Equations (16)-(18) (Wesson
[12]) suggest the definitions of quantities v
p(vacuum
pressure) and v p(vacuum density) that involves neither
the Einstein tensor of conventional theory nor the prop-
erties of conventional matter. These two quantities can
be obtained as:

1
2
22 22 2
44 44
40
2
22 kB v Bcepc
 
 

 (19)

1
2
22 22 2
44 44
40
2
22 kB v Bcepc
 



 (20)

1
2
22 22 4 44
40
2
32 kB v Bce c



 

  (21)
It is evident from Equations (19) and (20) that
42
0BBk

 since 40


 (22)
where 2 k is an integrating constant. Using Equation (22)
in Equations (19)-(21), the pressure and energy density
for vacuum case can be obtained as

B. MISHRA ET AL. 187
12
12
2
22 22 44 4
0
22 2 2
12kk
vkk
pce
ce
 








(23)
12
12
2
22 22 4
0
22 2 4
13kk
vkk ce
ce


 






(24)
Here v p and v p relate to the properties of vacuum
only in conventional physics. The definition of above
quantities is natural as regards to the scale invariant
properties of the vacuum. The total pressure and energy
density can be defined as
tmv
ppp (25)
tmv

  (26)
Using the aforesaid definitions of t
p and t p, the
field equations in scale invariant theory i.e. (16)-(18) can
now be written by using the components of Einstein ten-
sor (6)-(10) and the results obtained in Equations (22)-
(24) as:
1
22 22
4
kB
t
Bpce 

 (27)
1
22 22
44 4
2kB
t
BB pce 

 (28)
1
22 24
4
kB
t
Bce 

 (29)
3. Solution
From Equations (27) and (29), we obtained the equation
of state
2
tt
pc

 (30)
Using Equation (27) in Equation (28), we obtained
12
Bdtd (31)
where 1 d and 2 d are integrating constants.
Substituting Equation (31) in Equation (27) and Equa-
tion (29) respectively, the total pressure ‘t p’ and energy
density ‘t

’ can be obtained as:
1
2
21
22
1
tt dt
d
pccqe 



 




(32)
where 12
22kk
qe  is a constant. The reality condition
demands that 2
10d.
Using Equation (31) in Equations (23) and (24) respec-
tively and taking 1
ct
 , the pressure and energy den-
sity corresponding to vacuum case can be calculated as:
0
22
1 1
v
q
pcq t



 

(33)
0
42
3
1
v
q
cq t





(34)
In this case, when there is no matter and the gauge
function

is a constant, one recovers the relation
24
Λ
8
GR
vv
cc p
G


 i.e. 20
vv
cp  , which is the
equation of state for vacuum. Here 2
0GR

 = con-
stant, is the cosmological constant in general relativity.
Also v
pbeing dependent on the constants GR , c and G,
is uniform in all directions and hence isotropic in nature.
The cosmological model with this equation of state is
rare in literature and is known as

– vacuum or false
vacuum or degenerate vacuum model [18-21], the cor-
responding model in the static case is a well known
de-Sitter model.
Now the matter pressure and density can be obtained
as:
112
2
01
2222
1 1
mtv kdtd
qd
pppcqt e





 



 (35)
112
2
01
4222
3 1
mtv kdtd
qd
cqt e





 



 (36)
Now, we have m

 as 0t and m

as
t . Also when 0t

, constant
m

. It is inter-
esting to note that the model free from singularity.
So, the Einstein-Rosen cylindrically symmetric model
in scale invariant theory of gravitation is given by the
Equations (12), (31) and (32) and the metric in this case
is


11 21 21 2
2
22 22 22 2
222 222
2
1
W
kdtddtd dtd
dS
ec dtdrr ededz
ct

 



 


(37)
4. Some Physical Properties of the Model
The scalar expansion, ;3
iT
iQ
UQ   for the model
given by Equation (37) takes the form

121
1
1dt dk
de
c


 (38)
Thus, we find 21
1
1()
dk
de
c


 as 0t and
0

 as t .
If 0c, 10d

and 21
dk the model represents
expanding one for 12
1
1
()
kd
tt d

 .
It is also observed that as 2constant
m


 as t
and 2
m



 as 0t. Thus the universe confirms the
homogeneity nature of the space-time.

B. MISHRA ET AL. 188
Following Raychaudhuri [22], the anisotropy

can
be defined as
2
22
2
11,422,422,433,433,4 11,4
11 22223333 11
gggg gg
gggg gg
 

 


 

 

(39)
Consequently for the model (37),

2
12
80
3dt d
 .
So the shear scalar remains constant for 0t and be-
comes indefinitely large for t .
The ratio of anisotropy to expansion
2
2



12
2
22
80
3
kd
ce for 0t. Thus there is a singularity of
0tfor 12 22kd is not very large. Moreover, the
model is isotropy for finite t and does not approach iso-
tropy for large value of t.
It is observed that the vorticity ‘w’ vanishes which in-
dicates that i u is hypersurface orthogonal. As the acce-
leration
.
i
u found to be zero, the matter particle follows
geodesic path in this theory.
5. Conclusions
Every physical theory carries its own mathematical
structure and the validity of the theory is usually studied
through the exact solution of the mathematical structure.
In this theory black holes do not appear to exist. If the
existence of black holes in nature is confirmed, it will
represent a great success of general theory of relativity.
Since there is no concrete evidence at present for the
existence of black holes, one can take a stand point that
black holes represents a familiar concept of space time.
Therefore the scale invariant theory involves gauge theo-
ries as it relates to gravitational theories with an added
scalar field.
The significance of the present work deals with the
modification of gravitational and geometrical aspects of
Einstein’s equations. These are 1) scale invariant theory
of gravitation which describes the interaction between
matter and gravitation in scale free manner; and 2) the
gauge transformation, which represents a change of units
of measurements and hence gives a general scaling of
physical system. The nature of the cosmological model
with modified gravity that would reproduce the kinemat-
ical history and evolution of perturbation of the universe
is investigated.
Here, cylindrically symmetric static zeldovich fluid
model is obtained in the presence of perfect fluid distri-
bution in scale invariant theory of gravitation. As far as
matter is concerned the model does not admit either big
bang or big crunch during evolution till infinite future.
The model appears to be a steady state.
6. Acknowledgements
The authors are very much grateful to the referee for his
valuable suggestions for the improvement of the paper.
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