International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1305

ISSN 2229-5518

N-Dimentional Kaluza-Klein Barotropic Fluid Cosmological Model with Varying Gravitational Constant in Creation Field Theory of Gravitation

H. R. Ghate1, Sandhya S. Mhaske2

1,2Department of Mathematics, Jijamata Mahavidyalaya, Buldana (India) – 443 001

2e-mail: sandhyamhaske@yahoo.com

**Abstract**—N-dimensional Kaluza Klein cosmological model with varying gravitational constant for barotropic fluid in creation field theory of gravitation have been investigated. To get the deterministic model of the universe, we assume that G = Al , where A is a scale factor and l is the constant. The solution of the field equations are obtained for l = −1 in particular. Also the physical properties have been studied.

—————————— ——————————

The study of higher dimensional cosmological models acquire much significance as it gained attention for unifying gravita- tion and particle interaction, electromagnetism, gauge theories etc. which were first gracefully presented by Kaluza [1] and

Klein [2] independently. Kaluza has emphasized that general relativity when interpreted as a vacuum five dimensional theory contains four dimensional general relativity in the presence of electromagnetic field together with Maxwell’s electromagnetism. To do so, Kaluza supposed that the model should maintain Einstein’s vision that the nature is purely ge- ometric. Mathematics of general relativity is not modified but just extended to five dimensions and there is no physical de- pendence on fifth dimension. Daemi *et al. *[3] and Marciano [4] have suggested that the experimental detection of time varia- tion of fundamental constant could provide strong evidence

background, large scale structure, Hubbles law, but big bang theory fails to provide the explanation for initial conditions of the universe and is suffered from following problems: i) the model has singularity in the past and possibly one in the future, ii) The conservation of energy is violated, iii) it leads to the very small particle horizon, iv) no consistent scenario ex- ists that explains the origin, evolution and characteristics structures in universe at small scale, v) horizon problem. After that Bondi and Gold [8] introduced a most popular theory called as steady state theory. According to this theory, the uni- verse has no beginning and no end. The steady-state theory assumes that although the universe is expanding, it never change it’s appearance over the time. They visualized a very slow but continuous creation of matter for maintenance of uniformity of mass density in contrast to explosive creation at

for the existence of extra dimension. The resulting field equa-

t = 0

of the standard model. But the theoretical calculations

tions can be separated into further sets of equations which are equivalent to Einstein’s field equations, another set is equiva- lent to Maxwell’s field equations for electromagnetism and the final part an extra scalar field. Moraes and Miranda [5] have studied cosmology from Kaluza Klein gravitational model. Cosmological solutions and their properties of matter in Kaluza-Klein theory have been discussed by Liu and Wesson [6]. Li *et al. *[7] have studied inflation in Kaluza Klein theory: relation between fine structure constant and cosmological constant.

The Big Bang theory based on the Einstein’s field equations is

the leading explanation about evolution of universe. The key idea of big bang model is that the universe is expanding. According to big bang model, the beginning of universe is considered from the single point, where all matter in the uni- verse was contained in. Also the big bang theory provides comprehensive explanation for cosmic microwave

pointed out that under general relatively static universe was impossible. Also the discovery of cosmic microwave back- ground radiations gives the refutation of steady-state theory for most cosmologists. To overcome this difficulty, Hoyle and Narlikar [9] introduced a C-field theory in which there is no big-bang type singularity as in the steady-state theory. Accord- ing to Narlikar, matter creation is accomplished at the expense of negative energy C-field in which he solved horizon and flatness problem by big-bang model. Modeling repulsive grav- ity with creation have been discussed by Vishwakarma and Narlikar [10]. Bali and Saraf [11] have studied C- field cosmo- logical model for barotropic fluid distribution with varying Λ in FRW space-time. Recently, Ghate *et al. [*12] have investigat- ed LRS Bianchi Type V cosmological model for Barotropic fluid distribu- tion with varying Λ(t ) in creation field theory of gravitation.

Gravitational constant has much importance in general relativ- ity as it plays the role of coupling constant between geometry

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and matter. Dirac’s large number of hypothesis [13] is the origin of many theoretical explorations of time varying G. Some new concepts appeared after the original Dirac’s hy- pothesis and also some generalized theories of gravitation admitting variations of the effective gravitational coupling.

(m)T ij ; j =−(c)T ij ; j = fC iC j ; j , (4) i.e. the matter creation through a non-zero left hand side is possible while conserving the overall energy and momentum.

The above equation is identical with

dxi

Thereafter cosmological theories like Brans Dicke theory [14],

Hoyle-Narlikar theory [15] and the theory of Dirac [16] him-

mgij

ds − C j = 0 , (5)

self supported the idea of time decreasing gravitational con- stant. Solar evolutions in the presence of the varying gravita- tional constant have been studied by Pochoda and Schwarschild [17]. Dicke and Peebles [18] have shown that the

which indicates the 4-momentum of the created particle is

compensated by 4-momentum of the *C*-field. In order to main-

tain the balance, the *C*-field must have negative energy. Further, the *C*-field satisfies the source equation

i

importance of gravitation on large scale is due to short range

of strong and weak forces and because of global neutrality of

f C i;i

= J i;i

and

J i = ρ dx ds

= ρ *v*i , (6)

matter, electromagnetic forces become weak. Bali and Tikekar

[19] have studied C-field cosmological models for dust distri-

where ρ is the homogeneous mass density.

The conservation equation for *C*-field is given by

bution in flat FRW space-time with variable gravitational con- stant. Recently Bali and Kumawat [20] investigated cosmologi-

(8π GTi j

= 0 . (7)

cal models with variable *G *in C-field cosmology.

In this paper, we have investigated N-dimensional Kaluza

Klein cosmological model with variable *G *for barotropic per- fect fluid distribution in C-field cosmology. To obtain the

deterministic solution, we have assumed G = Al , where *A *is a

The physical quantities in cosmology are the expansion

scalar θ , the mean anisotropy parameter ∆ , the shear scalar

σ 2 and the deceleration parameter *q *are defined as

θ = (*n *− 1)*H *, (8)

1 (n −1) H

− *H * 2

scale factor and l is a constant in particular l = −1 .

∆ = (*n *− 1)

∑ i

H

, (9)

i =1

2 = 1 *n *−1 H 2 − (n − 1)H 2

σ ∑

The Einstein field equations are modified by Hoyle and

2 i =1 i

, (10)

Narlikar [9-11] through the introduction of a massless scalar

(n − 1) 2

field usually called Creation field *viz*. *C*-field. The modified field equations are

=

q = −

2

R / R

∆*H*

, (11)

j __ __1 j j j 2 2

Ri −

Rgi

= −8π*G **T*i + *T*i

, (1)

R / R

2 (*m*)

(*c*)

where *H *is a Hubble parameter.

where Ti j is a matter tensor for perfect fluid of Einstein’s theo-

(*m*)

ry given by

j j j

N-dimensionalKaluza Klein metric is given by,

2 2 2 *n *− 2 2 2 2

Ti = (ρ + p)vi v

(*m*)

− *pg*i

(2)

ds = dt − A

∑ *dx*i i =1

− *B dx*n −1 , (12)

and Ti

is a matter tensor due to C -field given by

Where A, B

are scale factors and are functions of cosmic

( *c *)

T j = − f C C j −

1 g j Cα C . (3)

time t .

It is assumed that creation field *C *is a function of time only

(*c*) 2

i.e.

C ( x, t ) = C (t )

Here ρ is the energy density of massive particles and *p *is the

and*T*i j = (ρ ,− *p*, − *p*, .............(*n *− 1)*times *) .

(*m*)

pressure. vi

are co-moving four velocities which obeys the

The field equations (1) for the metric (12) leads to

relation *v*i *v *j = 1 , vα = 0 ,

α = 1, 2, 3 . *f *> 0 is the coupling con-

dC

(n − 2)(n − 3) A 2

+ (*n *− 2)

A B

= 8π*G * ρ − 1*fC* 2

(13)

stant between matter and creation field and *C*i = .

dx

2 *A*2

AB 2

A B

A B

*n*2 − 7*n *+ 12 *A* 2

As T 00 has negative value (i.e. T 00 < 0 ), the *C*-field has nega- tive energy density producing repulsive gravitational field

(n − 3) +

A

+ (*n *− 3) +

B AB A2

, (14)

= 8π*G* 1 *fC *2 − *p *

which causes the expansion of the universe. Thus the energy

conservation law reduces to

2

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ISSN 2229-5518

(n − 2)(n − 3) A 2 (

2) *A*

1 2

, (15)

field: C;i = 0 leads to C = t for large r . Thus *C* = 1

2 *A*2

+ *n *− =

A

G fC

2

− *p *

Using *C* = 1 , (18) leads to

⋅

(*n *− 2)(*n *− 3) *A* 2

where the overdot

denotes partial differentiation with

8π*G*ρ = *h*(*n *− 1) +

2 *A*

+ 4π*Gf *. (21)

2

respect to *t.*

The conservation equation (7) for the metric (12) is

Using *C* = 1 and barotropic condition we have*p *= γρ in equation (19),

8π*G* ρ − 1 *fC* 2 +

=A

(n − 2)(n − 3) A 2

− + = π

− π γρ

22)

2

(*n *2)

A

2 *A*2

4 *Gf *8 *G *(

*A*

B

where 0 ≤ γ ≤ 1.

8π*G *

ρ − *fCC *+ ρ (*n *− 2)

*A *

= 0

(16)

Multiplying equation (21) by γ and adding with equation

*A*

B

A B

(22), we get

− *fC *2 (*n *− 2)

+ + *p* (*n *− 2)

+

*A *

A

(*n *− 2)

A +

*h*γ (*n *− 2) +

(n − 2)(n − 3)

(1 + γ )

*A* 2

A 2

*A*2

(23)

The field equations (13)-(15) are three independent equations

= (1 − γ )4π*Gf*

in five unknowns*A*, *B *, ρ , *p *and G . Hence two additional

To obtain the deterministic solution, we assume

conditions may be used to obtain the solution. We assume that the expansion θ in the model is proportional to scalar σ . This condition leads to

G = Al , (24)

where l is a constant and *A *is the scale factor. Using equation (24) in equation (23) leads to

B = Ah , h ≠ 1

where h is a proportionality constant.

(17)

2 *A* + [2*h*γ + (*h *− 3)(1 + γ )]

A 2

A

= (1 − γ )8π*f*

(*n *− 2)

Al +1

(25)

The motive behind assuming condition is explained with ref-

erence to Thorne [21], the observations of the velocity red-shift

Let us assume that *A* = *F *( *A*) ,

A = F F ' with *F *' = *dF *.

dA

relation for extra galactic sources suggest that Hubble expan-

Using this in equation (25), it reduces to

sion of the universe is isotropic today within

≈ 30

percent 2 2

(Kantowski and Sachs, [22]; Kristian and Sachs, [23]). To put

σ

more precisely, red shift place the limit ≤ 0.3 on the ratio of

*dF *+ [2*h*γ + (*n *− 3)(1 + γ )] *F*

dA A

= (1 − γ )8π*f*

(*n *− 2)

Al +1 , (26)

θ which on simplification gives

shear σ to Hubble constant *H *in the neighborhood of our

galaxy today. Collin *et al. *[24] have pointed that for spatially

F 2 =

8π*f *(1 − γ ) *A*l +2

(*n *− 2)[*l *+ 2*h*γ + (*n *− 3)(1 + γ ) + 2]

. (27)

homogeneous metric, the normal congruence to the homoge-

σ

For simplicity, the integration constant has taken to be zero. Hence equation (27) leads to

neous expansion satisfies that the condition

is constant.

θ

8π*f *(1 − γ ) *A*l +2

With the help of (17), the field equations (13)-(15) take the

form

F = . (28)

(*n *− 2)[*l *+ 2*h*γ + (*n *− 3)(1 + γ ) + 2]

(*n *− 2)(*n *− 3) *A*2

1

dA =

(1 − γ )8π*f*

(29)

*h*(*n *− 2) +

= 8π*G*(*t *) ρ −*fC* 2 , (18)

l + 2

(*n *− 2)[*l *+ 2*h*γ + (*n *− 3)(1 + γ ) + 2]

2

A2

2 *A*

(*n *− 2)

=A

+

(n − 2)(n − 3) A 2

2

= 8π*G*(*t * 1*fC* 2 − *p *

(19)

To obtain the determinate value of *A *in terms of cosmic

time t , we consider l = −1 .

A 2 A

2

Putting l = −1 inequation (29), we have

The conservation equation (16) takes the form

dA =

A

(1 − γ )8π*f*

(*n *− 2)[2*h*γ + (*n *− 3)(1 + γ ) + 1]

. (30)

8π*G* (ρ − 1 *fC* 2 ) +

2

On integration, equation (30) leads to*A *= (*at *+ *b*)2 , (31)

A

ρ − *fCC *+ (*h *+ *n *− 2)ρ

8π*G * *A*

= 0

(20)

where

1

(1 − γ )8π*f*

*A*

A a =

, (32)

− (*h *+ *n *− 2) *fC *2

+ (*h *+ *n *− 2) *p *

A

2 (*n *− 2)[2*h*γ + (*n *− 3)(1 + γ ) + 1]

Following Hoyle and Narlikar [11] , the source equation of C-

b = N . (33)

2

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ISSN 2229-5518

Here N is a constant of integration. Thus we have

G = A−1 = (at + b)−2 . (34)

Which on simplification gives

[(*h *+ *n *− 2)(1 + γ ) − 1]

h(n − 2) +

n 2 − 5n + 6

2

Using equations (31) and (34), equation (21) simplifies to

C =

1 +

2*h *+ 2*n *− 5 π*f*

(43)

8πρ = 4*a *2 *h*(*n *− 2) +

(n − 2)(n − 3)

+ 4π*f *. (35)

n 2 − 5n + 6

2

[(*h *+ *n *− 2)(1 + γ ) − 1]

h(n − 2) +

2

Using equation (31)in equation (12), we get

C = t

1 +

2*h *+ 2*n *− 5 π*f*

. (44)

n − 2

ds 2 = dt 2 − (at + b)4 dx 2 − (at + b)4

dx 2

2

∑ i

i =1

n n −1 . (36)

n

*h*(*n *− 2) +

− 5*n *+ 2

The equation of state for barotropic fluid distribution is given

Taking

π*f *= __ 2 __ , we get C = 1 , which

__ __2*h *+ 2*n *− 5

by *p *= γρ .

(*h *+ *n *− 2)(1 − γ ) − 1 − 1

Using

p = γρ

in equation (20) we get

8π (*G*ρ + *G* ρ ) − 4π*fG* *C* 2 − 8π*GfC**C* −

8π*G*(*h *+ *n *− 2) *fC* 2 *A* + 8π*G*(*h *+ *n *− 2)ρ *A* (1 + γ ) = 0 . (37)

agrees with the value used in the source equation. Thus crea-

tion field *C *is proportional to time *t *and the metric (12) for the constraints mentioned above, leads to

A A 2

n−2

2 4 2

4*n *2

With the help of equations (27) and (31), equation (34)

2

ds = dt − t

∑ dxi − t

i =1

dxn−1 . (45)

yields

dC

dt

+ (2*h *+ 2*n *− 5)

2*a*

(*at *+ *b*)

*C* 2 = [2(*h *+ *n *− 2) − 1] ρ

f

A

A

(38)

The homogeneous mass density ρ , Gravitational constant *G*, the scale factor *A *and the deceleration parameter *q *for the model (45) are given by

Using equations (31) and (35), equation (38) leads to

*n *2 − 5*n *+ 2

dC 2

dt

+ (2*h *+ 2*n *− 5)

2*a*

(*at *+ *b*)

*C* 2 =

8πρ = 4*h*(*n *− 2) +

+

2

[(*h *+ *n *− 2)(1 + γ ) − 1]×

*n *2

*h*(*n *− 2) +

− 5*n *+ 2

. (46)

4__ 2 __

*a *2 [*h*(*n *− 2) +

(n − 2)(n − 3)

]

2

2*a*

+ 1

(39)

__ __2*h *+ 2*n *− 5

(*h *+ *n *− 2)(1 − γ ) − 1

− 1

π*f*

*at *+ *b*

G = t −2

, (47)

To reach the deterministic value of *C* , we assume and *b *= 0 .

Thus equation (39) leads to

a = 1

A = t 2 , (48)

q = − 1 . (49)

2

2(*n *− 1)(*h *+ *n *− 2)

dC 2

+

dt

2(2*h *+ 2*n *− 5)

t

C 2

2[(*h *+ *n *− 2)(1 + γ ) − 1]

=

t

θ = (50)

t

2 2

[*h*(*n *− 2) +

×

(n − 2)(n − 3)

]

2

+ 1

(40)

(*h *+ *n *− 3) + (*n *− 2)

∆ =

(n − 1)(h + n − 2)2

(51)

π*f *

σ 2 =

2(*n *− 1)[(*h *+ *n *− 3)2 + (*n *− 2)2 ]

t 2

(52)

On integration equation (40) gives

*C* 2 *t *4h+4n−10 = 2[(*h *+ *n *− 2)(1 + γ ) − 1]×

We have considered the space-time geometry corresponding

*n *2

*h*(*n *− 2) +

− 5*n *+ 6

2

+ 1 ∫

1 4*h*+4*n*−10

t dt

(41)

to Kaluza-Klein type in Hoyle Narlikar ’s creation field theory

of gravitation. Kaluza-Klein universe in creation field cosmol- ogy has been investigated by Ghate and Mhaske [25] whose

π*f * *t*

work has been extended and studied in N-dimensions. We

have noted that all the results of Ghate and Mhaske [25] can be

From equation (41), we get

*h*(*n *− 2) +

n 2 − 5n + 6

obtained from our results by assigning appropriate values to the functions concerned.

*C* 2 = [(*h *+ *n *− 2)(1 + γ ) − 1] __ 2 __ + 1

(42)

2*h *+ 2*n *− 5 π*f*

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