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
IJSER © 2015 http://www.ijser.org
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1306
ISSN 2229-5518
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 vi , (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
= −8πG Ti + Ti
, (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)
− pgi
(2)
ds = dt − A
∑ dxi i =1
− B dxn −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
Ti 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 vi 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
= 8πG ρ − 1
fC 2
(13)
stant between matter and creation field and Ci = .
dx
2 A2
AB 2
A B
A B
n2 − 7n + 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)
= 8πG 1 fC 2 − p
which causes the expansion of the universe. Thus the energy
conservation law reduces to
2
IJSER © 2015 http://www.ijser.org
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1307
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 A2
+ 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
8πGρ = h(n − 1) +
2 A
+ 4πGf . (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),
8πG ρ − 1 fC 2 +
=A
(n − 2)(n − 3) A 2
− + = π
− π γρ
22)
2
(n 2)
A
2 A2
4 Gf 8 G (
A
B
where 0 ≤ γ ≤ 1.
8πG
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 +
hγ (n − 2) +
(n − 2)(n − 3)
(1 + γ )
A 2
A 2
A2
(23)
The field equations (13)-(15) are three independent equations
= (1 − γ )4πGf
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 + [2hγ + (h − 3)(1 + γ )]
A 2
A
= (1 − γ )8πf
(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 + [2hγ + (n − 3)(1 + γ )] F
dA A
= (1 − γ )8πf
(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 =
8πf (1 − γ ) Al +2
(n − 2)[l + 2hγ + (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
8πf (1 − γ ) Al +2
With the help of (17), the field equations (13)-(15) take the
form
F = . (28)
(n − 2)[l + 2hγ + (n − 3)(1 + γ ) + 2]
(n − 2)(n − 3) A2
1 
dA =
(1 − γ )8πf
(29)
h(n − 2) +
= 8πG(t ) ρ −
fC 2 , (18)
l + 2
(n − 2)[l + 2hγ + (n − 3)(1 + γ ) + 2]
2
A2
2 A
(n − 2)
=A
+
(n − 2)(n − 3) A 2
2
= 8πG(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 )8πf
(n − 2)[2hγ + (n − 3)(1 + γ ) + 1]
. (30)
8πG (ρ − 1 fC 2 ) +
2
On integration, equation (30) leads to
A = (at + b)2 , (31)
A
ρ − fCC + (h + n − 2)ρ
8πG A
= 0
(20)
where
1
(1 − γ )8πf
A
A a =
, (32)
− (h + n − 2) fC 2
+ (h + n − 2) p
A
2 (n − 2)[2hγ + (n − 3)(1 + γ ) + 1]
Following Hoyle and Narlikar [11] , the source equation of C-
b = N . (33)
2
IJSER © 2015 http://www.ijser.org
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1308
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 +
2h + 2n − 5 πf
(43)
8πρ = 4a 2 h(n − 2) +
(n − 2)(n − 3)
+ 4πf . (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 +
2h + 2n − 5 πf
. (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) +
− 5n + 2
The equation of state for barotropic fluid distribution is given
Taking
πf = 2 , we get C = 1 , which
2h + 2n − 5
by p = γρ .
(h + n − 2)(1 − γ ) − 1 − 1
Using
p = γρ
in equation (20) we get
8π (Gρ + G ρ ) − 4πfG C 2 − 8πGfCC −
8πG(h + n − 2) fC 2 A + 8πG(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
4n 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
+ (2h + 2n − 5)
2a
(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 − 5n + 2
dC 2
dt
+ (2h + 2n − 5)
2a
(at + b)
C 2 =
8πρ = 4h(n − 2) +
+
2
[(h + n − 2)(1 + γ ) − 1]×
n 2
h(n − 2) +
− 5n + 2
. (46)
4 2
a 2 [h(n − 2) +
(n − 2)(n − 3)
]
2
2a
+ 1
(39)
2h + 2n − 5
(h + n − 2)(1 − γ ) − 1
− 1
πf
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(2h + 2n − 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)
πf
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) +
− 5n + 6
2
+ 1 ∫
1 4h+4n−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
πf 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)
2h + 2n − 5 πf
REFERENCES
[1] T. Kaluza,“On the problem of unity in physics”,Sitzungsber,
IJSER © 2015 http://www.ijser.org
International Journal of Scientific & Engineering Research, Volume 6, Issue 4, April-2015 1309
ISSN 2229-5518
Preuss. Akad. Wiss. Berlin, Phys. Math. K 1, pp. 966-972, (1921).
[2] O. Klein, “Quantentheorie und fünfdimensionaleRelativitätstheorie”, Z. Phys. 37, pp. 895-906, (1926).
[3] S. Ranjibar-Daemi, A. Salam, J. Strathdec, “On Kaluza-Klein cosmolo
gy”, Phys. Lett. B., Vol. 135, Issue 5-6, pp. 388-392,(1984).
[4] W. J. Marciano, “Time Variation of the Fundamental "Constants and
Kaluza -Klein Theories” Phys. Rev. Letters., 52, 489, (1984).
[5] P. H. R. S. Moraes, O.D. Miranda, “Cosmology from Kaluza-Klein grav- itational model”, AIP Conf. Proc. 1483, 435-440,(2012).
doi 10.1063/1.4756990.
[6] H. Liu, P. S. Wesson, “Cosmological solution and their effective proper- ties of matter in Kaluza Klein theory”, Int. J. Phys. D 3, 627, (1994).
[7] L. X. Li, I. Gott, J. Richard, “Inflammation in Kaluza Klein theory: relation between fine structure constant and the cosmological constant”, Phys. Rev. D58, 103513, (1998).
[8] H. Bondi, T. Gold, “The steady state theory of the expanding
universe”, Mon. Not. R. Astron. Soc. 108, pp 252, (1948).
[9] F. Hoyle, J. V. Narlikar, “A new theory of gravitation”, Proc. Roy. Soc.A Math. Phys. Sci. Vol. 282, Issue 1389, pp 191-207, (1964).
[10] R. G. Vishwarkarma, J. V. Narlikar., “Modeling repulsive gravity with
creation”, J. Astrophys and Astronomy 28(1), pp.17-27, (2007).
[11] R. Bali, S. Saraf, “C Field Cosmological model for barotropic fluid distribution with varying Λ in FRW space time”, Int. J. Theor. Phys. 52: pp 1645-1653, (2013). DOI 10.1007/s10773-013-1486-6.
[12] H. R. Ghate, S. A.Salve, “LRS Bianchi Type V cosmological model for
DOI 10. 1007/s10773-010-0489-9.
[21] K. S. Thorne, “Primordial Element Formation, Primordial Mag- neFields, and the Isotropy of the Universe”, Astrophys. J., Vol. 148, pp. 51, (1967).
[22]Kantowski, R. K. Sachs, “Some spatially homogeneous anisotropic relativistic cosmological models”, Journal of Mathematical Physics, Vol. 7, No. 3, pp. 443, (1966).
[23]Kristian, R. K. Sachs, “Observations in Cosmology”, Astrophysical
journal, Vol. 143, pp. 379, (1966)
[24] C. B. Collins, E. N. Glass, D. A. Wilkinson, “Exact spatially homoge- neous cosmologies”, Gen. Relat. Grav., Vol. 12, No. 10, pp. 805-823, (1980). [25] H. R. Ghate, S. S. Mhaske, “Kaluza-Klein Barotropic Cosmological Models with Varying Gravitational Constant G In HoyleNarlikar’s Crea- tion field Theory of Gravitation.”, Accepted for publication in Global Jour. Sci. Front. Res. (F) , (2015)
Barotropic fluid distribution with varying Λ(t )
creation field theory of
gravitation”, Int. J. Sci. and Engg. Res., 5(6), pp 254-259, (2014).
[13] P.A.M. Dirac, “Large Number of Hypothesis”, Nature 139, pp323, (1937).
[14] C. Brans, R. H. Dicke, “New test of the equivalence principle from
lunar laser ranging”,Phys. Rev. 124(3), pp925, (1967).
[15] Narlikar, J. V., “Singularity and Matter Creation in Cosmological
Models”, Nat. Phys. Sci., 242: 135-136, (1973).
[16] P. A. M. Dirac, “In the past Decade in particle Theory”, E. C. G. Sudar- shan, Y. Ne’eman (eds), Gordan and Breach, London,s (1973).
[17] P. Pochoda, M. Schwarzschild, “Variation of the gravitational constant
and the evolution of the Sun”, Astrophys. J. 139, 587-593, (1964). [18] Dicke, R. H. and Peebles, P.J.E. 965 Space-science Rev. 4, 419.
[19] Bali, R. and Tikekar, R. S., “C-field cosmology with variable G in the
Flat Friedman-Robertson-Walker Model”, Chin. Phys. Lett.,Vol. 24, No. 11,
3290, (2007).
[20] R.Bali, M. Kumawat, “Cosmological Model with variable G in C- field
Cosmology”, Int J Theor. Phys 50, pp27- 34, (2011).
IJSER © 2015 http://www.ijser.org