Sam Ant A

Five Dimensional Bianchi Type-III String Cosmological Models in General Relativity

Ganeshwar MOHANTYa,b, Gauranga Charan SAMANTAc and Shantanu Kumar BISWALd a

703, Rameswarpatna, Bhubaneswar-751002, India.

b

P. G. Department of Mathematics, Sambalpur University, Jyoti Vihay-768019, India.

e-mail: [email protected] c

Department of Mathematics, Gandhi Institute for Technological Advancement (GITA),

Madanpur-752054, India. e-mail: [email protected] d

Department of Physics, Gandhi Institute for Technological Advancement (GITA), Madanpur-

752054, India. e-mail: [email protected]

Abstract: In this paper we have constructed some five dimensional Bianchi type-III cosmological models in general relativity when source of gravitational field is a massive string. We have obtained different classes of solutions by considering different functional forms of metric potentials. It is also observed that the models posses big bang singularity. The physical and kinematical behaviors of the models are discussed.

Key words: Five dimensions, massive strings and general relativity.

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1. Introduction The Kaluza-Klein theory is attractive because it has an elegant presentation in terms of geometry. In certain sense, it looks just like ordinary gravity in free space, except that it is phrased in five dimensions instead of four. Kaluza [1] and Klein [2, 3] attempted to unify gravitation and electromagnetism. An interesting possibility known as the “cosmological reduction process” is based on the idea that at very early stage all dimensions in the universe are comparable. Later, the scale of the extra dimension becomes so small as to be unobservable by experiencing contraction. Such cosmological models were investigated by Forgacs and Horvath [4]. Chodos and Detweller [5] showed that, the extra dimensions are unobservable due to dynamical contraction to very small scale, while the three other spatial dimensions expand isotropicaly as a consequence of cosmological evolution in the frame work of a pure gravitational theory of Kaluza-Klein. Freund [6] explained the smallness of the extra dimensions of the universe through the dynamical evolution in 10 and 16-dimensional super gravity model. Guth [7] and Alvarez and Gavela [8] observed that during the contraction process extra dimensions produce large amount of entropy, which provides an alternative resolution to the flatness and horizon problem, as compared to usual inflationary scenario. Gross and Perry [9] showed that the five dimensional Kaluza-Klein theory of unified gravity and electromagnetism admits soliton solutions. Further, they explained the inequality of the gravitational and inertial masses due to the violation of Birkoff’s theorem in Kaaluza-Klein theories, which is consistent with the principle of equivalence. Appelquist and Chodos [10] and Randjbar-Daemi et al. [11] claimed through solution of the field equations that there is an expansion of four dimensional space time while fifth dimension contracts to the unobservable plankian length scale or remains constant as needed for the real universe. Recently cosmic string theory is important in the early stages of the evolution of the universe before the particle creation. The study of cosmic strings in elementary particle physics arised from the gauge theories with spontaneous broken symmetry. After the big bang, it is believed that the universe might have experienced a number of phase transitions by producing vacuum domain structures such as domain walls, strings and monopoles. Cosmic strings may act as gravitational lenses and give rise to density 2

perturbations leading to formation of galaxies. Chatterjee [12] constructed massive string cosmological model in higher dimensional homogeneous space time. Krori et al. [13] constructed Bianchi type-I string cosmological model in higher dimension and obtained that matter and strings coexist throughout the evolution of the universe. Rahaman et al. [14] obtained exact solutions of the field equations for a five dimensional space time in Lyra manifold when the source of gravitation is massive strings. Recently Mohanty et al. [15] constructed five dimensional string cosmological models in Barber’s second self creation theory of gravitation and they showed that, one of the models degenerates into two different string cosmological models in Einstein’s theory corresponding to variable G and constant G. In this paper we have taken an attempt to study the role of string in five dimensional Bianchi type-III space time in general theory of relativity. 2. Metric and field equations Here we consider the five dimensional Bianchi type-III metric in the form ds 2 = dt 2 − A 2 dx 2 − B 2 e −2 ax dy 2 − C 2 dz 2 − D 2 dψ 2

(1)

where A(t), B(t), C(t) and D(t) are the scale factors. The universe is assumed to be filled with distribution of matter represented by energy momentum tensor of a cloud of massive strings Tij = ρu i u j − λwi w j

(2) u i u i = −w i wi = −1

(3) and

u i wi = 0

(4)

Where ρ is the rest energy density of the cloud of strings with particles attached to them,

λ is the tension density of the strings and ρ = ρ p + λ , ρp being the rest energy of the particles. The velocity ui describes the cloud 5-velocity and wi represents the direction of anisotropy. For the weak, strong and dominant energy conditions, one finds that ρ >0, 3

ρp >0 and the sign of λ is unrestricted. The fifth co ordinate is taken to be space like

and the co-ordinates are co-moving, where

u 0 = u 1 = u 2 = u 3 = 0, u 4 = 1

(5)

Without loss of generality, we choose w i = (0, 0, 1, 0, 0)

(6) The Einstein’s field equation is given by Rij −

1 Rg ij = −8πTij 2

(7) With the help of equations (2) – (6), the field equation (7) for the metric (1) can be written explicitly as

B ′′ C ′′ D ′′ B ′C ′ B ′D ′ C ′D ′ + + + + + =0, B C D BC BD CD

(8) A′′ C ′′ D ′′ A′C ′ A′D ′ C ′D ′ + + + + + =0, A C D AC AD CD

(9) A′′ B ′′ D ′′ A′B ′ A′D ′ B ′D ′ a 2 + + + + + − 2 = 8π λ, A B D AB AD BD A

(10) A′′ B ′′ C ′′ A′B ′ A′C ′ B ′C ′ a 2 + + + + + − 2 = 0, A B C AB AC BC A

4

(11)

A′B ′ A′C ′ B ′C ′ A′D ′ B ′D ′ C ′D ′ a 2 + + + + + − = 8π ρ, AB AC BC AD BD CD A2

(12) A′ B ′ = . A B

and

(13)

3. Solutions of the field equations From equation (13) it can be shown that A=kB, where k is an arbitrary constant. Therefore we have five equations in six unknowns. For deterministic solutions we need one assumption. We shall explore physically meaningful solutions of the field equations (8) – (13) by considering a simplifying assumption to the field variables A, B, C and D. 3.1. Classes of solutions Case:-1

A = B = tn

(14)

Where n is arbitrary constant. In this case equation (11) reduces to C ′′ 2nC ′ 2n(n − 1) + n 2 a 2 + + − 2n = 0 C tC t2 t

(15) Now equation (15) solvable for n=1. The equations (14) and (15) reduce to A= B =t

(16) t 2 C ′′ + 2tC ′ + (1 − a 2 )C = 0

and (17) Which on integration yields

5

C =t

−1+ 4 a 2 −3 2

(18a) or

C =t

−1− 4 a 2 −3 2

(18b) Using equations (16) and (18a) in the field equation (8) we get t 2 D ′′ + t (1 + 4a 2 − 3 ) D ′ +

(−1 + 4a 2 − 3 ) 2 D =0 4

(19)

Which on integration yields

D = t m1

(20a)

or

D = t m2

where

m1 =

− 4a 2 − 3 + 4a 2 − 3 − ( −1 + 4a 2 − 3 ) 2 2

and

m2 =

− 4a 2 − 3 − 4a 2 − 3 − ( −1 + 4a 2 − 3 ) 2 2

(20b)

Now the above solutions give two different set of models. Set:-1 Thus, the five dimensional string cosmological model corresponding to the solution (16), (18a) and (20a) is written as ds 2 = dt 2 − t 2 dx 2 − t 2 e −2 ax dy 2 − t −1+

4 a 2 −3

dt 2 − t 2 m1 dψ 2

(21)

The rest energy density ( ρ ), string tension density ( λ ), the particle density ( ρp ), the scalar of expansion ( θ ), the shear ( σ ), the spatial volume (V) and the deceleration parameter (q) for the model (21) are obtained as

6

 3 4a2 − 3 − (− 1+ 4a2 − )3 2 + 4a2 − 3  1  8π = 2 ρ 4t  2 2 2 2 2   + 4a − 3 4a − 3 − (− 1+ 4a − )3 − 4a + 3

(22)

2

m1 + m1 + 1 − a 2 8π λ= t2

where

ρp =

1 2 (2n1 + m1 + n1 m1 − m1 ) 8π t 2

n1 =

− 1 + 4a 2 − 3 2

θ=

n1 + m1 + 2 t

(23)

(24)

(25)

2 2 2 1   1 1   1 n1   1 m1   σ = 2 −  +  −  +  −   2   3 t   3 t   3 t   2

(26)

V = t n1 + m1 + 2 e − ax (27) and

q=

− (n1 + m1 + 1) n1 + m1 + 2

(28)

where

n1 + m1 +1 =

1 + 4a 2 − 3 − ( −1 + 4a 2 − 3 ) 2 2

and

n1 + m1 + 2 =

3 + 4a 2 − 3 − (−1 + 4a 2 − 3 ) 2 2

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The rest energy density, string tension density, expansion scalar and shear become infinite for t=0, which indicates that the universe starts at t=0. Hence the model (21) admits initial

σ singularity. The scalar of expansion θ → 0 as t → ∞ . Since lim 2 ≠ 0 , the model does not t →∞ θ 2

approach isotropy for large value of t. The spatial volume V is zero when t=0 and becomes infinite when t → ∞ . The deceleration parameter is negative. Therefore the model (21) is inflationary. Set:-2 The five dimensional string cosmological model corresponding to the solutions (16), (18a) and (20b) is written as ds 2 = dt 2 − t 2 dx 2 − t 2 e −2 ax dy 2 − t −1+

4 a 2 −3

dt 2 − t 2 m2 dψ 2

(29) The rest energy density ( ρ ), string tension density ( λ ), the particle density ( ρp ), the scalar of expansion ( θ ), the shear ( σ ), the spatial volume (V) and the deceleration parameter (q) for the model (29) are obtained as

 − 3 4a 2 − 3 − ( − 1 + 4 a 2 − 3 ) 2 + 4a 2 − 3  1  8π = 2 ρ 4t  2 2 2 2 2   − 4a − 3 4a − 3 − (− 1+ 4a − 3) − 4a − 3

(30)

2

m2 + m2 + 1 − a 2 8π λ= t2 ρp =

(31)

1 2 ( 2n1 + m 2 + n1 m1 − m 2 ) 8π t 2

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(32)

θ=

n1 + m2 + 2 t

(33)

2 2 2 1   1 1   1 n1   1 m 2   σ =  2 −  +  −  +  −   2   3 t   3 t   3 t   2

(34)

V = t n1 + m2 + 2 e − ax (35) and

q=

− (n1 + m2 + 1) n1 + m2 + 2

(36)

In this set the rest energy density ( ρ ) does not satisfy the reality condition i. e. ( ρ >0). Therefore the model (29) corresponding to the solutions (16), (18a) and (20b) are physically unrealistic. Hence the model (29) is not acceptable. Now using equations (16) and (18b) in the field equation (8) we get t 2 D ′′ + t (1 − 4a 2 − 3 ) D ′ +

(−1 − 4a 2 − 3 ) 2 D =0 4

(37)

Which on integration yields

D = t l1

(38a)

D = t l2

or (38b) where

l1 =

4a 2 − 3 + 4a 2 − 3 − (1 + 4a 2 − 3 ) 2 2

and

l2 =

4a 2 − 3 − 4a 2 − 3 − (1 + 4a 2 − 3 ) 2 2

The above solutions (16), (18b), (38a) and (38b) give two different set of models. Set:-1 9

The geometry of the string cosmological model corresponding to the solutions (16), (18b) and (38a) is described by the metric ds 2 = dt 2 − t 2 dx 2 − t 2 e −2 ax dy 2 − t −1−

4 a 2 −3

(39)

dt 2 − t 2l1 dψ 2

The physical and kinematical quantities for the model (39) have the following expressions:

 3 4a2 − 3 − (1+ 4a2 − )3 2 − 4a2 − 3  1  8π = 2 ρ 4t  2 2 2 2 2   − 4a − 3 4a − 3 − (1+ 4a − )3 − 8a + 3

(40)

2

l1 + l1 + 1 − a 2 8π λ= t2 ρp =

θ=

(41)

1 2 ( 2n2 + l1 + n 2 l1 − l1 ) 8π t 2

(42)

n 2 + l1 + 2 t

σ2 =

(43)

2 2 2 1   1 1   1 n 2   1 l1   2 −  +  −  +  −   2   3 t   3 t   3 t  

(44)

V = t n2 +l1 + 2 e − ax (35) q=

and (46) 10

− (n2 + l1 + 1) n 2 + l1 + 2

where

n 2 + l1 + 1 =

1 + 4a 2 − 3 − (1 + 4a 2 − 3 ) 2 2

and

n 2 + l1 + 2 =

3 + 4a 2 − 3 − (1 + 4a 2 − 3 ) 2 2

The rest energy density, string tension density, expansion scalar and shear become infinite for t=0, which indicates that the universe starts at t=0. Hence the model (39) admits initial

σ singularity. The scalar of expansion θ → 0 as t → ∞ . Since lim 2 ≠ 0 , the model does not t →∞ θ 2

approach isotropy for large value of t. The spatial volume V is zero when t=0 and becomes infinite when t → ∞ . The deceleration parameter is negative. Therefore the model (39) is inflationary. Set:-2 The geometry of the string cosmological model corresponding to the solutions (16), (18b) and (38b) is described by the metric ds 2 = dt 2 − t 2 dx 2 − t 2 e −2 ax dy 2 − t −1−

4 a 2 −3

dt 2 − t 2l2 dψ 2

(47)

The physical and kinematical parameters for this model are obtained as

 − 3 4a2 − 3 − (1+ 4a2 − )3 2 − 4a2 − 3  1  8π = 2 ρ 4t  2 2 2 2 2  + 4a − 3 4a − 3 − (1+ 4a − )3 − 8a + 3 

(48)

2

l2 + l2 + 1 − a 2 8π λ= t2

(49) 11

ρp =

θ=

1 2 ( 2n 2 + l 2 + n 2 l 2 − l 2 ) 2 8π t

(50)

n2 + l 2 + 2 t

(51)

2 2 2 1   1 1   1 n2   1 l 2   σ = 2 −  +  −  +  −   2   3 t   3 t   3 t   2

(52)

V = t n2 +l2 + 2 e − ax (53) and

q=

− ( n 2 + l 2 + 1) n2 + l 2 + 2

(54)

The reality condition ( ρ >0) is satisfied for 3 4a 2 − 3 − (1 + 4a 2 − 3 ) 2 + 4a 2 − 3 + 8a 2 < 4a 2 − 3 4a 2 − 3 − (1 + 4a 2 − 3 ) 2 + 3

The rest energy density, the string tension density, particle density, shear scalar expansion and deceleration parameter become infinite at t=0. Therefore we observed that the model (47) admits initial singularity at t=0. The spatial volume V=0 at t=0 and V →

∞ as t → ∞ .

Case:-2 In this case we consider A = B = C = tn

, n>0

(55)

Now using equation (55) in the field equation (8) we get D ′′ 2nD ′ 3n 2 − 2n + + =0 D Dt t2

Which on integration yields

12

(56)

D =tm

(57) where

m=

− ( 2n −1) ± ( 2n −1) 2 − 4(3n 2 − 2n) 2

The value m is real if (2n −1) 2 − 4(3n 2 − 2n) ≥0. The above solutions satisfies the field equation (11) if t 2 n −2 (3n(2n −1) − a 2 ) = 0 . In this case, the geometry of the string cosmological model is described by the metric ds 2 = dt 2 − t n dx 2 − t n e −2 ax dy 2 − t n dt 2 − t m dψ 2

(58)

The rest energy density and string tension density of strings are 8π ρ=

3n(n + m) a 2 − 2n t2 t

(59) and

8π λ=

Case2.1:- If n =

3n 2 + m 2 − 2n − m + 2nm a 2 − 2n t2 t

(60)

1+ 3 1− 3 then m = . 4 4

In this case the solution can be expressed as A=B=C =t

1+ 3 4

(61) and

D =t

(62) The physical and kinematical quantities have the following expressions:

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1− 3 4

8π ρ=

3(1 + 3 ) a2 − 1+ 3 8t 2 t 2

(63) a2

8π λ= − t

1+ 3 2

(64) 8π ρp =

3(1 + 3 ) 8t 2

(65) θ=

2+ 3 2t

(66)

2 2 1 1− 3   1  1 1+ 3        σ = 3 − + − 2  3 4t  3 4t        2

(67)

V =t

2+ 3 2

e −ax

(68) and

q=

− 3

(69)

2+ 3

The rest energy density, string tension density, scalar expansion, shear and particle density become infinity for t=0. Therefore we observe that the model (55) admit initial singularity. From equations (63), (64) and (65) we observed that ρ >0, λ <0 and ρp >0. Thus at early era strings exist with negative λ but particle exist with positive ρp . The spatial volume becomes zero at t=0 and this shows the expansion of the model with time. As t → ∞ , V → ∞ , θ → 0 , A, B, C →∞ but the extra dimension D → 0 . The deceleration parameter is negative. Therefore the model is inflationary. 4. Conclusions 14

In this paper we have constructed five different five dimensional string cosmological models out of which one is physically unrealistic. Further we observed that all models are inflationary and posses initial singularity.

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