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Vol.2, No.5, 484-488 (2010) Natural Science http://dx.doi.org/10.4236/ns.2010.25060 Copyright © 2010 SciRes. OPEN ACCESS Higher dimensional bianchi type-V universe in creation-field cosmology Kishor S. Adhav*, Shivdas D. Katore, Abhijit S. Bansod, Prachi S. Gadodia Department of Mathematics, Sant Gadge Baba Amravati University, Amravati, India; *Corresponding Author: ati_ksadhav@yahoo.co.in Received 16 December 2009; revised 6 March 2010; accepted 15 March 2010. ABSTRACT We have studied the Hoyle-Narlikar C-field cos- mology with Bianchi type-V non static space- time in higher dimensions. Using methods of Narlikar and Padmanabham [1], the solutions have been studied when the creation field C is a function of time t only as space time is non static. The geometrical and physical aspects for model are also studied. Keywords: Bianchi Type-V Space-Time, Creation Field Cosmology, Cosmological Model of Universe, Higher Dimensions 1. INTRODUCTION The study of higher dimensional physics is important because of several prominent results obtained in the de- velopment of the super-string theory. In the latest study of super-strings and super-gravity theories, Weinberg [2] studied the unification of the fundamental forces with gravity, which reveals that the space-time should be dif- ferent from four. Since the concept of higher dimensions is not unphysical, the string theories are discussed in 10- dimensions and 26-dimensions of space-time. Because of this, many researchers are inspired to study the higher dimensional to explore the hidden knowledge of the uni- verse. Chodos and Detweller [3], Lorentz-Petzold [4], Ibanez and Verdaguer [5], Gleiser and Diaz [6], Banerjee and Bhui [7], Reddy and Venkateswara [8], Khadekar and Gaikwad [9], Adhav et al. [10] have studied the multi-dimensional cosmological models in general rela- tivity and in other alternative theories of gravitation. The three important observations in astronomy namely the phenomenon of expanding universe, primor- dial nucleon-synthesis and the observed isotropy of cosmic microwave background radiation (CMBR) were supposed to be successfully explained by big-bang cos- mology based on Einstein’s field equations. However, Smoot et al. [11] revealed that the earlier predictions of the Friedman-Robertson-Walker type of models do not always exactly meet our expectations. Some puzzling results regarding the red shifts from the extra galactic objects continue to contradict the theoretical explana- tions given from the big bang type of the model. Also, CMBR discovery did not prove it to be a out come of big bang theory. Infact, Narlikar et al. [12] have 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 [13], Bondi and Gold [14] 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 model. But the principle of conservation of matter was violated in this formalism. To overcome this difficulty Hoyle and Narlikar [15] adopted a field theo- retic approach by introducing a mass less and charge less 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 singu- larity as in the steady state theory of Bondi and Gold [14]. A solution of Einstein’s field equations admitting radiation with negative energy mass less scalar creation fields C was obtained by Narlikar and Padmanabhan [1]. The study of Hoyle and Narlikar theory [15-17] to the space-time of dimensions more than four was carried out by Chatterjee and Banerjee [18]. The solutions of Ein- stein’s field equations in the presence of creation field have been obtained for Bianchi type-V universe in four dimensions by Singh and Chaubey [19]. Here, we have considered a spatially homogeneous and anisotropic non static Bianchi type-V cosmological model in Hoyle and Narlikar C-field cosmology with five dimensions. Therefore, we have assumed that the creation field C is a function of time t only i.e. K. S. Adhav et al. / Natural Science 2 (2010) 484-488 Copyright © 2010 SciRes. OPEN ACCESS 485 485 tCtxC ,. This study is important because of the fact that the resulting cosmological model is considered to be ame- nable to the model obtained by Singh and Chaubey [19]. 2. HOYLE AND NARLIKAR C-FIELD COSMOLOGY Introducing a mass less scalar field called as creation field namely C-field, Einstein’s field equations are mo- dified. Hoyle and Narlikar [15-17] field equations are ij c ij m ijij TTRgR 8 2 1 (1) where ij mT is matter tensor of Einstein theory and ij cT is matter tensor due to the C-field which is given by k k ijjiij cCCgCCfT 2 1 (2) where 0f and i ix C C . Because of the negative value of 0 0000 TT , 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 j ji j ijc j ijmCfCTT ;;; (3) i.e. matter creation through non-zero left hand side is possible while conserving the over all energy and mo- mentum. Above equation is similar to 0 j i ij C ds dx mg (4) which implies that the 4-momentum of the created parti- cle 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 i i i iJCf ;; and i i iv ds dx J , where is homogeneous mass density. 3. METRIC AND FIELD EQUATIONS The five-dimensional Bianchi-Type-V line element can be written as 22 2 4 22 2 3 22 2 2 2 2 1 22 dueadzeadyeadxadtds mxmxmx (5) where 1 a, 2 a, 3 a and 4 a are functions of t only and m is constant. Here the extra coordinate is taken to be space like. The above space time is non static, hence, we have assumed that creation field C is function of time t only i.e. tCtxC , and ppppdiagT i j m ,,,, (6) We have assumed that velocity of light and gravita- tional constant are equal to one unit. Now, the Hoyle-Narlikar field Eq.1 for metric (5) with the help of Eqs.2, 3, and 6 can be written as 2 2 1 2 43 43 42 42 32 32 41 41 31 31 21 21 2 1 8 6 Cf a m aa aa aa aa aa aa aa aa aa aa aa aa (7) 2 2 1 2 43 43 42 42 32 32 4 4 3 3 2 2 2 1 8 3 Cfp a m aa aa aa aa aa aa a a a a a a (8) 2 2 1 2 43 43 41 41 31 31 4 4 3 3 1 1 2 1 8 3 Cfp a m aa aa aa aa aa aa a a a a a a (9) 2 2 1 2 42 42 41 41 21 21 4 4 2 2 1 1 2 1 8 3 Cfp a m aa aa aa aa aa aa a a a a a a (10) 2 2 1 2 32 32 31 31 21 21 3 3 2 2 1 1 2 1 8 3 Cfp a m aa aa aa aa aa aa a a a a a a (11) 4 4 3 3 2 2 1 1 3a a a a a a a a (12) C a a a a a a a a CCf p a a a a a a a a 4 4 3 3 2 2 1 1 4 4 3 3 2 2 1 1 (13) where dot )( indicates the derivative with respect to t. From Eq.12, we get 432 3 1aaaa (14) Assume that V is a function of time t defined by 4321 aaaaV (15) From Eqs.14 and 15, we get K. S. Adhav et al. / Natural Science 2 (2010) 484-488 Copyright © 2010 SciRes. OPEN ACCESS 486 4 1 1Va (16) Above Eq.13 can be written in the form VCV dV d VCfpV dV d (17) In order to obtain a unique solution, one has to specify the rate of creation of matter-energy (at the expense of the negative energy of the C-field). Without loss of gen- erality, we assume that the rate of creation of matter en- ergy density is proportional to the strength of the exist- ing C-field energy-density. i.e. the rate of creation of matter energy density per unit proper-volume is given by VgCpV dV d2222 (18) where is proportionality constant and we have de- fined VgVC . Substituting it in Eq.17 , we get Vg dV d VfgpV dV d (19) Comparing right hand sides of Eqs.18 and 19, we get Vg f gV dV d Vg 2 2 (20) Integrating , which gives 1 1 2 f VAVg (21) where 1 A is arbitrary constant of integration. We consider the equation of state of matter as p (22) Substituting Eqs.21 and 22 in the Eq.18, we get 12 2 1 2 2 f VAV dV d (23) which further yields 12 2 2 1 2 2 12 f V f A (24) From Eq.22, we get 12 2 2 1 2 2 12 f V f A p (25) Subtracting Eq.8 from Eq.9, we get 0 4 4 3 3 2 2 1 1 2 2 1 1 2 2 1 1 a a a a a a a a a a a a a a a a dt d (26) Now, from Eqs.15 and 26, we get 0 2 2 1 1 2 2 1 1 V V a a a a a a a a dt d Integrating, which gives V dt xd a a 11 2 1exp , 1 d=constant, 1 x=constant (27) Subtracting Eq.9 from Eq.10, we get 0 3 3 2 2 3 3 2 2 V V a a a a a a a a dt d Integrating, we get V dt xd a a 22 3 2exp , 2 d=constant, 2 x=constant (28) Subtracting Eq.10 from Eq.11, we get 0 4 4 3 3 4 4 3 3 V V a a a a a a a a dt d which on integration gives V dt xd a a 33 4 3exp , 3 d=constant, 3 x=constant (29) Subtracting Eq.8 from Eq.11, we get 0 4 4 1 1 4 4 1 1 V V a a a a a a a a dt d Integrating, we get V dt xd a a 44 4 1exp , 4 d= constant, 4 x= constant (30) where 3214 dddd , 3214 xxxx and 4321 aaaaV . Using Eqs.27, 28, 29 and 30, the values of ta1, ta2, ta3 and ta4 can be written explicitly as V dt XVDta 1 41 11 exp (31) V dt XVDta 2 41 22 exp (32) V dt XVDta 3 41 33 exp (33) K. S. Adhav et al. / Natural Science 2 (2010) 484-488 Copyright © 2010 SciRes. OPEN ACCESS 487 487 V dt XVDta 4 41 44exp (34) where the relations 1 4321 DDDD and 21 XX 0 43 XX are satisfied by 4,3,2,1iDi and 4,3,2,1iX i. From Eqs.10 and 31, we get 1 1 D and 0 1X. Adding Eqs.8, 9, 10, 11 and 4 times Eq.7, we get p a m aa aa aa aa aa aa aa aa aa aa aa aa a a a a a a a a 3 3212 2 2 1 2 43 43 42 42 41 41 13 13 32 32 21 21 4 4 3 3 2 2 1 1 (35) From Eq.15, we have 43 43 42 42 41 41 13 13 32 32 21 21 4 4 3 3 2 2 1 1 2aa aa aa aa aa aa aa aa aa aa aa aa a a a a a a a a V V (36) From Eqs.35, 36 and 22, we get 1 3 3212 2 1 2 a m V V (37) Substituting Eq.16 in Eq.37, we get 1 3 3212 21 2 V m V V (38) Substituting Eq.24 in Eq.38, we get 12 2 22 21 2 2 12 1 3 3212 f V f A V m V V (39) which further gives t kVmV f fA dV f 1 2322 2 2 16 123 132 2 (40) where 1 k is integration constant. For 1 (Zeldovich fluid or Stiff fluid) and 0 1 k, the above equation gives 44tmV (41) Substituting Eq.28 in Eq.21, we get 1414 1 22 ff tmAg (42) Also, from equation VgVC , we get 34 2 3414 1 22 f tmA C ff (43) Substituting Eq.41 in Eq.24, the homogeneous mass density becomes 1818 2 1 22 2 1ff tfmA (44) Using Eq.25 and 1 , pressure becomes 1818 2 1 22 2 1ff tfmAp (45) From Eqs.38 and 39, it is observed that for 2 f, there is no singularity in density and pressure. Using Eq.41 in Eqs.31, 32, 33 and 34, we get mtta 1 (46) 34 2 22 1 3 exp tm X mtDta (47) 34 3 33 1 3 exp tm X mtDta (48) 34 4 44 1 3 exp tm X mtDta (49) 4. PHYSICAL PROPERTIES The expansion scalar is given by H4 t 4 (50) The mean anisotropy parameter is given by 4 1 4 1 i i H H A 68 2 4 2 3 2 2 4tm XXX (51) The shear scalar 2 is given by 22 4 1 2 2 2 4 4 2 1AHHH ii 88 2 4 2 3 2 2 2tm XXX (52) The deceleration parameter q is given by 1 1 Hdt d q=0 (53) where HHH ii and H is the Hubble parameter. K. S. Adhav et al. / Natural Science 2 (2010) 484-488 Copyright © 2010 SciRes. OPEN ACCESS 488 For large t, the expansion scalar and shear scalar tend to zero. Further, if 2 f, for large t, the model re- duces to the vacuum case. 5. CONCLUSIONS In this paper, we have considered the space-time geome- try corresponding to Bianchi type-V in Hoyle-Narlikar [15-17] creation field theory of gravitation with five di- mensions. Bianchi type-V universe in creation-field cos- mology has been investigated by Singh and Chaubey [19] whose work has been extended and studied in five di- mensions. An attempt has been made to retain Singh and Chaubey’s [19] forms of the various quantities. We have noted that all the results of Singh and Chaubey [19] can be obtained from our results by assigning appropriate values to the functions concerned. REFERENCES [1] Narlikar, J.V. and Padmanabhan, T. (1985) Creation-field cosmology: A possible solution to singularity, horizon, and flatness problems. 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