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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 ahoomaku@r 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. Copyright © 2010 SciRes. JMP 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. Copyright © 2010 SciRes. JMP 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. Copyright © 2010 SciRes. JMP 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 0tfor 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. 7. References [1] C. H. Brans and R. H. Dicke, “Mach’s Principle and a Relativistic Theory of Gravitation,” Physical Review A, Vol. 124, No. 3, 1961, pp. 925-935. [2] K. Nordverdt, Jr., “Post Newtonian Metric for a General Class of Scalar—Tensor Gravitational Theories and Ob- servational Consequences,” The Astrophysical Journal, Vol. 161, 1970, pp. 1059-1067. [3] R. V. Wagoner, “Scalar—Tensor Theory and Gravita- tional Waves”, Physical Review D, Vol. 1, No. 2, 1970, pp. 3209-3216. [4] D. K. 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