THICK DOMAIN WALLS WITH BULK VISCOSITY IN EINSTEIN ROSEN CYLINDRICAL SYMMETRIC SPACE-TIME

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1 AIJREAS VOLUME 3, ISSUE (08, FEB) (ISSN ) ONLINE THICK DOMAIN WALLS WITH BULK VISCOSITY IN EINSTEIN ROSEN CYLINDRICAL SYMMETRIC SPACE-TIME PURUSHOTTAM D. SHOBHANE Rajiv Ghi College of Engineeing Reseach, Wanadongi, Nagpu- 440, India, ABSTRACT: In this pape, we have examined thic domain walls with bul viscous fluid in Einstein-Rosen cylindical symmetic space-time in the pesence of cosmological constant Λ. Also, we have discussed the popeties of the solutions obtained. Keywods - Cylindical symmetic space-time, thic domain walls, bul viscosity. INTRODUCTION The topological defects such as domain walls, stings monopoles have an impotant ole in the fomation of ou univese. These topological defects ae (often) stable configuations of matte pedicted by some theoies to fom at phase tansitions in the vey ealy univese. Depending upon the natue of symmety beadown, vaious solutions ae believed to have fomed in the ealy univese accoding to the Higgs-Kibble mechanism. Hill et al. (989) pointed out that the fomation of galaxies is due to domain walls poduced duing a phase tansition afte the time of ecombination of matte adiation. Domain walls ae twodimensional objects that fom when a discete symmety is spontaneously boen at a phase tansition. Afte symmety beaing, diffeent egions of the univese can settle into diffeent pats of the vacuum with domain walls foming the boundaies between these egions. Domain walls have some athe peculia popeties, fo example, the gavitational field of a domain wall is epulsive athe than attactive [Padhan et al. (005), Padhan (009)]. A lot of wo has been done on domain walls. Padhan (009) evaluated geneal solutions fo plane-symmetic thic domain walls in Lya geomety in SHAILENDRA D. DEO N. S. Sc. Ats College, Bhadawati, Chapu , India, s_deo0@yahoo.in pesence of bul viscous fluid. Patil et al. (00) obtained plane symmetic cosmological models of domain wall with viscous fluid coupled with electomagnetic field in geneal elativity by using equation of state p = (γ )ρ. Yilmaz (006) obtained Kaluza-Klien cosmological solutions fo qua matte coupled to the domain wall in the context of geneal elativity by using anisotopy featue of the univese. Deo (0) studied cylindical symmetic inhomogeneous univese with thic domain walls in biometic theoy of elativity. Mahanta Biswal (0) constucted cosmological models fo domain walls coupled with qua matte in Lya geomety. Deo (0) studied spheically symmetic Kantowsi-Sachs space-time in the context of Rosen s biometic theoy with the souce matte domain walls. Deo Queshi (04) studied spheically symmetic space-time with the matte domain wall coupled with massive meson in the context of Rosen s biometic theoy of elativity. The effect of bul viscosity on cosmological evolution has been investigated by a numbe of authos in the context of geneal theoy of elativity othe paallel theoies of gavity. Humad et al. (06) investigated bul viscous fluid Bianchi-I sting cosmological model in geneal elativity by assuming the equation ξ θ = M (constant), whee ξ is the coefficient of bul viscosity θ is the scala of expansion. Shi Ram Piyana Kumai (04) pesented nonsingula Bianchi type-i V cosmological models in the pesence of 8

2 AIJREAS VOLUME, ISSUE (06, FEB) (ISSN ) ONLINE bul viscous fluid within the famewo of f(r,t) gavity theoy. Tiwai (04) obtained exact solutions fo anisotopic Bianchi-I model with bul viscosity, vaiable gavitational constant G cosmological constant Ʌ by taing vacuum density Ʌ popotional to Hubble paamete H found that Ʌ is positive deceasing function of time. Shi Ram et al. (00) pesented Hypesufacehomogeneous cosmological models containing a bul viscous fluid with time vaying G Λ, whee the bul viscosity coefficient is assumed to be a powe function of the enegy density. Shi Ram (009) investigated a spatially homogeneous anisotopic Bianchi type V model filled with an impefect fluid with both viscosity heat conduction within the famewo of Lya's geomety, whee two diffeent physically viable models of the univese ae pesented in two types of cosmologies, one with powe-law expansion othe one with exponential expansion. Cosmological model with powe-law expansion has an initial bigbang type singulaity at t = 0, wheeas the model with exponential expansion has a singulaity in the infinite past. Cylindical symmetic space-time plays an impotant ole in undesting some essential featues of the univese such as fomation of galaxies duing ealy stages of thei evolution. Katoe et al. (0) investigated cylindical symmetic Einstein-Rosen cosmological model with bul viscosity zeo-mass scala field in Lya geomety whee the cosmological models ae obtained with the help of the special law of vaiation fo Hubble s paamete poposed by Bemann (983). Katoe et al. (00) studied the solutions of Einstein field equations fo domain walls with cosmological constant heat flow in Einstein-Rosen cylindical symmetic space-time when stange qua matte nomal matte attached to the domain walls discussed some physical inematical featues of the obtained cosmological model. Mete et al. (06) obtained mesonic pefect fluid solutions with the aid of Einstein-Rosen cylindical symmetic space time investigated static vacuum model a non-static cosmological model coesponding to pefect fluid, whee the cosmological tem Λ is found to be a deceasing function of time which is suppoted by the esult found fom ecent type Ia Supenovae obsevations. Shaih (06) studied cylindical symmetic Einstein-Rosen cosmological models with linea equation of state p = aρ + b, whee a b ae constants discussed some physical geometic popeties of the model along with physical acceptability of the solutions. This pape is devoted to the study of thic domain walls with bul-viscous fluid in cylindical symmetic Einstein-Rosen space-time in the pesence of cosmological constant Λ. The pape is outlined as follows: In Section II, we have obtained Einstein field equations fo thic domain wall with bul viscous fluid in Einstein- Rosen cylindical symmetic space-time in the pesence of cosmological constant Λ. In Section III, the solutions of the field equations ae obtained fo thic domain wall with bul viscous fluid. In Section IV, the popeties of the solutions ae discussed with concluding emas.. FIELD EQUATIONS The non-static cylindical symmetic Einstein-Rosen metic [Mete Ela (06)] is given by ds = e (α β) dt d e β d e β dz, () whee ae functions of t x,,3,4 =,, z, t. Einstein's field equations in the pesence of cosmological constant ae given by R ij Rg ij + g ij = 8 G T ij. () 83

3 AIJREAS VOLUME, ISSUE (06, FEB) (ISSN ) ONLINE Hee, we shall use geometized units so that 8 G = c =. The enegy momentum tenso T ij fo thic domain wall with bul viscous fluid has the fom [Padhan (009)] T ij = ρ g ij + w i w j + pw i w j, w i w i =, (3) whee i p = p ξ w ;i = p ξθ (4) Hee ρ, p, p ξ ae the enegy density, the pessue in the diection nomal to the suface of the wall, effective pessue bul viscous coefficient espectively, w i is unit i space-lie vecto in the same diection θ = w ;i is the scala expansion. Using (3), the field equations () fo the metic () educe to e (α β) β + β 4 α = p (5) e (α β) α 44 α β + β 4 = ρ (6) e α β β β 44 + β α e (α β) β + β 4 α + α 44 β + β 4 = ρ (7) + = ρ (8) β β 4 α 4 = 0, (9) whee the suffixes 4 st fo patial deivatives w.. t. t espectively. Equations (6) (7) gives equations (6) (8) give β β 44 = 0 (0) α 44 α + β 4 α = 0 (). SOLUTIONS OF FIELD EQUATIONS Assume = A() + B(t) = ma() + nb(t). () Then (0) () give o A B + A = 0 i. e. A + A = B (3) nb ma + B ma m A + A = nb + B, (4) whee pimes dots denote odinay deivatives w.. t. t espectively. Equations () (3) give nb + B = m o m n B = B. (5) B The geneal solution of (3) is given by whee l = m n B = l ln = 0, q(> 0) (6) m n q > 0, > 0 ae the constant of integation. 84

4 AIJREAS VOLUME, ISSUE (06, FEB) (ISSN ) ONLINE Equation (9) gives A B nb = 0 o A = n. ( fo B 0) (7) The solution of equation (7) is given by A = n ln c, (8) whee c(> 0) is the constant of integation. Theefoe equations () give α = mn ln c + n l ln (9) β = n ln c + l ln (0) Theefoe α β = n(m ) ln c + (n ) ln l () Using (9) (0) in (5), (6) (8) the quantities p ρ can be witten in the fom: (0) n(n m) (α β) p = e 4 + () () (nl + ) n(n m) (α β) ρ = e 4 (3) n(m n) (α β) ρ = e 4 whee ( ) is given by equation (0). Using (3) (4), we get nl = o n = l., (4) () l = gives m = 0. m n Theefoe, using (9) - (4), we get α = ln l, (5) β = ln c (6) l Using () (3), we obtain α β = l ln c ( l)/l. (7) 85

5 AIJREAS VOLUME, ISSUE (06, FEB) (ISSN ) ONLINE Theefoe p = p ξθ = e (α β) l + (). p = e (α β) l + + ξθ (8) () ρ = e (α β) l + (), (9) whee ( ) is given by equation (7). Fom (9), it is clea that the condition ρ 0 e (α β) l + (), (30) that is, the cosmological constant should be negative. Assume that the pessue p density satisfy the baotopic equation of state [Patil et al. (00]. p = γ ρ, 0 γ. (3) Then using equation (3), equations (8) (9) give γ = e (α β) γ l + () ξθ γ. (3) Theefoe, fom (8) (9), we obtain γ p = e (α β) γ l + ξθ (33) () ρ = γ The scala expansion is given by ξθ e (α β) l + (). (34) i θ = w ;i = β α e (α β), o l + θ = e (α β), (35) l Fo > 0, we equie l (,0) i.e. < l < 0. Putting l = c., whee 0 < c <. Then using (33) (34), the quantities p ae given by γ p = e (α β) γ c + ξθ, (36) (q c t) whee ρ = γ θ = ξθ e (α β) c + (q c t) (37) c c α β = c ln c e (α β), 0 < c < (38) q c t 3. PARTICULAR CASES In most of the investigations involving bul viscosity, it is assumed that (+c )/c. (39) 86

6 AIJREAS VOLUME, ISSUE (06, FEB) (ISSN ) ONLINE ξ = ρ n, (40) whee n ae constants [Padhan (009)]. Hee, we conside two models: (I) ξ = ρ (II) ξ =, whee is constant. Model I: Let ξ = ρ. (4) Then equation (37) gives ρ = θ γ e α β c + q c t (4) fom (36), we get ρ = γ θ γ e (α β) c + (q c t) (43) whee the quantities ( ) ae given by equations (36) (39) espectively. Hee, we conside thee cases : i γ = 0 (dust distibution), ii γ =, iii γ = 4 disodeed adiation iv γ = stiff fluid. 3 Case (i) : Fo γ = 0, equations (4) (43) give p = ρ = θ e (α β) c + (q c t). (44) That is, p + = 0. Case(ii): Fo γ =, equations (4) (43) give ρ = θ e (α β) c + (q c t) (45) p = 0. (46) Case iii : Fo γ = 4, equations 4 43 give 3 That is, = 3p. p = 3 ρ = 3 θ 4 e (α β) c + (q c t) (47) Case (iv) : Fo γ =, equations (4) (43) give p = ρ = θ e (α β) c + (q c t) (48) which gives stiff fluid solution. Model II: Let ξ = (constant). Then fom equation (6) (7), we get p = γ γ e (α β) c + (q c t) θ, (49) 87

7 AIJREAS VOLUME, ISSUE (06, FEB) (ISSN ) ONLINE ρ = γ θ e (α β) c + (q c t), (50) whee the quantities ( ) ae given by equation (38) (39) espectively. Case(i): Fo γ = 0, fom (49) (50), we get p. Case(ii): Fo γ =, equations (49) (50) give p = 0, (5) ρ = γ θ e (α β) c + (q c t), (5) Case iii : Fo γ = 4, equations 5 5 give 3 p = 3 ρ = 4 θ e (α β) c + (q c t). (53) Case (iv): Fo γ =, equations (6.4.0) (6.4.) give p = ρ = θ e (α β) c + (q c t) (54) 4. CONCLUSION In this chapte, we have studied thic domain walls with bul viscous fluid in Einstein-Rosen cylindical symmetic space-time in the pesence of cosmological constant Λ. The physical quantities, spatial volume the scala expansion have the following expessions fo line element (): Spatial volume is V = g = e (α β), c c c q c t that is, V =. c Fom above expession it is clea that the spatial volume is infinite, when t zeo, when t q c. The line element () taes the fom: ds = c T c c c dt d c c c T c dϕ c c T c dz, whee T = q c t. Fom the above line element, it is clea that ou model has singulaity at finite past. Scala expansion θ is given by c c θ = K c c, whee K T > 0. This implies that θ tends to zeo as T. A netwo of domain walls acceleates the expansion of univese. Futhe it is obseved that the cosmological constant is negative. At 88

8 AIJREAS VOLUME, ISSUE (06, FEB) (ISSN ) ONLINE evey instant, the domain wall density pessue pependicula to the suface of domain wall deceases both sides of wall away fom symmety axis as T. REFERENCES [] A. Padhan, P. Vema, axiv: gqc/ v 0, 005. [] A. Padhan,, Commun. Theo. Phys. (Beijing, China) 5 (), 009, 378. [3] V. R. Patil, D. D. Pawa, A. G. Deshmuh, Romanian Repots in Physics, Vol. 6 (4), 00, 7. [4] I. Yilmaz, Gen. Rel. Gav., Vol. 38, 006, 397. [5] S. D. Deo, Intenational Jounal of Mathematical Achive, (), 0,. [6] K. L. Mahanta, A. K. Biswal, Jounal of Moden Physics, 3, 0, 479. [7] S. D. Deo, Intenational Jounal of Applied Computational Science Mathematics, (), 0, 3. [8] S.D. Deo, A. Queshi, Achives of Applied Science Reseach, 6 (4), 04, 9. [9] V. Humad, H. Naga, S. Shimali, IOSR Jounal of Mathematics, (), 06,. [0] Shi Ram, Piyana Kumai, Cent. Eu. J. Phys.,(0), 04,744. [] R.K. Tiwai, The Afican Review of Physics, 04, 9:0053. [] Shi Ram, Vema, M. K., Astophysics Space Science, 330(), 00, 5, [3] Shi Ram, Vema, M.,K., Mohd Zeyauddin, Moden Physics Lettes A, 4(3), 009, 847. [4] S. D. Katoe, A.Y. Shaih, M. M. Sancheti, J. L. Pawade, Pespacetime Jounal, 3(), 0, 83. [5] M. S. Beman, Nuovo Cimento, 74B,983, 8. [6] S. D. Katoe, R. S. Rane, K. S. Wanhade,, Int. J. Theo. Phys., 49, 00, 99. [7] V. G. Mete, V. D. Ela, Intenational Jounal of Astonomy Astophysics, 6 (), 06, Aticle ID: 655,6 pages [8] A.Y. Shaih, The Afican Review of Physics, 06, :000 89