Nonexistence of Plane Symmetric Micro Model in Presence of Cosmological Constant in Barber s Modified Theory of General Relativity

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1 International Journal of Adanced Research in Physical Science (IJARPS) Volume, Issue, February 05, PP -0 ISSN (Print) & ISSN (Online) Nonexistence of Plane Symmetric Micro Model in Presence of Cosmological Constant in Barber s Modified Theory of General Relatiity R.C.Sahu Department of Mathematics, K.S.U.B. College, Bhanjanagar, Odisha, India rcsahu@rediffmail.com S.P.Misra Department of Mathematics, Sri Jagannath Mahaidyalaya, Rambha, Ganjam, Odisha sibaprasada_misra@rediffmail.com Ritarani Swain Department of Physics, U.P.S.C. College, Sheragada, Ganjam, Odisha, Abstract: In Barber s modified theory of general relatiity (Gen. Rel. Gra., 7(98)), the homogeneous and anisotropic plane symmetric space-time in presence of micro matter field with cosmological constant- is considered. It is shown that the cosmological constant generate solutions and models only when it is positie. But the models generated by cosmological constant are not of micro models. Some physical and geometrical features of the models are also studied. PACS: 0.50.Kd Keywords: Barber s modified theory of general relatiity, micro matter field, cosmological constant-. INTRODUCTION Einstein s theory of general relatiity seres as a basis for constructing mathematical models of the unierse though the theory has some controersies and lapses. To oercome these controersies and lapses, arious alternatie and modified theories of it hae been proposed by authors from time to time to unify graitation and matter fields in arious forms. So in an attempt to produce continuous creation theory, Barber (98) modified the Einstein s theory of general relatiity to a ariable G-theory, which creates the unierse out of self contained graitational and matter fields in arious forms. This theory predicts local effects which are within the obserational limits and the scalar field does not graitate directly but simply diides the matter tensor acting as a reciprocal to graitational constant G. The micro matter fields represented by scalar meson fields represent matter fields with spin less quanta. There are two types of scalar fields iz. zero rest mass scalar fields and massie scalar fields. The zero rest mass scalar fields describe long range interactions, whereas massie scalar fields describe short range interactions. The study of scalar meson fields in general relatiity has a focal point for the researchers due to its physical importance in particle physics. The micro matter field in relatiistic mechanics yields some significant results as regards to the singularities. The micro matter field being a field of a single ariable (say) is the special case of general field and the expression gien by T ij i j k gij k m ARC Page

2 R.C.Sahu et al. is the energy-momentum tensor of Yukawa (95) fields (spin zero meson particle) for the metric of (+) signature in flat space time. The Klein-Gorden equation takes the form g ij ; ij m 0 where is the real scalar field and m is the rest-mass of scalar meson field. Here (;) semicolon followed by an index denotes coariant differentiation. When m = 0, the scalar field is known as massless scalar field or micro matter field. The cosmological constant has been introduced in 97 by Einstein to modify his own equations of general relatiity. Now this term remains a focal point of interest in the context of quantum field theories, quantum graity, super graity theories, Kaluza-Klein theories and in the inflationary unierse scenario. A number of obserations suggest that the unierse possess a nonzero cosmological constant (Krauss and Turner,(995)). The cosmological term which is a measure of the energy of empty space, proides a repulsie force opposing the graitational pull between the galaxies. If it exists, then the energy it represents counts as mass because Einstein has shown that mass and energy are equialent. Moreoer, if the cosmological constant is large alue then the energy inoled due to the matter in the unierse, can sum up to the number that inflation predicts. In inflationary era it is ery important to study on the role of cosmological constant. But recent research suggests that the cosmological costant corresponds to a ery small alue of the order 0-58 cm - (Jhori and Chandra, (98)). Some of the authors those hae studied second self-creation theory in arious angles taking different space times in presence of different graitating fields are Pimentel (985), Soleng (987a,b), Venkateswarlu and Reddy(990), Shanti and Rao(99), Caralho(996), Shri Ram and Singh(998),Mohanty et al.(000,00,00,00-05),panigrahi and Sahu(00,00,00), Sahu and Panigrahi(00), Sahu and Mohanty (006), Sahu and Mahapatra (009a,b) and recently Sahu et al.(00,0). But to our knowledge none of the authors has studied the plane symmetric space-time in the context of second self-creation theory when the graitational field in presence of cosmological constant is a micro matter field represented by massless scalar field. So in the present paper we hae considered this problem to study the role of (cosmological constant) for deriing mesonic solutions.the work presented in this paper is treated as an extension work of Sahu and Panigrahi(00).. FIELD EQUATIONS IN BARBER S MODIFIED THEORY OF GENERAL RELATIVITY We consider the homogeneous and anisotropic plane symmetric metric in the form where A and B are functions of cosmic time t. The field equations in Barber s modified theory of general relatiity for micro matter field represented by massless scalar field with cosmological term gij may be written as Gij Rij gijr gij 8 T and where ; k ; k ij is inariant D Alembertian, T is the trace of the energy-momentum tensor T ij, is the Barber s scalar and is the coupling constant to be ealuated from the experiment. The measurements of the deflection of light restricts the alue of coupling to. In the limit 0 this theory approaches the Einstein s general relatiity theory in eery respect. () () () The energy momentum tensor taken as T ij for micro matter field representing massless scalar field is International Journal of Adanced Research in Physical Science (IJARPS) Page 5

3 Nonexistence of Plane Symmetric Micro Model in Presence of Cosmological Constant in Barber s Modified Theory of General Relatiity together with where is the massless scalar field, is the source density of massless scalar field and (;) is the coariant differentiation. Also and are both function of cosmic time.here the equation () connects the distribution of matter and energy with geometry of the space-time by relating energy Tij to the fundamental metric tensor and its deriaties. It is the business of momentum tensor relatiistic mechanics to inestigate with the help of this equation, the principle which goern the energy momentum tensor and hence determines the behaior of energy and matter. By using eqn. () the field equations () and () for the metric () are obtained as A A B B - - -, A AB B A A - - A A -, (7) A B A B (9) where the subscript denotes the ordinary differentiation with respect to time. Adding two times of eqn. (6) with eqn.(7) and three times of eqn.(8), we obtain ( A B) ( A B) 0. SOLUTIONS AND MODELS In order to find exact and explicit solutions of A and B from four field equations (6) to (9), we consider the following three cases : (i) = 0 (ii) > 0 (iii) < 0 Case i: When = 0. In this case the result reduces to that of already studied by Panigerahi and Sahu (00). Case ii: When > 0. In this case equation (0) after integration yields A B e e () t t where and are constants of integration. From eqn. (), we can write the explicit form of A and B as t A e e and t B e e where t t n n are real constants and satisfies the relation n n. () () (5) (6) (8) (0) () International Journal of Adanced Research in Physical Science (IJARPS) Page 6

4 R.C.Sahu et al. Here the oer determinacy for determining three unknowns A and B from three field eqns. (6)-(8) can be settled by actual substitution of the alues of A and B from eqn.() in eqn.(8). Thus we obtain As n i, i =, are real constants, so also n () n n is a real constant. But this relation cannot be hold good in eqn. () as its L.H.S part is constant but R.H.S part is a function of t. To make the relation consistent in eqn. (), we take the following two sub cases: Sub case. When 0, = 0 and =. (5a, b, c) By use of (5), eqn () reduces to n nn. (6) Thus from eqn. (5c), we obtain = α, (7) where α is the constant of integration. Soling () and (6), we find n n. (8) Now use of (8) and (5b), eqn. () yields t A B e (9). Use of (9) and (7) in eqn. (9) and then integrating, we get = t. +, (0) where 0 and are the constants of integration. The energy density associated with (Anderson,(967) is gien as. () The source density σ of the scalar field gien by eqn.(5) is () Using eqn.(7) in () and () separately, we obtain 0 () and σ = 0. () Thus the corresponding metric of our solution can be expressed as (5) International Journal of Adanced Research in Physical Science (IJARPS) Page 7

5 Nonexistence of Plane Symmetric Micro Model in Presence of Cosmological Constant in Barber s Modified Theory of General Relatiity Sub case. when = 0, 0 and By use of (6), eq n () reduces to n =. (6a,b,c) n n. (7) From (6a, b, c), (7), () and () as soled in preious case-, we obtain t = α, n n A B e. (8a,b,c) Use of (8 a,c) in eqn. (9) and then integrating, we get and = t. + (9) where 0 and are the constants of integration. As in sub case-, here also we can find 0 and σ = 0. Thus the geometry of our unierse for space time () can be written as (0) Case iii. When < 0, say =. In this case eqn. (0) reduces to A B ( A B) 0 Soling (),we find () A B cos t sin t () where and are the constants of integration. From (), A and B can be expressed in explicit form as A t t m ( cos sin ), B ( cos t sin t) m () where and are real constants such that =. () Use of () in eqn. (8), we get In eqn. (5), L.H.S is a real constant but R.H.S is a function of time though st term of R.H.S can be considered as constant. Thus the relation in (5) is not possible and hence the solution cannot be determined. So for < 0, the micro model of the unierse does not exist in Barber s second self creation theory. (5) International Journal of Adanced Research in Physical Science (IJARPS) Page 8

6 R.C.Sahu et al.. SOME PHYSICAL AND GEOMETRICAL FEATURES OF THE MODELS Case i: Here we intended to study the following properties of the model (5).. The spatial olume in the model is gien by V g / = ( A B) e. t Now V as t 0 and V as t. Thus the model of the unierse starts expanding from a constant olume and becomes infinite large or blows up at infinite future.. The scalar expansion i V u; i, V where V is the olume element and and the anisotropy are defined by (Raychoudhuri, (955) as g, g, g, g, g, g, σ. g g g g g g Here the scalar expansion and anisotropy in the model are found as 0 and σ 0. Thus Thus it is eident from the aboe result that the unierse is expanding in nature with constant rate of expansion. Also the unierse is non-shearing and isotropic throughout the eolution.. The massless scalar field and Barbers scalar φ are gien by (7) and (0) respectiely. Thus a constant as t 0 and a constant as t subject to the condition 5 0 or 5 = 0. So Barber's scalar has no singularity in this model. As the massless scalar field in this model is found to be constant, so micro model of the unierse does not exist.. The Kretschmann curature inariant defined by L = R hijk R hijk, where R hijk is the Riemann curature tensor. Here L is found to be L ( t e ) Now L a constant as t 0 and L 0 as t. Since L is constant, the result confirms that the model has no geometrical singularity. 5. As the Hubble s parameter H is constant, so the model is of steady state model. Case ii: Here we intended to study the following properties of the model (0).. The spatial olume in the model is gien by V. t e Here V as t 0 and V 0 as t. Thus the model of the unierse starts expanding from a constant olume but olume of the unierse gradually decreases as time increases. So at infinite future, olume of the unierse becomes zero and the unierse may collapse.. As in case-, here also the scalar expansion and anisotropy in the model are found as 0 and σ 0. Thus. Thus it is eident from the aboe result that the expansion of the unierse is contracting in nature and isotropic throughout the eolution.. The massless scalar field and Barbers scalar φ are gien by (7) and (9) respectiely. As in case- here also the micro model of the unierse does not exist.. The Kretschmann curature inariant is found to be International Journal of Adanced Research in Physical Science (IJARPS) Page 9

7 Nonexistence of Plane Symmetric Micro Model in Presence of Cosmological Constant in Barber s Modified Theory of General Relatiity L 5 t.( e ). L a constant as t 0 and L 0 as t. Since L is constant, the result confirms that the model has no geometrical singularity. 5. As the Hubble s parameter H is constant, the model gien by (0) is of steady state model. 5. CONCLUSION In this paper we hae described the plane symmetric space-time in modified theory of general relatiity in presence of massless scalar field with cosmological constant as source matter. It is obsered that plane symmetric cosmological models of the unierse exist only when the cosmological constant is positie. The model found in subcase- is expanding in nature, isotropic throughout the eolution, steady state and free from geometrical singularity. Howeer the model found in sub case- is contracting in nature, isotropic throughout the eolution, steady state and free from geometrical singularity. It is interesting that massless scalar field does not surie in both the models. Hence micro model of the plane symmetric unierse does not exist in modified theory of general relatiity. REFERENCES Barber, G.A.:98, Gen. Rel. Gra., 7-6. Yukawa, H.:95, Modern Physics (edited by J.C.Slater), Asian Student Edition, McGraw Hill Book Inc., 0. Krauss, L.M. and Turner, M.S.:995, Gen. Rel.Gra., 7, 7. Jhori, V.B. and Chandra, R.: 98,J. Math. Phys.Sci., 7, 7. Pimentel, L.O.:985, Astrophys. & Space Sci. 6, 95. Soleng, H.H.:987a, Astrophys. & Space Sci. 7, 7. Soleng, H.H.:987b, Astrophys. & Space Sci. 9,. Venkateswarlu, R. and Reddy, D.R.K.:990, Astrophys. & Space Sci. 68, 9. Shanti, K. and Rao, V.U.M.:99, Astrophys. & Space Sci. 79, 7. Caralho, J.C.:996, Int. Jour. Theor. Phys., 5, 09. Shri Ram and Singh, C.P.:998, Astrophys. & Space Sci. 57,. Mohanty, G.,Mishra, B.and Das, R.:000, Bull. Inst. Math. Aca. Sinica (ROC), 8, -50. Mohanty, G., Panigrahi, U.K. and Sahu, R.C.:00, Astrophys. & Space Sci.8, Mohanty, G.,Sahu, R.C. and Panigrahi,U.K.:00,Astrophys. & Space Sci., 8, Mohanty, G., Sahu, R.C. and Panigrahi, U.K.:00-05, Orissa Math.Soc., -, 9-0. Panigrahi, U.K. and Sahu, R.C..: 00, National Aca. Sc.Lett, 5, No. and,-9. Panigrahi, U.K. and Sahu, R.C.:00, Theo. and Appl.Mech. 0, No., Panigrahi, U.K. and Sahu, R.C.:00, Czech. Jr. Phys., 5, No.5, Sahu, R.C. and Panigrahi, U.K.:00, Astrophys. & Space Sci. 88, Sahu, R.C. and Mohanty, G.:006, Astrophys. & Space Sci. 06,79-8. Sahu, R.C. and Mahapatra, L.K.:009a, Jour.Int.Aca.Phy.Sci.,, No.,9-7. Sahu, R.C. and Mahapatra, L.K.:009b, Jour.Indi.Aca.Math,,07-6. Sahu, R.C., Mahapatra,L.K.and Mohanty,G.:00,Romanian Reports in Phys.,6, 9-6. Sahu,R.C.; Nayak,B.K.and Mishra,S.P.:0,The Jour. Ind. Aca. Math., No., Anderson, J.:967, Principle of Relatiity Physics, Academic Press, New York, p-89. Raychaudhuri, A.K.:955,.Physical Re., 98,. International Journal of Adanced Research in Physical Science (IJARPS) Page 0