Plane gravitational waves with mesonic perfect fluid in bimetric relativity

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1 Avalable onlne atwww.scholarsresearchlbrary.com Scholars Research Lbrary Archves of Aled Scence Research, 015, 7 (8):-31 (htt://scholarsresearchlbrary.com/archve.html) ISSN X CODEN (USA) AASRC9 Plane gravtatonal waves wth mesonc erfect flud n bmetrc relatvty Sulbha R. Sule 1 and S. D. Deo 1 Deartment of Mathematcs, Rashtrasant Tukadoj Maharaj Nagur Unversty, Nagur, Inda Deartment of Mathematcs, N. S. Scence and Arts College, Bhadrawat, Dst-Chandraur, (M.S.), Inda ABSTRACT In ths aer, we wll study Z = tye lane gravtatonal waves wth erfectflud and scalar meson matter feld resectvely. We observed that massve scalar feld couled wth erfect flud n lane gravtatonal waves doesnot exst n bmetrc theory of gravtaton formulated by Rosen.Only a vacuum model can be constructed. Keywords: - Plane gravtatonal waves, Scalar Meson feld, Perfect flud, Bmetrc Relatvty AMS Code-83C05 (General relatvty) INTRODUCTION A new theory of gravtaton called the Bmetrc theory of gravtaton,was roosed by Rosen[1][]to modfy the Ensten s general theory of relatvty by assumng two metrc tensors,vz.,a Remannan metrc tensor g j and a background metrc tensor j.the metrc tensor g j determnes the Remannan geometry of the curved sacetme whch lays the same role as gven n the Ensten s general relatvty and t nteracts wth matter. The background metrc tensor nertal forces.ths metrc tensor j refers to the geometry of the emty(free from matter and radaton) unverse and descrbes the t nteracts wth g j but not drectly wth matter.onecan regard were no matter.in the absence of matter one would have g j = j has no drect hyscal sgnfcance but aears n the feld equatons.therefore j j as gvng the geometry that would exst f there.moreover,the bmetrc theory also satsfed the covarance and equvalence rncles:the formaton of general relatvty. The theory agrees wth the resent observatonal facts ertanng to general relatvty.thus at every ont of sace-tme there are two lne elements: ds = g dx dx (1.1) j j And dσ = dx dx (1.) j j Where ds s the nterval between two neghborng events as measured by means of a clock and a measurngrod. Scholars Research Lbrary

2 Sulbha R. Sule et al Arch. Al. Sc. Res., 015, 7 (8):-31 The nterval dσ s an abstract or geometrcal quantty not drectly measurable. One can regard t as descrbng the geometry that would exst f no matter were resent. Plane gravtatonal waves are usually dscussed as a secal case of the well-establshed lane fronted gravtatonal waves wth arallel rays, the so called - waves.the method of secalzaton s qute techncal, e.g. the curvature tensor must be comlex recurrent wth a recurrence vector whch s collnear wth a real null vector. H Takeno (191) [3] roounded a rgorous dscusson of lane gravtatonal waves, defnedvarous terms by formulatng a meanngful mathematcal verson and obtaned numerous results. A farly general case of lane" gravtatonal wave s reresented by the metrc ds = Adx Ddxdy Bdy dz + dt (1.3) both for weak feld aroxmaton and for exact solutons of Ensten feld equatons.reformulatng Takeno s (191) [3] defnton of lane wave, we wll use here, Z = t tye lane gravtatonal waves by usng the lne elements, 3At ds = x dx + y dy + z dz Bdu Cdv + Adt (1.4)Mohsen, Tucker and Wang [4] have ( ) ( ) studed the moton of snnng test artcles n lane gravtatonal waves. S Kessar, D Sngh et al [5], analyzed the moton of electrcally neutral massve snnngtest artcle n thelane gravtatonal and electromagnetc wave background. The theory of lane gravtatonal waves have beenstuded by many nvestgators Takeno []; Pandey [7]; Lal and Shafullah [8];Lu Huqng [9]; Bond, H.et.al.[10],Torre,C.G.[11]; Hogan, P.A.[1];Deo and Ronghe[13],[14] ;Deo and Sule [15],[1],[17]and they obtaned the solutons. In ths aer,we wll study Z = tye lane gravtatonal wave wth macro and mcro matter feldcouledwth erfect flud and wll observe the result n the context of Bmetrc theory of relatvty.. FIELD EQUATIONS IN BIMETRIC RELATIVITY: Rosen N. has roosed the feld equatons of Bmetrc Relatvty from varatonrncle as j j 1 j j K = N N g = 8π κ T (.1) j 1 α β h j Where N = g g h α g = det (g ), = det( ) (.4) and j j (.) N N α α β =, Where a vertcal bar ( ) denotes a covarant dfferentaton wth resect to. j And, T the energy momentum tensor for macro matter feld lke Perfect fluds gven by j κ g = (.3) Scholars Research Lbrary 7

3 Sulbha R. Sule et al Arch. Al. Sc. Res., 015, 7 (8):-31 T =( ) j j j j j j T = u u g flow vector of the flud havng and ρ asroer ressure and energy densty resectvely. ρ + (.5) together wth 1 g u u = where u s the 3. Z = tye lane gravtatonal wave wth Perfect Flud: For Z = lane gravtatonal waves, we have the lne element 3 A t d s = x d x + y d y + z d z B d u C d v + A d t (3.1) ( ) ( ) Where A = A(Z), B = B(Z), C = C(Z) and Z = Corresondng to the equaton (3.1), we consder the lne element for background metrc as j ( ) d σ = d x + d y + d z + d u + d v + d t Usng equatons (.1) to (.5) wth (3.1) and (3.), We get the feld equatons as A A 1 B B 1 C C (3.3) + + 1πκ = A A B B C C A A 1 B B 1 C C (3.4) D + 1πκ = A A B B C C A A 1 B B 1 C C (3.5) A A 1 B B 1 C C + = 1πκ = π κ ρ A A B B C C A A B B C C (3.)where t ( ) D and = ( ) A A B B C C A =, A =, B =, B =, C =, C = Z Z Z Z Z Z Usng equaton (3.3) to (3.), we get + ρ = 0 (3.7) Ths equaton of state s known as false vacuum.in vew of realty condtons > 0, ρ > 0 Equaton (3.7) mmedately mles that = 0, ρ = 0.ematter feld lke erfect flud does not exst n Z = t (3.). lane gravtatonal waves n Rosen s Bmetrc theory of relatvty. Hence for vacuum case = 0 = ρ, the feld equaton reduced to A A 1 B B 1 C C (3.8) = A A B B C C A A 1 B B 1 C C D + = 0 (3.9) A A B B C C as Scholars Research Lbrary 8

4 Sulbha R. Sule et al Arch. Al. Sc. Res., 015, 7 (8):-31 A A 1 B B 1 C C (3.10) + 0 = A A B B C C Solvng equatons (3.8) to (3.10), we have K Z A = K e (3.11) 1 = K 4 Z 3 (3.1) B K e C K e K Z = (3.13) 5 where K 1, K, K 3, K 4, K 5 and K are the constants of ntegraton.thus substtutng the value of (3.11) and (3.13) n (3.1), we get the vacuum lne element as KZ 3K1e t K4Z KZ KZ ds = x dx + y dy + z dz K 3e du K5e dv + K1e dt (3.14) ( ) ( ) Thus, t s found that n lane gravtatonal wavez =, the macro matter feld lke erfect flud does not survve n Bmetrc theory of relatvty and only vacuum model can be constructed. 4.Z = tye lane gravtatonal wave wth Scalar Meson Feld: In ths secton, we consder the regon of the sace-tme flled wth massve scalar feld whose energy momentum tensor s gven by s j j j 1 j k T = T = V V, g ( V kv,,, m V ), (4.1) together wth σ = g V + where m s the mass arameter and σ s the source densty of the meson j j ; m V, feld.here afterwards the suffx(,) and semcolon (;) after a feld varable reresent ordnary and covarant dfferentaton wth resect to x and g res. j Usng equatons (.1) to (.5) wth (3.1) and (3.) wth energy momentum tensor (4.1) are obtaned as A A 1 B B 1 C C (4.) D + + 8πκ = ( V m V ) A A B B C C A A 1 B B 1 C C (4.3) D + 8πκ = ( V m V ) A A B B C C A A 1 B B 1 C C (4.4) D + 8πκ = ( V m V ) A A B B C C A A 1 B B 1 C C (4.5) + + 8πκ = ( V + m V ) A A B B C C Usng (4.) and (4.5), we get 1 π κ V = 0 e V = 0 e V = constant (4.) Scholars Research Lbrary 9

5 Sulbha R. Sule et al Arch. Al. Sc. Res., 015, 7 (8):-31 Thus for the sace-tme(3.1) the Scalar Meson feld wth or wthout mass arameter does not survve n Bmetrc theory of relatvty.in both cases source densty becomes constant. 5.Coulng of Scalar Meson Feld wth Perfect Flud: The energy momentum tensor for a mxture of erfect flud and scalar meson feld together s gven by j j j T = T + T (5.1) s By the use of co-movng co-ordnate system,the feld equaton (.1) to (.4) for the metrc (3.1) and (3.) corresondng to the energy momentum tensor (5.1) can be wrtten as A A 1 B B 1 C C 1 D + + 8πκ V = + m V A A B B C C (5.) A A 1 B B 1 C C 1 D + 8πκ V = + m V A A B B C C (5.3) A A 1 B B 1 C C 1 D + 8πκ V = + m V A A B B C C (5.4) A A 1 B B 1 C C 1 D + + 8πκ ρ V = + + m V A A B B C C (5.5) Usng (5.) and (5.5),we obtan ( ) ρ + + V = 0 (5.) In vew of the realty condtons.e. > 0, ρ > 0, the above equaton mles that 0, 0 and V= constant. CONCLUSION In the study of Z = tye lane gravtatonal waves, there s nl contrbuton of MesoncPerfect flud n Bmetrc theory of relatvty resectvely.it s observed that the matter felds ether massve scalar feld or erfect flud cannot be a source of gravtatonal feld n the Rosen s Bmetrc theory but only vacuum model exsts.the concluson arrved at vz.,v = constant, σ = constant, = 0, ρ = 0 are nvarant statements and hold n all coordnate systems even though we have derved these n co-movng coordnate system. Acknowledgement The authors are thankful to Dr. R. D. Gr, Prof. Emertus, and P.G.T.D.(Mathematcs), R. T. M. N. U., Nagur, Inda for hs constant nsraton. REFERENCES [1] N.Rosen, Phys. Rev. 1940, 57, 147. [] N.Rosen, Rela. Grav. 1973, 04, [3] H.Takeno,The mathematcal theory of lane gravtatonal waves n General Relatvty.Scentfc reort of Research Insttute fortheoretcal Physcs, Hroshma Unversty, Hroshma, Ken, Jaan (191). Scholars Research Lbrary 30

6 Sulbha R. Sule et al Arch. Al. Sc. Res., 015, 7 (8):-31 [4] M.Mohsen,;R. W.Tucker, ; C.Wang, Quantum Grav.001, 18, [5] S.Kessar, ;D.Sngh, et al,. Quant.Grav.,00, 19, [] H.Takeno, Prog.Theor.Phys., 1958,0, 7-7 [7] S. N. Pandey,. Theo. Math. Phys. 1979,39, [8] K. B.Lal, ;Shafullah,,On lane wave solutons of non symmetrc feld equatons of unfed theores of Ensten Bonner and Schrödnger. Annal de Mathematcaedura Alcata.,1980,1, [9] Lu Huqung, Astronomy Astrohys , [10] H.Bond; F.A.E.Pran, and Robnson, I. Proc. Roy.Soc.Lond.A3, 1959, 5, [11] C.G.Torre, Gen. Rela. Grav.,00, 38,53- [1] P.A.Hogan, Math. Proc. Roy. Irsh Acad. 1999, 99A, [13] A.K.Ronghe and S.D.Deo, JVR, 011,, 1-11 [14] A.K. Ronghe and S.D.Deo, Internatonal Journal of Mathematcal Archve- (3) Mar [15]S.D.Deo and S.R.Sule, Asan Journal of current Engneerng and Maths,013,,, [1]S.D. Deo and S.R.Sule, Mathematca Aeterna, Vol. 3, 013, no., [17]S.D.Deo and S.R.Sule, Internatonal Journal of Mathematcs Trends and Technology,014, 8,1,51-55 Scholars Research Lbrary 31