A Comparative Study between Einstein s Theory and Rosen s Bimetric Theory through Perfect Fluid Cosmological Model

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1 Internatonal Journal of dvanced Research n Physcal Scence (IJRPS) Volume, Issue 5, May 05, PP -9 ISSN (Prnt) & ISSN (Onlne) www. arcjournals. org omparatve Study between Ensten s Theory and Rosen s metrc Theory through Perfect Flud osmologcal Model P.N.Samant Department of Mathematcs, erhampur Unversty, erhampur, Odsha, Inda dr.pns.math@gmal.com R..Sahu Department of Mathematcs, K.S.U.. ollege, hanjanagar, Odsha, Inda rcsahu@redffmal.com.ehera Department of Mathematcs, U.N.ollege, Soro, alasore, Odsha, Inda. benudharbhr@gmal.com bstract: Spatally-homogeneous and ansotropc anch type- space tme s nvestgated n Ensten s theory of general relatvty and ts alternatve theory Rosen s [Gen. Rel. Grav., vol. (97)5] bmetrc theory of relatvty, when source of the gravtatonal feld s perfect flud. onsderng gamma law equaton of state, false vacuum models of the unverse are determned n general relatvty whch are sotropc, nflatonary and de-stter unverses. However, the same false vacuum model of the unverse does not survve n Rosen s bmetrc theory of relatvty but only vacuum model of the unverse exsts. PS: 0.0,-q ; 0.50.Kd; 0.50,-h Keywords: false vacuum, perfect flud, gamma law, general relatvty, bmetrc theory. INTRODUTION.. Ensten s Theory of General Relatvty General theory of relatvty developed by Ensten s the only coordnate nvarant theory whch lad foundaton for constructng mathematcal models of the unverse. The feld equatons n general theory of relatvty gven by Ensten are R j ½ g j R = -8 T () j where the unts are chosen such that G= = and R j s the Rcc tensor, R s the Rcc scalar, g j s the metrc tensor and T j s the Energy momentum tensor of the matter... Rosen s metrc Theory of Relatvty It s known that most of the cosmologcal models based on general theory of relatvty developed by Ensten contan ntal sngulartes (the bg-bang) from whch the unverse expands. ut ths theory has some controverses and lapses for whch authors have proposed varous alternatve and modfed theores of t to unfy gravtaton and matter felds n varous forms. Thus to get rd of sngulartes n the sad cosmologcal models, Rosen [] proposed a new theory of relatvty known as bmetrc theory of relatvty.ths theory conssts of two metrc tensors at each pont of the space tme whose role s to determne the physcal stuatons. The frst metrc tensor g j determnes the Remannan geometry of the curved space tme whch plays the same role as n general relatvty and t nteracts wth matter. The back ground metrc tensor γ j refers to the geometry of the empty unverse and descrbes the nertal forces. lso t has no drect physcal R Page

2 R..Sahu et. al. sgnfcance but appears n the feld equatons. Moreover, t nteracts wth g but not drectly j wth matter. One can regards γ j as gvng the geometry that would exst f there were no matter. Ths theory satsfes the covarant and equvalence prncples and agrees wth the theory of general relatvty up to the accuracy of observatons made tll the date. The spatally homogeneous and ansotropc cosmologcal models have sgnfcant role n the descrpton of the Unverse n the early stages of ts evoluton. lso a perfect flud satsfactorly descrbes the dstrbuton of matter due to large-scale dstrbuton of galaxes n our unverse. Therefore, we consder to nvestgate the anch type- homogeneous model of the Unverse wth ansotropc background n presence of perfect flud correspondng to Ensten s theory and Rosen s bmetrc theory. The basc am of comparatve study between both the theores of gravtaton s to determne the percentage of resemblances of the models constructed n each theory wth the physcal unverse. The feld equatons of bmetrc theory of gravtaton proposed by Rosen [] are N j Nδ j = 8πkT j where () N j = ab (g h g hj a ) b and N = N j, (, j=,,, ) ; k = g together wth g = determnant of g j and = determnant of γ j. Here the vertcal bar ( ) denotes the covarant dfferentaton wth respect to γ j and T j energy momentum tensor of the matter. s the. SPE TIME ND PERFET FLUID DISTRIUTION The anch type- space tme descrbed by ds = -dt + dx + dy + dz (5) wth,, as functons of cosmc tme t. Ths ensures that the space-tme s ansotropc and spatally homogenous. The energy momentum tensor for perfect flud dstrbuton for the space-tme (5) s gven by T j = (ρ+p)u u j + pg j (6) where ρ, p and u are respectvely the energy densty, pressure and unt flow vector of the flud satsfyng u u = -. (7) For the sake of smplfcaton of feld equaton and to get the vable soluton we consder the gamma law equaton of state as p= (γ-)ρ, 0 γ. (8). EINSTEIN FIELD EQUTIONS ND SOLUTIONS Usng co-movng coordnate system, Ensten feld equatons () correspondng to eqn. (6) and (7) for the metrc (5) take the followng explct forms : - 8 p, (9) - 8 p, (0) Internatonal Journal of dvanced Research n Physcal Scence (IJRPS) Page 5

3 omparatve Study between Ensten s Theory and Rosen s metrc Theory through Perfect Flud osmologcal Model - 8 p () and 8. () Here and afterwards the subscrpt after a feld varable represents ordnary dfferentaton wth respect to tme t. For the metrc (5), the energy conservaton equaton of general relatvty T j j 0 () ; takes the form ( p )( ) 0. Takng γ = 0, equaton (8) reduces to the form p + ρ = 0 whch s known as false vacuum or de-generate vacuum (lome and Prster[]). Usng eqn.(5) and addng eqns.(9),(0)and()wth three tmes of eqn.(),we get () (5) ( ). (6) pplyng relaton (5) n equaton () and then ntegratng, we obtan K = -p, (7) where K( 0) s the constant of ntegraton. On substtuton (7), equaton (6) reduces to ( K, (8) where ) K K. On ntegraton, (8) yelds ] K ( ) [( ) K (9) where K s a constant of ntegraton. For exact ntegraton of (9), put K = 0 and then ntegratng we fnd K t K e, (0) whch can be expressed n the form K t K n K t K n K t K n ( e ), ( e ), ( e ) () where n,=,, are real constants satsfyng the condton n. () Here the over determnacy for determnng the feld varables, and from the feld equatons (9) to () can be settled by actual substtuton of the solutons() n the feld equatons.thus we obtan Internatonal Journal of dvanced Research n Physcal Scence (IJRPS) Page 6

4 R..Sahu et. al., j n n j. () j Now equatons () and () yeld an explct relaton n and n n n. () Now subject to restrcton (), eqn. () yelds the admssble soluton K K = = = ( t e ). (5) Therefore the metrc (5) correspondng to the soluton (5) can be wrtten as ds Hence the spatally-homogeneous ansotropc anch-type- model reduces to spatallyhomogeneous, sotropc and false vacuum model n Ensten s theory. It s nterestng to note that the cosmologcal model (6) s a de-stter unverse and hence an Ensten space... Some Physcal and Geometrcal spects of the Model The physcal and knematcal parameters nvolved n the models (6) are as follows: The energy densty and pressure are gven by ρ = ( p) = 0. Snce both energy densty and pressure are ndependent of tme, so the model has no sngularty at t=0 and the space tme reduces to a flat space tme. The spatal volume s found to be V=( g) K K = = e t. Now V a constant as t 0 and V as t. Thus t s nferred that the model starts wth a fnte volume and expands contnuously wth tme. s the model has exponental expanson, the expanson n the unverse never ends, whch was earler suggested by Wllem destter of Leydon. gan the space tme s flat space tme and has exponental expanson, so the model of the unverse obtaned s nflatonary-unverse. lso the unverse undergoes strongly frst-order phase transton. s the unverse super cools nto a false vacuum phase, the false-vacuum energy densty acts as an effectve cosmologcal constant whch trggers an epoch of de-stter (exponental) expanson. The magntude of scalar expanson θ s gven by θ =u ; dt = V V =K (a constant). Hence the model has fxed expanson throughout the evoluton of the unverse. The shear scalar σ s calculated as σ = σ j σ j = 0 and hence σ = 0. Ths ndcates that the model reman sotropc and non-shearng throughout the evoluton. Lm t ς θ e K ( dx = 0 onfrms that the model of the unverse s pont wse sotropc (Szafron, 977). The generalzed mean Hubble parameter t K dy H = (H + H + H )= ( + + )= K ( a constant). dz ). s H s found to be constant and not a functon of tme,so the model s a steady state model. (6) Internatonal Journal of dvanced Research n Physcal Scence (IJRPS) Page 7

5 omparatve Study between Ensten s Theory and Rosen s metrc Theory through Perfect Flud osmologcal Model The scale factor n the model s gven by S K K = = e t. Hence S ncreases as tme ncreases. n mportant observatonal quantty s the deceleraton parameter q whch can be defned as q = - VV V =-. The sgn of q ndcates whether the model nflates or not. The +ve sgn of q corresponds to the standard deceleratng model, whereas the negatve sgn of q ndcates nflaton. s the result found here s q=-, so the model corresponds to an nflatonary model of the unverse. The rotaton ω gven by the vortcty tensor ω j as ω = ω j ω j. Here ω = 0 mples ω =0.Snce the rotaton of the model found to be zero, the model s nonrotatng n nature. The Kretshmann curvature nvarant L n model s found to be L= 5 7 K. e K t+ K. So when t 0, L a constant and when t, L 0.The above results confrms that the models posses no geometrcal sngulartes at t=0. We know that when the Rcc tensor R j s proportonal to the metrc tensor g j the space tme s called Ensten space(petrov,969). Here R j = ¼R g j s true so the space tme s an Ensten space. It s found that ρ θ =0.Ths ndcates that the model approaches homogenety n a pont wse (Szafron, 977).. ROSEN S FIELD EQUTIONS ND SOLUTIONS The background flat space -tme correspondng to the metrc (5) s dς = dt + dx + dy + dz. (7) y use of co-movng coordnates and equatons (6) and (7), Rosen s feld equatons () for the metrcs (5) and (7) can be wrtten as ( ) ( ) ( ) = 6πkp, (8) ( ) ( ) + ( ) = 6πkp, (9) ( ) + ( ) ( ) = 6πkp, (0) ( ) + ( ) + ( ) = 6πkρ. () From feld equatons (8) to (0), we have ( ) = ( ) = ( ). () ddng eqns.(9) and (0), we get ( ) = 6πkp, () Now use of eqn.() and ()n eqn.(),we can fnd ρ + p = 0. () On substtuton ρ = p from (5) n eqn.(), we get Internatonal Journal of dvanced Research n Physcal Scence (IJRPS) Page 8

6 R..Sahu et. al. p = 0 and ρ = 0. (5) s p=0 and ρ = 0, so the anch type- cosmologcal model n presence of perfect flud do not exst n bmetrc theory. Usng eqn.(5) n eqn.()and then puttng the value n eqn.(),the metrc potental are found as === e K 8t, (6) where K 8 s the constant of ntegraton. Thus n vew of eqn.(6),the metrc (5) takes the form ds = dt + e K 8t dx + dy + dz. (7) Thus t s observed that anch type- cosmologcal perfect flud model does not survve n bmetrc theory but only the vacuum model of the unverse exsts. It s nterestng that the model (7) s spatally homogeneous, sotropc and has no sngularty at t=0. 5. ONLUSION It s observed that cosmologcal false vacuum model exst n Ensten s theory whereas vacuum model only survves n Rosen s bmetrc theory. The model whch found n Ensten s theory s unformly expandng, non-rotatng, non-shearng, sotropc and has no geometrcal sngularty. s Edwn Hubble and M.L.Humason (mercan astronomers) who after studyng the red shft (see Doppler effect) n the spectral lnes of the dstant galaxes have concluded that the unverse s expandng, thus our nvestgaton and result found here reveals that Ensten theory s more vable than Rosen s bmetrc theory. It s also nterestng to conclude that as the model n Rosen s bmetrc theory does not admt sngularty whch s of physcal nature, so Rosen s am n developng hs own theory has been fulflled n ths artcle. REFERENES []. G..arber: Gen. Rel. Grav., 7(98). []. N.Rosen: Gen. Rel. Grav., 5(97). []. J.J. lome and W. Prester: Naturewssenshaften, 7, 58 (98). []. H.H.Soleng: strophys. Space Sc. 7, 7(987). [5]. H.H.Soleng: cta Physca, Hungarca, 8 (988). [6]. D..Szafron,.: Inhomogeneous osmologes: New exact solutons and ther evoluton. J.Math.Phys., 8, 67(977). [7]..Z.Petrov,.: Ensten Spaces, Pergamon Press(969). [8].. Enten.:nn.Physk, 9,769(96). Internatonal Journal of dvanced Research n Physcal Scence (IJRPS) Page 9