Bianchi Type I Magnetized Cosmological Model in Bimetric Theory of Gravitation

Transcription

1 Avalable at Appl. Appl. Math. ISSN: Vol. 05 Issue (December 00) pp (Prevously Vol. 05 Issue 0 pp ) Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) Banch Type I Magnetzed osmologcal Model n Bmetrc Theory of Gravtaton M. S. Borkar Post Graduate Department of Mathematcs Mahatma Jotba Fule Educatonal ampus Amravat Road R. T. M. Nagpur Unversty Nagpur 0033 (Inda) borkar.mukund@redffmal.com; borkar.mukund@yahoo.com S. S. haran Department of Mathematcs Late K. Z. S. Scence ollege Bramhan (kalmeshwar) 50 Ds. Nagpur Inda sscharan@redffmal.com Receved: September 9 00; Accepted: November Abstract In ths paper we attempt to construct a Banch Type I magnetzed cosmologcal model n Rosen s bmetrc theory of gravtaton by usng the technques of Leteler and Stachel under the A where n 0 n Rosen s feld equatons. The physcal and geometrcal sgnfcance of the model are dscussed. It s mportant to note the added dmenson of ths paper nclude the ntroducton of a magnetc feld on the model. Pror to ths all t has been s a vacuum model n bmetrc gravtaton. condton that n B Keywords: Bmetrc theory; cosmc strng; magnetc feld; Banch Type I MS (00) No.: F05 563

2 56 M.S. Borkar and S.S. haran. Introducton In the early phase of the unverse there s no defnte evdence that the present day unverse (FRW unverse) was of the same type. Therefore t s mportant to study dfferent Banch Type models n the context of the early phase of the unverse. In ths regards we plan to study varous Banch Type models n bmetrc theory of gravtaton. EarlerBorkar and haran (009 00) have developed the models lke Banch Type I strng dust cosmologcal model wth magnetc feld n bmetrc relatvty (009) The charged perfect flud dstrbuton n bmetrc theory of relatvty (00) LRS Banch Type I strng dust magnetzed cosmologcal models n bmetrc theory of relatvty (00) and Banch Type I bulk vscous flud strng dust cosmologcal model wth magnetc feld n bmetrc relatvty (00). Several new theores of gravtaton have been formulated whch are consdered to be alternatves to Ensten s theory of gravtaton. The most mportant among them s Rosen s (977) bmetrc theory of gravtaton. The Rosen s bmetrc theory s the theory of gravtaton based on two metrcs. One s the fundamental metrc tensor g descrbes the gravtatonal potental and the second metrc refers to the flat space tme and descrbes the nertal forces assocated wth the acceleraton of the frame of reference. The metrc tensor g determne the Remannan geometry of the curved space tme whch plays the same role as gven n Ensten s general relatvty and t nteracts wth matter. The background metrc refers to the geometry of the empty unverse (no matter but gravtaton s there) and descrbe the nertal forces. The metrc tensor has no drect physcal sgnfcance but appears n the feld equatons. Therefore t nteracts wth wth matter. One can regard g but not drectly as gvng the geometry that would exsts f there were no matter. In the absence of matter one would have g =. Moreover the bmetrc theory also satsfed the covarance and equvalence prncples; the formaton of general relatvty. The theory agrees wth the present observatonal facts pertanng to general relatvty [ For detals one may refer Karade (980) Katore and Rane (006) and Rosen (97 977)]. Thus at every pont of space tme there are two metrcs ds g dx dx () d dx dx () The feld equatons of Rosen s (97) bmetrc theory of gravtaton are N N 8 kt (3)

3 AAM: Intern. J. Vol. 05 Issue (December 00) [Prevously Vol. 05 Issue 0 pp ] 565 where N pr s g g s p r k g N N together wth g det( g ) and det( ) stands for covarant dfferentaton and T s the energy momentum tensor of matter felds.. Here the vertcal bar Several aspects of bmetrc theory of gravtaton have been studed by Rosen (97 977) Karade (980) Israelt (98) Katore and Rane (006) Khadekar and Tade (007). In partcular Reddy and Venkateswara Rao (998) have obtaned some Banch Type cosmologcal models n bmetrc theory of gravtaton. The purpose of Rosen s bmetrc theory s to get rd of the sngulartes that occur n general relatvty that was appearng n the bg bang of the cosmologcal model; ths has rekndled a lot of nterest and study of n cosmologcal models related to Rosen s bmetrc theory of gravtaton. In bmetrc theory accordng to Rosen (97) the background metrc tensor should not be taken as descrbng an empty unverse but one n whch the cosmologcal prncples hold. Hence Rosen proposed that the metrc be taken as the metrc tensor of exactly ths knd of unverse. In accordance wth ths prncple the large scale structure of the unverse s deally assumed to the same aspect from everywhere n space and at all tmes. Ths prncple s not qute vald on a small scale structure due to rregulartes n the matter dstrbuton and s equally nvald on large scale structure due to evoluton of the matter. We however assume the dal case of bndng cosmologcal prncple. It does not apply to g and the matter n the unverse but to the metrc. Hence descrbes a space tme of constant curvature. In the context of general relatvty cosmc strngs do not occur n Banch Type models (see K. D. Kror et al. (99)). In t some Banch Type cosmologcal models two n four and one n hgher dmensons are studed by Kror et al (99). They show that the cosmc strngs do not occur n Banch Type V cosmology. Bal and Dave (003) Bal and Upadhaya (003) Bal and Sngh (005) Bal and Pareek (007) have all nvestgated Banch Type IX I and V strng cosmologcal models under dfferent physcal condtons n general relatvty. Ra Bal and Anal (006) have nvestgated Banch Type I magnetzed strng cosmologcal model n general relatvty by ntroducng the condton A B n where n 0 n Ensten feld equatons whereas Ra Bal and Umesh Kumar Pareek (007) have deduced Banch Type I strng dust cosmologcal model wth magnetc feld n general relatvty by mposng the condton A N B n where n 0 and N s proportonalty constant n Ensten feld equatons. Further M. S. Borkar and S. S. haran (009) have examned Banch Type I strng dust

4 566 M.S. Borkar and S.S. haran cosmologcal model wth magnetc feld n bmetrc relatvty under the condton n where n 0 and N s proportonalty constant n Rosen s feld equatons. Researchers lke Bal et al. (003006) Pradhan et al. (007) Wang (00 (006) Rothore et al. (008) Pradhan (009) Borkar and haran (00900) developed the models n the feld of bulk vscous flud solutons and Banch Type strng models whch are the most useful as well as n bmetrc theory of gravtaton. In order to acheve our magnetzed model n bmetrc theory of gravtaton we have freely emphaszed the technque and symbols of Bal et al. and Borkar et al. (00). A N B. Solutons of Rosen s Feld Equatons We consder Banch Type I metrc n the form ds d t A dx B dy dz () where A B and are functons of t alone. Here B = otherwse we get LRS Banch Type I model. The flat metrc correspondng to metrc () s d dt dx dy dz (5) The energy momentum tensor T for the strng dust wth magnetc feld s taken as T ( p ) p g x x E (6) wth x x (7) 0. (8) x In ths model and denote the rest energy densty and the tenson densty of the system of strngs respectvely p s the pressure s the flow vector and x the drecton of strngs. The electromagnetc feld E s gven by Lchnerowcz (967) E h g h h (9)

5 AAM: Intern. J. Vol. 05 Issue (December 00) [Prevously Vol. 05 Issue 0 pp ] 567 The four velocty vector s gven by g (0) and s the magnetc permeablty and h s the magnetc flux vector defned by h g kl kl F () where F kl s the electromagnetc feld tensor and kl s the Lev vta tensor densty. Assume the comovng coordnates and hence we have 3 0. Further we assume that the ncdent magnetc feld s taken along x axs so that h 0 h h h 0. 3 The frst set of Maxwell s equaton F k 0 () yelds F 3 = constant = H (say). Due to the assumpton of nfnte electrcal conductvty we have F F F 0. 3 The only non vanshng component of F s F 3. So that h AH B (3) and h H. () B

6 568 M.S. Borkar and S.S. haran From equaton (9) we obtan E E E 3 3 E H B (5) Equaton (6) of energy momentum tensor yeld H 3 H H T p T T3 p T. B B B (6) The Rosen s feld equatons (3) for the metrc () and (5) wth the help of (6) gves A B A B H 6 AB p A B A B B A B A B H 6 AB p A B A B B A B A B H 6 AB p A B A B B A B A B H 6 AB A B A B B (7) (8) (9) (0) da db d where A B etc. dt dt dt From equatons (8) and (9) we obtan B B. () B B Equatons (7) and (8) leads to B A A B K 6 AB B A A B B () where H K.

7 AAM: Intern. J. Vol. 05 Issue (December 00) [Prevously Vol. 05 Issue 0 pp ] 569 Equatons (0) and () after usng strng dust condton [Zel dovch (980)] lead to B 3 A 3 A B 6 A K. B A A B B (3) Equatons (7) to (0) are four equatons n sx unknowns A B and p and therefore to deduce a determnate soluton we assume two condtons. Frst s that the component of shear tensor s proportonal to the expanson whch leads to n A B where n 0 () and second s Zel dovch condton. Usng the frst condton [equaton ()] n equaton (3) we wrte B B (5) B B n ( 3 n) (3n ) (3n ) (3n ) 6 K( B). Now the equaton () rewrte as B B B B B B (6) whch on ntegratng we get B LB (7) where L s the constant of ntegraton. B Assume B v then B v. In vew of these relatons equaton (7) v becomes v L. v (8) Now equaton (5) after usng () and assumptons B B and v leads to v v v v n 3n 3n 6 K. (9)

8 570 M.S. Borkar and S.S. haran The expressons (8) and (9) yeld 3 n K (30) 3n whch reduces to n d f 3 K f d 3n (3) where f ( ). The dfferental equaton (3) has soluton f P 3 K 3 nn ( ) n (3) where P s the constant of ntegraton. From equaton (8) we wrte Ld log v log b (33) n 3 K P 3 nn ( ) where b s the constant of ntegraton. Now usng f ( ) and expresson (3) the metrc () wll be d ds dx v dy dz n n 3 K v P 3 nn ( ) (3) where v s determned by equaton (33). After sutable transformaton of co ordnates T x X y Y z Z.e. d dt dx dx dy dy dz dz.

9 AAM: Intern. J. Vol. 05 Issue (December 00) [Prevously Vol. 05 Issue 0 pp ] 57 Above metrc can be re wrtten as dt T ds T dx T v dy dz n n 3 KT v PT 3 nn ( ). (35) Ths s the Banch Type I magnetzed cosmologcal model n bmetrc theory of gravtaton. 3. Observatons The energy densty the strng tenson densty the pressure p for the model (35) are n K n 0 (36) 3n T K p. (37) 3 T The strong energy condtons of Hawkng and Ells (973): p 0 and 0 p must be satsfed by our model (35). For ths model (35) these condtons are specfcally and 0 or n n (38) n or n whch contradct the assumpton that n 0. The expanson gven by A B A B becomes ( n) f T for the model(35) whch [after usng (3)] reduces to n 3 KT ( n) P. 3 nn ( ) (39)

10 57 M.S. Borkar and S.S. haran The components of shear tensor are gven by 3 A A B B or n n 3 K T P (0) 3 3n( n ) L () 3 L 3 () 0. (3) In the absence of magnetc feld.e. for K = 0 we have 0 and p 0 () n P (5) n P 3 n 6 P L 3 n L 3 P (6) Geometrcal and Physcal Sgnfcance From equatons (36) and (37) t s seen that the pressure p the rest energy densty and the strng tenson densty n the model (35) are all negatve showng that such a Banch Type I magnetzed cosmologcal model (35) does not exst n bmetrc theory of gravtaton under the condton A = (B) n where n > 0 n Rosen s feld equatons.

11 AAM: Intern. J. Vol. 05 Issue (December 00) [Prevously Vol. 05 Issue 0 pp ] 573 The strong energy condtons p 0 and 0 leads to p stated by Hawkng and Ells (973) and n 0 or n n or n from whch t s observed that n s negatve for whch the strong energy condtons of Hawkng and Ells (973) satsfes. But ths volet the assumng condton n > 0 n A = (B) n and therefore t s concludes that the strong energy condtons of Hawkng and Ells (973) does not satsfed for n > 0. From the results (39) () t s concludes that the expanson and shear of the model (35) are magnary when P < 0 whch support the non exstence of the model n the presence of magnetc feld. In the absence of magnetc feld.e. for K = 0 we have p 0 and therefore we get vacuum model and there s no expanson and no shear n the model for constants P and L. Hence our models (35) strongly suggest a vacuum unverse wthout expanson and shearng and no magnetc feld snce forcng t to be n magnetc feld eopardzed ts very exstence. Acknowledgement The Authors would lke to convey ther sncere thanks and grattude to the referees. REFERENES Bal R. and Anal (006). Banch Type I magnetzed strng cosmologcal model n general relatvty Astrophys. Space Sc Bal R. and Dave S. (003). Banch Type -IX strng cosmologcal models wth bulk vscous flud n general relatvty Astrophys. Space Sc Bal R. and Pareek U.K. (007). Banch Type I strng dust cosmologcal model wth magnetc feld n general relatvty Astrophys. Space Sc Bal R. and Sngh D. (005). Banch type V bulk vscous flud strng dust cosmologcal model n general relatvty Astrophys. Space Sc Bal R. and Upadhaya R. D. (003). LRS Banch Type I strng dust magnetzed cosmologcal models Astrophys. Space Sc Borkar M. S. and haran S. S. (009) Banch Type I strng dust cosmologcal model wth magnetc feld n bmetrc relatvty IJAM (3) 5-56 Bulgara.

12 57 M.S. Borkar and S.S. haran Borkar M. S. and haran S. S. (00). The charged perfect flud dstrbuton n bmetrc theory of relatvty (n press) J. Ind. Acad. math 3 No.. Borkar M. S. and haran S. S. (00). LRS Banch Type I strng dust magnetzed cosmologcal models n bmetrc theory of relatvty Tensor N.S. 7() Borkar M. S. and haran S. S. (00). Banch Type I bulk vscous flud strng dust cosmologcal model wth magnetc feld n bmetrc theory of gravtaton An Int. J. AAM 5() Hawkng S. W. and Ells G.F. R. (973). The Large Scale Structure of Space-Tme ambrdge Unversty Press ambrdge p. 9. Isrelt M. (98). Sphercally symmetrc felds n Rosen s bmetrc theores of gravtaton general relatvty and gravtaton 3(7) Karade T. M.(980) Sphercally symmetrc space tmes n bmetrc relatvty theory I Indan J. Pure-appl. Math (9) Katore S. D. and Rane R. S. (006) Magnetzed cosmologcal models n bmetrc theory of gravtaton Pramana J. Phys. 67() Khadekar G. S. and Tade S. D. (007). Strng cosmologcal models n fve dmensonal bmetrc theory of gravtaton Astrophys. Space Sc. DoI 0.007\S Kror K. D. houdhur T. and Mahanta. R. (99). Strng n some Banch Type cosmologes General Relatvty and Gravtaton 6(3) Leteler P. S. (979). louds of strngs n general relatvty Physcal revew D 0(6)) Leteler P. S. (983). Strng cosmology Physcal revew D8 (0) -9. Lchnerowcz A. (967) Relatvstc hydrodynamcs and Magnetohydrodynamcs Benamn Newyork p. 3. Rathore G.S. Bagora A. and Gandh S.(008). A magnetc ttled homogenous cosmologcal model wth dsordered radatons. Adv. Studes Theor. Phys. No Reddy D. R. K. and Rao N. V. (998). On Some Banch Type cosmologcal models n bmetrc theory of gravtaton Astrophys. Space Sc Rosen N. (97). A Theory of Gravtaton Annls of Phys Rosen N. (977). A Topcs n theoretcal and experental gravtaton physcs edted by V. D. Sabtta and J. Weber. Plenum press London Stachel John (980). Thckenng the strng I The strng perfect dust Phy. Revew D (8) 7-8. Zel dovch Y. B. (980). osmologcal fluctuatons near sngularty Mon. Not. R. Astron. Soc