Exact Solutions of Axially Symmetric Bianchi Type-I Cosmological Model in Lyra Geometry

Transcription

1 IOSR Joura of ed Physcs (IOSR-JP) e-issn: Voume 5, Issue 6 (Ja. ), PP -5 ac Souos of ay Symmerc ach Tye-I Cosmooca Mode Lyra Geomery. sar, M. sar earme of Mahemacs, Norh-aser H Uversy, Permae Camus, Sho- 79, Mehaaya, Ida. bsrac: I hs aer we have obaed aay symmerc ach ye-i cosmooca modes for erfec fud dsrbuo he coe of Lyra s mafod. ac souos of he fed equaos are obaed by assum he easo he mode s rooroa o he shear. Ths eads o he codo ad are scae facors ad ( ) s a cosa. Some emaca ad hysca arameers of he mode have bee dscussed. The souos are comabe wh rece observaos. Keywords: ach ye modes, Cosmooy, Lyra eomery. I. Iroduco se roosed hs eera heory of reavy, whch ravao s descrbed erms of eomery ad movaed he eomerzao of oher hysca feds. Oe of he aem hs dreco was made by Wey [] who roosed a more eera heory whch ravao ad eecromaesm s aso descrbed eomercay. u hs heory was o acceed as was based o o-eraby of eh rasfer. Laer Lyra [] roduced a aue fuco.e. dsaceme vecor Remaa sace-me whch removes he o-eraby codo of a vecor uder arae rasor. Ths modfed eomery was amed as Lyra s eomery. I couao of vesaos, Se [] ad Se ad u [] roosed a ew scaar-esor heory of ravao. They cosruced a aao of he se s fed equaos based o Lyra eomery. Haford [5] oed ou ha he cosa dsaceme vecor fed Lyra s eomery ays he roe cosmooca cosa he orma eera reavsc reame. I s show by Haford [6] ha he scaaresor reame based o Lyra s eomery redcs he same effecs, wh observaoa ms as he se s heory. Recey, severa auhors [7-] suded cosmooca modes based o Lyra s eomery varous coes. Se ad Vasoe [], hamra [], Karade ad orar [], Kayashe ad Wahmode [], Reddy ad Iaah [5], Reddy ad Veaeswaru [6] have suded varous cosmooca modes Lyra s eomery wh a cosa dsaceme fed. However, hs resrco of he dsaceme fed o be cosa s merey oe of coveece ad here s o a ror reaso for. Soe [7] has oed ou ha he cosmooes based o Lyra s mafod wh cosa aue vecor w eher cude a creao fed ad be equa o Hoye s creao fed cosmooy [8-] or coa a seca vacuum fed, whch oeher wh he aue vecor erm may be cosdered as a cosmooca erm. I he aer case he souos are equa o he eera reavsc cosmooes wh a cosmooca erm. eesham [] cosdered FRW modes wh me deede dsaceme fed. He has show ha by assum he eery desy of he uverse o be equa o s crca vaue, he modes have eomery. ay symmerc cosmooca modes have bee suded boh Remaa ad Lyra eomeres. I coe of eera reavy heory, by ada he comov coordae sysem, hese modes wh sr dus coud source are suded by haacharaya ad Karade []. They show ha some of hese modes are suar free eve a a a eoch. I he coe of Lyra s eomery hese modes are suded he resece of cosmc source ad hc doma was [] ad he resece of erfec fud dsrbuo []. The urose of hs wor s o aayse eera feaures of aay symmerc ach ye-i cosmooca modes wh me deede dsaceme vecor he framewor of Lyra eomery. Ths aer s orazed as foows: I seco II, we dscuss he merc ad fed equaos. I seco III, we dscuss souos of he fed equaos. I seco IV, we dscuss some hysca ad emaca arameers of he mode. Fay, seco V, dscusso ad cocud remars are ve. Pae

2 ac Souos of ay Symmerc ach Tye-I Cosmooca Mode Lyra Geomery Pae II. The Merc ad Fed equaos We cosder aay symmerc ach ye-i sace -me ) ( dz dy d d ds, () ), ) are he cosmc scae facors. The fed equaos orma aue for Lyra s mafod, as obaed by Se [] are T R R, () s he dsaceme vecor fed defed as ),,, (. Here ), G 8 ad oher symbos have her usua mea as Remaa eomery. We ae a erfec fud form for he eery momeum esor u u T ) (, () ad are he eery desy ad ressure of he cosmc fud resecvey oeher wh comov coordaes u u, ) (,,, u. The fed q. () oeher wh () for he merc () reduces o, (), (5). (6) The eery coservao equao T eads o ) ( (7) ad R R. (8) q. (8) eads o. (9) q. (9) s auomacay sasfed for,,. For, q. (9) eads o, () whch eads o. () Here do deoes dffereao wh resec o cosmc me.

3 ac Souos of ay Symmerc ach Tye-I Cosmooca Mode Lyra Geomery III. Souos of he Fed quaos qs. ()-(6) are hree equaos fve uows vz.,,, ad. I order o oba ec eac souos, we assume ha whch eads o, ( ) s a cosa. () ad he equao of sae s m, m () Now he se of qs. (), (5), (6), () ad () adm a eac souos ve by, (), (5) ( ), c( ) ad ad c are cosas of erao. ( ) Thus he eomery of he aay symmerc ach ye-i cosmooca mode s descrbed by he merc ds d d dy dz. (6) IV. Some Physca ad Kemaca Parameers of he Mode From qs. (), (5) ad () we have ( ) m( ) ( ) ( m) ( ), (7) ( )( ) ( ) ( m) ( ), (8) ( m) ( m) ( )( ) ( ) ( m)( ) m., he mode (6) becomes fa ad he ressure ( ), eery desy ), (9) ( ad aue fuco have fe vaues. Furher, as me creases, he scae facors ad crease defey. The easo scaar ( ) whch deermes he voume behavour of he fud ve by. () he a eoch, s fe ad whe. Hece here s fe easo he mode. so, Hubbe arameer (H) s ve by The voume eeme (V ) s ve by H. () V S, () S s he averae scae facor. The equao () shows ha he voume creases as he me creases, ha s, he mode (6) s ead wh me. Shear scaar ( ) s ve by H, resecvey. H H ( ) H H, () ( ) ( ) are he drecoa Hubbe s arameers he drecos of, y ad z Pae

4 ac Souos of ay Symmerc ach Tye-I Cosmooca Mode Lyra Geomery From qs. () ad (), we oba m ( ) ( ) cos a Therefore he mode does o aroach soroy for are vaues of. eceerao arameer (q) s ve by. () q. (5) so, asoroy arameer ( m ) for he mode s obaed as m ( ) cos a. (6) ( ) Therefore he mode has cosa asoroy arameer hrouhou he evouo of he uverse. I hs mode arce horzo ess because s a covere era. We have observed ha a d V d o, he saa voume vashes ad creases wh cosmc me. For he mea asoroy arameer vashes ad he drecoa scae facor (7), ( ) ( ). (8) Therefore, soroy s acheved he derved mode for. For hs arcuar vaue of, we observe ha ( ) ( ) S( ). Therefore he merc () reduces o fa FRW sace-me. Thus, he derved mode acqures faess for. u he same sr, he shear ( ) vashes for. Hece we cao choose he derved mode. V. scusso ad Cocud Remars I he rese sudy we have obaed eac souos of Se s equaos he resece of erfec fud for aay symmerc ach ye-i cosmooca modes orma aue for Lyra s mafod. The easo veocy S dveres as. Hece he easo of he uverse s fe as we aroach he a. For he mode (6) he above hysca quaes e Hubbe arameer (H), easo scaar ( ) ad he shear scaar ( ) are dveres as. Thus he uverse sars wh a fe rae of easo ad measure of asoroy. Ths behaves e he b-ba mode of he uverse. The easo ceases ad he voume becomes fey are a are vaue of (.e. ). Hece he rae of easo of he uverse decreases wh crease of me. Sce deceerao arameer q, hus we fd ha he mode (6) rereses a deceera uverse., he ressure ad eery desy rema udeermed. hese hysca quaes rema fe ad hysca sfca a fe reo. Sce cos a mode does o aroach soroy for are vaues of. Ths mode aso has a o ye suary a ( m ad has cosa asoroy arameer ) hrouhou he evouo of he uverse. For, herefore he, he derved mode reduces o fa FRW sace-me. I has a arce horzo. s, he shear des ou ad he easo so. Thus he aue fuco ) s are he be of he mode bu decays couousy dur s evouo. Smar resus ca be obaed for Hoye s creao fed [8] f he creao fed s me deede. Pae

5 ac Souos of ay Symmerc ach Tye-I Cosmooca Mode Lyra Geomery Refereces [] H. Wey, Sber. Preusssche ademe der Wsseschafe zu er, 98, 65. [] G. Lyra, Über ee Modfcao der Remasche Geomere, Mahemasche Zeschrf 5, 95, 5-6. []. K. Se, sac cosmooca mode, Zeschrf fur Phys C 9, 957, -. []. K. Se ad K.. u, scaar-esor heory of ravao a modfed Remaa mafod, Joura of Mahemaca Physcs, 97, [5] W.. Haford, Cosmooca heory based o Lyra s eomery, usraa Joura of Physcs, 97, [6] W.. Haford, Scaar-esor heory of ravao a Lyra mafod, Joura of Mahemaca Physcs, 97, [7]. Pradha, Cydrcay symmerc vscous fud uverse Lyra eomery, Joura of Mahemaca Physcs 5, 9, 5-5. [8] S. Kumar ad C. P. Sh, eac ach ye-i cosmooca modes Lyra s mafod, Ieraoa Joura of Moder Physcs, 8, 8-8. [9] V. U. M. Rao, T. Vuha ad M. V. Sah, ach ye-v cosmooca mode wh erfec fud us eave cosa deceerao arameer a scaar esor heory based o Lyra mafod, srohyscs Sace Scece, 8, -6. [] J. K. Sh, ac souos of some cosmooca modes Lyra eomery, srohyscs Sace Scece, 8, []. K. Se ad J. R. Vasoe, O Wey ad Lyra mafod, Joura of Mahemaca Physcs, 97, [] K. S. hamra, cosmooca mode of cass oe Lyra s mafod, usraa Joura of Physcs 7, 97, [] T. M. Karade ad S. M. orar, Thermodyamc equbrum of a rava shere Lyra s eomery, Geera Reavy ad Gravao 9, 978, -6. [] S.. Kayashe ad.. Wahmode, sac cosmooca mode se-cara heory, Geera Reavy ad Gravao, 98, 8-8. [5]. R. K. Reddy ad P. Iaah, ae symmerc cosmooca mode Lyra mafod, srohyscs Sace Scece, 986, 9-5. [6]. R. K. Reddy ad R. Veaeswaru, sac coformay fa cosmooca mode Lyra s mafod, srohyscs Sace Scece 6, 987, [7] H. H. Soe, Cosmooes based o Lyra s eomery, Geera Reavy ad Gravao 9, 987, -6. [8] F. Hoye, ew mode for he ead uverse, Mohy Noces of he Roya sroomca Socey, 8, 98, 7-8. [9] F. Hoye ad J. V. Narar, Tme symmerc eecrodyamcs ad he arrow of he me cosmooy. Proceeds of he Roya Socey of Lodo. Seres 77, 96, -. [] F. Hoye ad J. V. Narar, O he avodace of suares C-fed cosmooy, Proceeds of he Roya Socey of Lodo Seres 78, 96, []. eesham, FLRW cosmooca modes Lyra s mafod wh me deede dsaceme fed, usraa Joura of Physcs, 988, 8-8. [] S. haacharya ad T. M. Karade, Uform asoroc cosmooca mode wh sr source, srohyscs Sace Scece, 99, []. R. K. Reddy ad M. V. S. Rao, ay symmerc cosmc srs ad doma was Lyra eomery, srohyscs Sace Scece, 6, [] V. U. M. Rao ad T. Vuha, ay symmerc cosmooca modes a scaar esor heory based o Lyra mafod, srohyscs Sace Scece 9, 9, Pae