Bianchi Type II Stiff Fluid Tilted Cosmological Model in General Relativity
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1 Ieraoal Joural of Mahemacs esearch. IN Volume 6, Number (0), pp. 6-7 Ieraoal esearch Publcao House hp:// Bach ype II ff Flud led Cosmologcal Model Geeral elay B. L. Meea Deparme of Mahemacs, Goerme P. G. College, ok (aasha.), Ida com elephoe No Absrac I hs paper, we hae esgaed led L Bach ype II cosmologcal model for sff flud dsrbuo wh hea coduco Geeral elay. o ge he deermsc soluo erms of cosmc me, we hae also assume a supplemeary codos = merc poeal ad, beg a cosa. he physcal ad geomercal aspecs of he model are also dscussed. Key Words: led, Bach II, Hea Coduco, ff Flud. Iroduco paally homogeeous ad asoropc cosmologcal models whch he flud flow s o ormal o he hyper-space of homogeey creae a more eres he sudy. hese models are called led models. Kg ad Ells [], Ells ad Kg [] hae suded he geeral dyamcs of led uerse. Bradley ad ese [] hae show ha hea flow s expeced for led cosmologcal models wh hea flux hae bee esgaed by umber of auhors lke oy ad Baeree [], Baeree ad aos [5], Coley adupper [6], Bal ad harma [7], Bal ad Meea []. ecely Bal ad Baeree [9] hae esgaed spaally homogeeous ad L (Locally oaoally ymmerc) Bach ype II space-me for perfec flud dsrbuo Geeral elay. o ge he deermsc soluo erms of cosmc me, we hae also assumed a codo where s he shear ad he expaso he model. We cosder he Bach ype II merc he form ds d dx dy x dz) dz ()
2 6 B. L. Meea where ad are fucos of aloe. he eergy momeum esor for perfec flud dsrbuo wh hea coduco s ake o he form ge by Ells [0] as p) p () ogeher wh g ( 0 () 0 (5) where p s he soropc pressure, desy ad he hea coduco ecor sh orhogoal o 00. Ese s feld euaos g For he merc () leads o p)sh p sh (6) p (7) p) p sh () p) sh 0 (9) where he suffx afer ad deoes dffereao wh respec o. Euao (6) (9) are four euaos sx ukows,,, p, ad. For he complee deermao of hese uaes, we assume wo exra codos. We assume he model s flled wh sff flud whch leads o = p (0) ad = () Euaos (6) ad () lead o p () for sff flud = p
3 Bach ype II ff Flud led Cosmologcal Model Geeral elay 65 0 ( () ad ( lead o 0 () whch leads o f f f 0 (5) where f, f df/ds (5) leads o (a (6) From () ad (6) (a (7) where a Hece merc () reduces o he form ds )d dx dy XdZ) dz () where l s he cosa of egrao. ome Physcal ad Geomercal Feaures From euaos (7), (6), (7), we hae 6p (9) (0) p sh ad 6p () where
4 66 B. L. Meea ) () Euaos (0) ad () lead o ) ( ( ) ( ( () ) ) ( sh (5) ) ( ) ( 6 ( (6) ) ( ) ( (7) ) ( ( () ) ( ( 6 ) ( (9) We kow 0 Usg (6), (7), () ad (9), we hae
5 Bach ype II ff Flud led Cosmologcal Model Geeral elay 67 ) ( ) ( 6 ( 6 ) ( ) ( ) ( ) ( ( ) ( ( = 0 (0) ) ( ( b a sce a ) ( a ( ) ( a(a a ) ( (a a(a a (
6 6 B. L. Meea a ) ( a ( a ( a ) ( a ) ( ) ( a(a ( ) ( ) ( 7) ( ( ce s) Usg euaos (6), (7), () ad (, we hae ) ( ( ) ( 7) ) ( ( ( ) ( ) ( (
7 Bach ype II ff Flud led Cosmologcal Model Geeral elay 69 ( ) ( 7) ( ( ) ( ) ( ) ( ( ) ( ) ( 7) ( ) ( ( ( sh Pug alues of (6), (7), (), (5) ad (, we hae ) ( ( ) ) ( ) ( 7) ( ) ( (
8 70 B. L. Meea ( ) ( ) ) ( ) ( 7) ( ) ( ( ( ) ( ( ) ( ) ( ) ( 7) ( ) ( ( (5) Whe addg ( ad (5), we hae 0 we kow
9 Bach ype II ff Flud led Cosmologcal Model Geeral elay 7 From (), ( ad (6), (7), we hae ( 7) ( ) ( ) ( ) ( ( ) ( ) ( ( ) ( 7) ( ) (6) Dscusso he model () sars wh a bg bag a = 0 ad expaso he model decreases as me creases where + > 0. he maer desy 0 whe ad + > 0. he flud elocy sasfes race free codo = 0 ad w = 0. he model geeral represes led, shearg ad o-roag uerse. he model () has Po ype sgulary a = 0 (MacCallum [97][]. efereces [] Kg, A.. ad Ells, G. F.., led Homogeeous Cosmologcal Models Cmmu. Mah. Phys., 09-, (97. [] Ells, G. F.. ad Kg, A.., Dyamcs of led Models, Commu. Mah. Phys., 9 (97). [] Bradley, J. M. ad ess, E., : ome oag me Depede Bach ype VIII Cosmologes wh Hea Flow, Ge. el. Gra. 6, 9 (9). [] oy,.. ad Baeree,. K. led Bach ype I Baroropc Cosmologcal Model Asrophys. pace c. 50, No., -. (9)
10 7 B. L. Meea [5] Baeree, A., aos, N. O. ad Das,.. Isoropc Homogeeous Uerse wh Vscous Flud J. Mah. Phys. 6, 7 (95). [6] Coley, A. A. ad upper, B. O. J. Coformal Kllg Vecors ad FW pace mes Geeral. elay ad. Graao., (990). [7] Bal,. ad harma, K., Bach ype I Mageged led Imperfec Baroropc Flud Asrophys. pace c., (00. [] Bal,. ad Meea, B. L., Asrophys. pace c., 565 (00). [9] Bal,. ad Baeree, aa, L Bach ype II space me for perfec flud dsrbuo Geeral elay J. a. Acad. Phys. c. 7, 55 (00) [0] Ells, G. F.., Geeral elay ad Cosmology,. K. achs[ed] New York Academc Press, pp. 6, [97] [] MacCallum, M. A. H. (97), Comm. Mah. Phys. 0, 57.
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