J. Bsi. Appl. i. Res. 4-5 4 4 TeRod Publiion IN 9-44 Journl of Bsi nd Applied ienifi Reserh www.erod.om Clindrill mmeri Mrder Universe nd Is Proper Teleprllel Homohei Moions Amjd Ali * Anwr Ali uhil Khn Muhmmd Iqbl eprmen of Bsi ienes nd Islmi Universi of ngineering & Tehnolog Peshwr Khber Pukhoonkhw Pkisn eprmen of Mhemis Universi of Peshwr Peshwr Khber Pukhoonkhw Pkisn eprmen of isis Universi of Peshwr Peshwr Khber Pukhoonkhw Pkisn Reeived: Jul 4 Aeped: Oober 9 4 ABTRACT This pper is onerned o find proper eleprllel homohei veor fields in lindrill smmeri Mrder universe. Our mehod omprised of some lgebri nd dire inegrion ehnique. I is shown h in he bsene of urvure nd in he presene of orsion he bove speime dmis one proper eleprllel homohei veor field for priulr hoie of he meri funions. KYWOR: Torsion ire inegrion ehnique Teleprllel Lie derivive Weienbök onneions Terd fields Teleprllel homohei veor fields. INTROUCTION The role of smmeries in generl relivi is obvious for differen purposes. In relivi heor mn uhors hve sudied differen smmeries of well known speimes like Killing homohei onforml nd self-similr veor fields [-]. These smmeries were sudied in he presene of urvure in he speimes. There re oher heories of grviion whih do no onsider urvure in he speime nd orsion is onsidered responsible for grviionl inerion []. mmeries in he presene of orsion were ignored o sud unil he disover of eleprllel Lie derivive b M. hrif nd M. J. Amir []. This new definiion of Lie derivive in he presene of orsion opened door for he sud of differen smmeries. In he pioneer pper [] he uhors eplored eleprllel Killing veors for insein Universe. Ler on his sud ws rried ou on more omplied speimes nd differen well known speimes hve been lssified ording o heir Killing [4578] homohei[9] nd onforml veor fields [45]. Keeping in mind he eensive use of smmeries in differen geomeril nd phsil perspeives of speime we re ineresed o sud proper homohei veor fields in lindrill smmeri Mrder Universe. TLPARALLL THORY: ome Bsis In his seion brief desripion of eleprllel heor is given. In generl relivi heor ovrin derivive is needed insed of pril derivive whih is oordine dependen. This ovrin derivive for seond rnk ensor in eleprllel heor is defined s [] B B Γ ν B Γ µ B where omm snds for pril derivive nd µ ν Γ ν represens Weienbök onneions defined b [] Γ O O µ ν ν where O is non-rivil erd field nd O µ represens inverse field. This erd field sisfies relion ν ν µ O µ O δ µ O µ Ob δ b. The Riemnnin meri n be genered hrough he bove erd field nd is inverse s b g O µ O ν 4 µ ν η b *Corresponding Auhor: Amjd Ali eprmen of Bsi ienes nd Islmi Universi of ngineering & Tehnolog Peshwr Khber Pukhoonkhw Pkisn mil: mjd_pue@hoo.om
Ali e l.4 where η b dig is he Minkowski meri. Torsion omponens re bsill he differene of Weienbök onneions i.e. T Γ νµ Γ. 5 Torsion is ni smmeri in lower indies. In he presene of orsion in speime eleprllel Killing equion is defined s [] L T g g T g ν µ g µ ν gνt µ g µ ν. In order o find proper eleprllel homohei moions in Mrder universe we need o use he definiion in is eended form s L T g g gν µ gµ ν gνt µ gµ T ν g. 7 where onsn nd when equion 7 will redue o equion. MAIN RULT: The line elemen for lindrill smmeri Mrder universe in he usul oordine ssem is given b ds [ d d ] d d 8 where nd re no-where ero funions of onl. Also he omponens of erd nd is inverse re obined b using he relion 4 s s O µ µ O. The non vnishing orsion omponens b using 5 re T T T T T T where prime snds for derivive wih respe o. Now using 8 nd in we ge he eleprllel Killing equions s follows: 4 5 9
J. Bsi. Appl. i. Res. 4-5 4 7 8 Now inegring equions nd 5 we ge 4 d 9 where nd 4 re funions of inegrion whih re o be deermined. I is imporn o remind he reders h when onsn Mrder universe beomes speil se of Binhi pe I speimes nd he resuls of whih n be seen in []. Here we re ineresed o obin eleprllel proper homohei veor field onl for he se when onsn. Resul is wrien direl here nd lengh deils re omied. We obined he eleprllel homohei veor fields s follows: ] [ 4 8 5 4 d d d where. 5 4 8 R The line elemen for lindrill smmeri Mrder universe fer suible resling of nd kes he form ] [ d d d d ds where. R The bove speime dmis nine linerl independen eleprllel homohei veor fields in whih eigh re eleprllel Killing veor fields given s G G nd where d G nd one is proper eleprllel homohei veor field. Proper eleprllel homohei veor field fer subring eleprllel Killing veor fields from is given s d CONCLUION In his pper we eplored proper eleprllel homohei veor fields in lindrill smmeri Mrder universe. In order o find proper homohei veor fields in one of eleprllel heor we pplied some lgebri nd dire inegrion ehnique. This sud is rried ou in he presene of orsion nd i omes ou h Mrder universe dmis one proper eleprllel homohei veor field for priulr hoie of he meri funions. The meri whih dmi proper homohe is given in equion.
Ali e l.4 RRRNC [] A. Ali T. Hussin nd J. Khn. Isomeries in Binhi pes VI o nd VII o spe-imes Adv. ud. in Theo. Phs. : 7-. [] A. Ali Kmrn nd Z. Minullh. Clssifiion of spherill smmeri non si speimes ording o heir Killing veor fields App. Mh. i. : 8-8. [] A. Ali J. Khn nd T. Hussin. Proper homohei smmer in Binhi pe IV spe-imes Adv. ud. in Theo. Phs. : 87-9. [4] A. Ali Z. Minullh nd Kmrn. Proper homohei veor field in Binhi pe-v speimes Adv. ud. in Theo. Phs. : 9-. [5] G. hbbir M. Rmn nd A. Ali 9. Clssifiion of non-si spherill smmeri speimes ording o heir proper onforml veor fields Uni. of Polih. Buh. i. Bull. eries A: App. Mh. nd Phs. 7: -8. [] G. hbbir nd A. Ali 9. A noe on proper onforml smmer in Binhi pes VIII nd I spe-imes Adv. ud. in Theo. Phs. : 9-97. [7] G. hbbir nd A. Ali 9. Clssifiion of spill homogeneous roing spe-imes ording o heir onforml veor fields App. Mh. i. : 89-87. [8] G. hbbir nd. Khn. A noe on self similr veor fields in plne smmeri si speimes TWM J. of Pure nd App. Mh. : 5-5. [9] G. hbbir nd. Khn. A noe on self-similr veor fields in si spherill smmeri speimes Uni. of Pol. Buh. i. Bull. eri. A App. Mh. nd Phs 74: 77-8. [] G. hbbir nd. Khn. A noe on self-similr veor fields in lindrill smmeri si spe-imes TWM J. of pure nd App. Mh. 4: 8-4. [] R. Aldrovndi nd J. G. Pereir 995. An inroduion o geomeril phsis World ienifi. [] M. hrif nd M. J. Amir 8. Teleprllel Killing Veors Of The insein Universe Mod. Phs. Le. A : 9-99. [] G. hbbir nd. Khn. A noe on lssifiion of Binhi pe I spe-imes ording o heir eleprllel Killing veor fields Mod. Phs. Le. A 5: 55-. [4] G. hbbir nd. Khn. A noe on Killing veor fields of Binhi pe II spe-imes in eleprllel heor of grviion Mod. Phs. Le. A 5: 7-74. [5] G. hbbir A. Ali nd. Khn. A noe on eleprllel Killing veor fields in Binhi pe VIII nd I spe-imes in eleprllel heor of grviion Chinese Phs. B : 74-. [] G. hbbir nd. Khn. Clssifiion of lindrill smmeri si spe-imes ording o heir Killing veor fields in eleprllel heor of grviion Mod. Phs. Le. A 5: 55-5. [7] G. hbbir. Khn nd A. Ali. A noe on lssifiion of spill homogeneous roing spe-imes ording o heir eleprllel Killing veor fields in eleprllel heor of grviion Comm. in Theo. Phs. 55: 8-7. [8] G hbbir. Khn nd M. J. Amir. A noe on lssifiion of lindrill smmeri non si spe-imes ording o heir Killing veor fields in eleprllel heor of grviion Br. J. of Phs. 4: 84-94. [9] G. hbbir nd. Khn. Clssifiion of Binhi pe I spe-imes ording o heir proper eleprllel homohei veor fields in he eleprllel heor of grviion Mod. Phs. Le. A 5: 45-5. [] G. hbbir nd. Khn. Clssifiion of eleprllel homohei veor fields in lindrill smmeri si spe-imes in eleprllel heor of grviion Comm. in Theo. Phs. 54: 75-78. 4
J. Bsi. Appl. i. Res. 4-5 4 [] G. hbbir nd. Khn. A noe on proper eleprllel homohei veor fields in non si plne smmeri mnifold Rom. J. of Phs. 57: 57-58. []. Khn T. Hussin nd G. A. Khn. A noe on proper eleprllel homohei moions of well-known speime using non digonl erd Life i. J. s: 87-9. []. Khn T. Hussin G. A. Khn nd A. Ali 4. Proper eleprllel homohei veors from non digonl erd of some well known si speime Life i. J. 5s: 7-. [4] G. hbbir A. Khn nd. Khn. Teleprllel onforml veor fields in lindrill smmeri si spe-imes In. J. of Theo. Phs. 5: 8-87. [5] G. hbbir nd H. Khn 4. A noe on lssifiion of eleprllel onforml veor fields in Binhi pe I speimes in he eleprllel heor of grviion Romnin Journl of Phsis 59: 79-85.