A NOTE ON TELEPARALLEL LIE SYMMETRIES USING NON DIAGONAL TETRAD

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1 A OTE O TELEPARALLEL LIE SYETRIES USIG O DIAGOAL TETRAD SUHAIL KHA * TAHIR HUSSAI and GULZAR ALI KHA Depamen of ahemais Abdul Wali Khan Univesiy adan KPK Pakisan Depamen of ahemais Univesiy of Peshawa KPK Pakisan * suhail_7pk@yahoo.om Reeived Febuay This pape onsides elepaallel heoy of gaviaion and exploes Killing and pope homohei veo fields fo non diagonal ead of Kanowski-Sahs spaeime. Fo he pupose die inegaion ehnique has been used. I uns ou ha he numbe of Killing veo fields is eihe fou o seven. The seven geneaos of elepaallel Killing veo fields esponsible fo onsevaion of enegy linea and angula momenum ae eoveed and ohe hee geneaos esponsible fo spin angula momenum ae los. The effe of osion on Killing symmey has also been deemined fo he spaeime unde onsideaion. oe ineesingly hee exiss no elepaallel pope homohei veo field fo he hoie of non-diagonal ead of Kanowski-Sahs spaeime. Key wods: Telepaallel heoy Tead fields Telepaallel Killing veo fields.. ITRODUCTIO In geneal elaiviy heoy symmeies pefom a key ole in disoveing geomeial and physial feaues of he spaeime. I has been emained an ineesing opi o find and analyze soluions of Einsein s field equaions hough diffeen symmeies. Sine symmeies have wide ange of appliaions in undesanding and evealing he hidden myseies of ou univese a lage body of lieaue is available on he opi. Fo insane in [] he auho obained anonial foms of a eal funion whih geneae he mei of ype- Robinson- Tauman spaeime unde he assumpion ha he spaeime admis a leas one uvaue ollineaion. The ovaian haaeizaion of self-similaiy of seond ype fo spheial disibuion of mae is disussed in []. The Rii and mae ollineaion ae found in [] whee auhos inodued mahemaial desipion of diffeeniabiliy and dimensionaliy fo hese symmeies. Vaious physial and mahemaial popeies of he spaeimes admiing kinemai self-similaiy ae disussed in []. In [5] all spheially symmei spaeimes ae lassified aoding o hei kinemai self-simila veos of seond infinie and zeoh kinds. Rom. Joun. Phys. Vol. 59 os. 5 6 P Buhaes

2 Telepaallel Lie symmeies using non diagonal ead 89 Self-simila soluions of fis seond infinie and zeoh kinds fo Bianhi ype III and sai spheially symmei spaeimes ae obained in [6-7]. Ugu Cami e al. [8] lassified Bianhi ype II spaeime aoding o is mae ollineaion fo degeneae and non-degeneae enegy momenum enso. The auho in [9] pesened exa ylindially symmei soluions of he Einsein field equaions wih supe fluid as a soue. Geneal elaiviy is no he only heoy of gaviy. Telepaallel heoy also desibes gaviy and is onsideed an equivalen heoy of gaviaion o he geneal heoy of elaiviy. The geneal heoy of elaiviy is based upon uvaue of he spaeime ha is gaviaional ineaion is dependen upon uvaue wihin spaeime. In elepaallel heoy he spaeime does no have uvaue ahe i has osion whih ompels he pailes o feel gaviaion. oeove elepaallel heoy uses Weizenbök onneions insead of osion less Chisoffel symbols. The ole of symmeies in elepaallel heoy may no be negleed in deemining he esuls fo gaviaional enegy momenum and angula momenum. A deep insigh of he effe of osion on symmeies of he spaeime is also needed. In ode o addess hese issues he auhos in [ ] exploed Killing homohei and onfomal veo fields fo diffeen spaeimes in onex of elepaallel heoy of gaviaion and agued ha osion has a song effe on symmeies. The main poblem one faes in elepaallel heoy is he hoie of an appopiae ead field fo diffeen appliaions. Two diffeen ypes of eads fo he same spaeime may give diffeen esuls fo example see [5] whee auhos obained wo diffeen equaions of moion fo diffeen hoies of eads fo he same spaeime. In suh senaio symmeies in elepaallel heoy may help us in hoosing igh ead. The ole of symmeies in deeminaion of igh ead fo diffeen appliaions an be seen in [6]. In his pape he auho showed ha a non diagonal ead podues moe geneaos fo elepaallel Killing veo fields han a diagonal ead. oeove he same non diagonal ead podues bee esuls fo enegy momenum ieduible mass and angula momenum. Keeping in mind he advanage of non diagonal ead ove diagonal one we ae ineesed o find Killing and pope elepaallel homohei veo fields fo he hoie of non diagonal ead fo Kanowski-Sahs spaeime. Wih he help of podued ouome he effe of osion on Killing symmey will be deemined.. TELEPARALLEL THEORY (A OVERVIEW) The elepaallel ovaian deivaive defined as [7] ρ of a ovaian enso of ank is

3 9 Suhail Khan Tahi Hussain Gulza Ali Khan = Γ Γ (.) ρ Fµν Fµ ν ρ ρν Fµ µ ρfν whee omma denoe paial deivaive and defined as [7] whee saisfies he elaions Γ ρν ae Weizenbök onneions a Γ = W W (.) µ ν a ν µ a W µ is he non-ivial ead field. Is invese field is denoed by W ν a and a W W ν ν a a a = δ W W µ µ b = δ b (.) µ µ The Riemannian mei an be geneaed fom he ead field as a b g = η W W (.) µ ν ab µ ν whee η ab is he inkowski mei given by η ab = diag( ). The Weizenbök and Levi-Civia onneions have he elaion Γ µ ν =Γ µ ν + S µ ν (.5) whee [ S µν = Tµ ν + Tν µ T µν ] (.6) is enso quaniy alled he onoion enso and Γ µ ν is he Levi-Civia onneion defined as ( ). g σ Γ g g g µν = σνµ + σµν µνσ (.7) Tosion in ems of Weizenbök onneions is defined as T µν νµ µ ν = Γ Γ (.8) his is ani symmei wih espe o is lowe indies. The Riemann uvaue enso in ems of Weizenbök onneion in elepaallel heoy is given as R λ λ σµν σν µ σµ ν λµ σν λν σµ. =Γ Γ +Γ Γ Γ Γ (.9) ow using equaion (.5) in (.9) he elepaallel Riemann uvaue enso vanishes i.e. whee R σ µν σ σ σ R = R + D = (.) µν µν µν epesens Riemann uvaue enso in geneal elaiviy and

4 Telepaallel Lie symmeies using non diagonal ead 9 σ σ σ λ σ λ σ D = S S S S + S S (.) µν µ ν ν µ ν λµ µ λν is a enso quaniy based on Weizenbök onneion only. In [8] he auhos defined Killing equaion in elepaallel heoy fo he veo field as T X ρ ρ ρ ρ L gαβ = gαβ ρ X + gρβ X α + gαρ X β + X ( gβt αρ + gα T βρ ) = (.) T whee L epesens Lie deivaive in elepaallel heoy. Fo finding elepaallel X pope homohei veo fields we shall use he above definiion in he exended fom as T X L g = αg α R. (.) µν µν. KILLIG VECTOR FIELDS OF KATOWSKI-SACHS SPACETIE Kanowski-Sahs spaeime in spheial oodinaes ( φ ) is defined as [9] ds = d + ( )d + ( ) (d + sin d φ ) (.) whee and ae funions of only whih ae nowhee zeo. The fou independen Killing veo fields of (.) in geneal elaiviy ae given as [9] φ os φ o sin φ sin φ + o os φ. φ φ (.) a Following a well-known poedue given in [] he ead W µ is invese W µ a a Weizenbök onneions Π b and osion omponens T αβ fo Kanowski-Sahs spaeime ae obained as osφsin ososφ sinsinφ W a µ = sin sin φ os os φ os φsin os sin os φsin sin sin φ os µ Wa = os os φ os sin φ sin ossinφ ososφ (.) (.)

5 9 Suhail Khan Tahi Hussain Gulza Ali Khan 5 Π = Π = Π = Π = Π = Π = o Π = sin Π = os sin Π = whee do epesens deivaive wih espe o. (.5) T = T = T = T = T =. (.6) A veo field X is alled a elepaallel homohei veo field if i saisfies equaion (.). Expanding equaion (.) wih he help of equaions (.) and (.6) he following en non linea oupled paial diffeenial equaions ae obained X = X =α (.7) X X X (.8) + = X X X (.9) + = sin X X + sin X = (.) X + X X = (.) X + sin X sin X = (.) X + X =α (.) X + sin X = (.) o X + X + X =α (.5) Fis solving equaions (.7)-(.5) fo elepaallel Killing veo fields by aking α=. Inegaing equaion (.7) and aking α= we ge Also equaion (.) gives X E X Eφ = ( φ ) = ( φ ) (.6) = ( φ ) + ( φ ). (.7) X E E φ φ ow using equaions (.6) and (.7) in equaion (.5) we ge

6 6 Telepaallel Lie symmeies using non diagonal ead 9 = ( φ ) + o ( φ ) o ( φ ) + ( ) (.8) X E E E E The above funions E ( φ) E ( φ) E ( φ ) and E ( ) ae funions of inegaion. The values of hese funions will be deemined by using he emaining six equaions. In he following esuls have been given diely jus o avoid exensive deails of simplifiaion. Case I: In his ase he wo mei funions ae diffeen i.e. () (). The esuling line elemen fo Kanowski-Sahs spaeime is given in equaion (.). Telepaallel Killing veo fields fo spaeime (.) beomes X = X = os + sin ( sin φ+ os φ) () () = sin + os ( sin φ+ os φ) () () X (.9) = ose ( os φ+ sin φ ) () X whee R. The fou geneaos of he elepaallel Killing veos fo he spaeime (.) ae = = sin sin φ os sin φ () () oseos φ = os sin () φ () () = sin os φ + os os φ osesin φ. Compaing () () () φ he obained esuls wih he geneaos fo Poinae symmey algeba of inkowski spaeime given in [] he fou geneaos esponsible fo onsevaion of enegy and linea momenum ae eoveed and he geneaos whih yield onsevaion of angula and spin momenum ae los. Case II: In his ase he mei funions ake he fom = onsan and = (). Afe a suiable esaling of equaion (.) beomes ds d d ( )(d sin d ) and = φ (.) Telepaallel Killing veos fo (.) ae given as

7 9 Suhail Khan Tahi Hussain Gulza Ali Khan 7 X = X = os + sin ( sin φ + os φ) = sin + os ( sin φ + os φ) () () X (.) = ose ( os φ+ sin φ ) () X whee R. The fou geneaos of he elepaallel Killing veos fo he spaeime (.) ae = = sin sin φ os sin φ () oseos φ = os sin () φ () = sin os φ + os os φ osesin φ. Resuls show ha () () φ geneaos yielding angula and spin angula momenum ae los fo he mei (.) and ohe geneaos ae eoveed as in Case I. Case III: In his ase he spaeime akes he fom ds d ( )d (d sin d ) and = + +η + φ (.) Telepaallel Killing veo fields fo he spaeime (.) ae given as X = X = os + sin ( sin φ+ os φ) () () = sin + os ( sin φ+ os φ) η η X (.) = ose ( os φ+ sin φ ) η X whee R. The fou geneaos of he elepaallel Killing veos fo he spaeime (.) ae = = sin sin φ os sin φ () η oseos φ = os sin and η φ () η = sin os φ + os os φ os esin φ. Same geneaos () η η φ

8 8 Telepaallel Lie symmeies using non diagonal ead 95 like Case I fo he elepaallel Killing veo fields ae obained wih a sligh hange ha = η in his ase. Case IV: In his ase he mei funions ake he fom = β β R \{} and =η η R \{} β η. The esuling line elemen afe suiable esaling of is given as ds d d (d sin d ) = + +η + φ (.) Telepaallel Killing veo fields fo spaeime (.) ae given as X = X = os + sin ( sin φ+ os φ) η X e 5 6 = sin + os ( sin φ+ os φ ) + ( os φ+ sin φ) η η ose ( os sin ) o η X e ( 5 sin 6 os ) η = φ+ φ + φ+ φ +η7e η (.5) whee R. The seven geneaos of he elepaallel Killing veos fo spaeime (.) ae = = os sin η = sinsinφ os sinφ ose os φ η η φ = sin osφ + os osφ ose sin φ η η φ = e η (os φ o sin φ ) 5 = e η (sin φ + o os φ ) and 6 =η e η. φ φ φ Compaing hese esuls wih he geneaos fo Poinae symmey algeba of inkowski spaeime given in [] i omes ou ha he fou geneaos esponsible fo onsevaion of enegy and linea momenum and hee geneaos 5 6 yielding onsevaion of angula momenum ae eoveed and he geneaos esponsible fo onsevaion of spin angula momenum ae los fo he spaeime (.). Case V: In his ase he esuling mei akes he fom ds d ( )(d d sin d ). = φ (.6) Telepaallel Killing veo fields fo he mei (.6) beomes

9 96 Suhail Khan Tahi Hussain Gulza Ali Khan 9 X = X = os + sin ( sin φ+ os φ) () () = sin + os ( sin φ+ os φ ) + () () X e + ( 5 sinφ+ 6 os φ) () (.7) e = os ( os φ+ sin φ ) + o ( os φ sin φ ) + () () X e e () whee R. The seven geneaos of he elepaallel Killing veos fo spaeime (.6) ae = = (sin os φ + os os φ ose sin φ ) () φ = (os sin ) () = (sin sin φ + os sin φ + oseos φ ) () φ e e = (sin φ + o os φ ) 5 = (os φ o sin φ ) and () φ () φ e 6. = Resuls show ha fo he mei (.6) he geneaos yielding only () φ spin angula momenum ae los and all ohe geneaos ae eoveed. I is also impoan o noe ha he ase when () = () =η whee η R \{}. The esuling mei akes he fom ds d (d d sin d ) = +η + + φ (.8) The geneaos of elepaallel Killing veos fo he mei (.8) ae he same as in Case V wih a sligh hange ha () = η.. PROPER TELEPARALLEL HOOTHETIC VECTOR FIELDS OF KATOWSKI-SACHS SPACETIE In his seion equaions (.7) (.5) ae solved ompleely when. α Solving equaion (.7) we ge

10 Telepaallel Lie symmeies using non diagonal ead 97 X E X E =α + ( φ ) =α + ( φ ) (.) Using equaion (.) in equaion (.) and solving we ge X =α α E ( ) E ( ) φ + φ (.) ow using equaions (.) and (.) in equaion (.) and solving we ge ose X = E φ ( φ ) d E ( ). + φ (.) B The above funions E ( φ) E ( φ) E ( φ ) and E ( φ ) ae inegaion funions. We will deemine he values of hese funions by using he emaining six equaions. Thus we need o deemine he unknown funions involved in he following sysem of equaions: X E X E =α + ( φ ) =α + ( φ) X =α α E ( φ ) + E ( φ) ose X = E φ ( φ ) d E ( ). + φ B (.) Using sysem of equaions (.) in equaion (.8) implies E ( φ ) E ( φ ) +α + E ( φ ) = (.5) Solving equaion (.) afully by fis diffeeniaing wih espe o and hen inegaing he esuling equaion gives E ( φ ) = + F ( φ ) + F ( φ ) () = + and α whee R. Subsiuing bak hese values in equaion (.5) and solving we ge E ( φ ) = F ( ) d φ + + F ( φ ) + F ( φ ) whee F ( φ) F ( φ) F ( φ ) and F ( φ ) ae funions of inegaion. Refeshing he sysem of equaions (.) by using he above infomaion we ge

11 98 Suhail Khan Tahi Hussain Gulza Ali Khan X =α + + F ( φ ) + F ( φ) X =α + F ( φ ) d F ( ) + φ X =α α F ( φ) d F ( ) φ (.6) F ( φ ) + E ( φ) ose X = [ Fφ ( φ ) + Fφ ( φ )] d E ( ). + φ Subsiuing equaion (.6) in equaion (.) and hen diffeeniaing he esuling equaion wih espe o and we eah o an equaion α () = whih simply implies ha α=. Thus we onlude ha Kanowski-Sahs spaeime do no admi pope elepaallel homohei veo fields fo he hoie of nondiagonal ead. 5. SUARY AD DISCUSSIO In his pape a non diagonal ead is fomed fo Kanowski-Sahs spaeime. Telepaallel Lie deivaive has been applied o obain elepaallel Killing veo fields along wih hei espeive meis. Invesigaion is also exended o pope elepaallel homohei veo fields. I has shown ha dimension of he elepaallel Killing veo field is eihe fou o seven. The geneaos of elepaallel Killing veos have been ompaed o he geneaos of Poinae symmey algeba of inkowski spaeime. This ompaison eveals when Kanowski-Sahs spaeime admi fou elepaallel Killing veo fields i loses six geneaos esponsible fo onsevaion of angula and spin angula momenum. oeove when Kanowski- Sahs spaeime admi seven elepaallel Killing veo fields i also loses hee geneaos esponsible fo onsevaion of spin angula momenum. The pupose of his sudy was also o see ha he hoie of non-diagonal ead allows Kanowski-Sahs spaeime o admi any exa symmey ohe han elepaallel Killing symmey. Ineesingly he hoie of non-diagonal ead esied he spaeime o exhibi only Killing symmey and hee exis no pope elepaallel homohei veo fields. I is impoan o noe ha in elepaallel heoy of gaviy he geomey of spaeime emains fla. I was expeed ha Kanowski-Sahs spaeime wih fla spaeime suue would exhibi en elepaallel Killing veo fields. To ou supise i admis only fou o seven elepaallel Killing veo fields. Thus pesene of osion edued he numbe of Killing symmeies. I has also been

12 Telepaallel Lie symmeies using non diagonal ead 99 obseved ha he pesene of osion in Kanowski-Sahs spaeime have an effe upon he angula and spin angula momenum only while onsevaion of enegy and linea momenum says unaffeed. Aknowledgmens. The fis auho aknowledges finanial suppo povided by Highe Eduaion Commission of Pakisan hough is saup eseah gan pogam. REFERECES. E. G. L. R. Vaz and C. D. Collinson Cuvaue ollineaion fo ype- Robinson-Tauman spaeimes Geneal Relaiviy and Gaviaion (98).. J. P. de Leon Self-simila symmey in geneal elaiviy Geneal Relaiviy and Gaviaion (99).. G. S. Hall I. Roy and E. G. L. R. Vaz Rii and mae ollineaion in spae-imes Geneal Relaiviy and Gaviaion 8 99 (996).. A. A. Coley Kinemai self-similaiy Classial and Quanum gaviy 87 8 (997). 5. H. aeda T. Haada H. Iguhi and. Okuyama Classifiaion of spheially symmei kinemai self-simila pefe-fluid soluion Pogess of Theoeial Physis (). 6. G. Shabbi and S. Khan Self-simila soluions of Bianhi ype III spae-imes using paial diffeenial equaions Jounal of he Assoiaion of Aab Univesiies fo Basi and Applied Sienes (9). 7. G. Shabbi and S. Khan Classifiaion of spheially symmei sai spae-imes aoding o self-simila veo field U. P. B. Siene Bullein Seies A (). 8. U. Cami and E. Sahin ae ollineaions lassifiaion of Bianhi ype II spaeime Geneal Relaiviy and Gaviaion 8 6 (6). 9. V. A. Popov Cylindially symmei saionay soluion o he Einsein equaions fo a supefluid Gaviaion and Cosmology 68 7 (8).. G. Shabbi and S. Khan A noe on lassifiaion of Bianhi ype I spae-imes aoding o hei elepaallel Killing veo fields oden Physis Lees A ().. G. Shabbi S. Khan and. J. Ami A noe on lassifiaion of ylindially symmei non sai spae-imes aoding o hei elepaallel Killing veo fields in he elepaallel heoy of gaviaion Bazilian Jounal of Physis 8 9 (). G. Shabbi and S. Khan Classifiaion of elepaallel homohei veo fields in ylindially symmei sai spae-imes in he elepaallel heoy of gaviaion Communiaions in Theoeial Physis ().. G. Shabbi and S Khan A noe on pope elepaallel homohei veo fields in non-sai plane symmei Loenzian manifolds Romanian Jounal of Physis ().. G. Shabbi A. Khan and S. Khan Telepaallel onfomal veo fields in ylindially symmei sai spae-imes Inenaional Jounal of Theoeial Physis (). 5.. H. Daouda. E. Rodigues and. J. S. Houndjo Inhomogeneous Univese in f() heoy axiv: 5.565v. 6. G. G. L ashed Bane wold blak holes in elepaallel heoy equivalen o geneal elaiviy and hei Killing veos enegy momenum and angula momenum Chinese Physis B 9 (). 7. R. Aldovendi and J. G. Peeia An Inoduion o Gaviaion Theoy (pepin). 8.. Shaif and. J. Ami Telepaallel Killing veos of he Einsein Univese oden Physis Lees A (8). 9. R. Kanowski and R. K. Sahs Some spaially homogeneous anisoopi elaivisi osmologial model Jounal of ahemaial Physis 7 6 (966).. J. G. Peeia T. Vagas and C.. Zhang Axial veo osion and he elepaallel Ke Spaeime Classial Quanum Gaviy (.

Scholars Research Library. Archives of Applied Science Research, 2014, 6 (5):36-41 (

Scholars Research Library. Archives of Applied Science Research, 2014, 6 (5):36-41 ( Available online a wwwsholaseseahlibayom Ahives o Applied Siene Reseah 4 6 5:6-4 hp://sholaseseahlibayom/ahivehml ISSN 975-58X CODEN USA AASRC9 Einsein s equaions o moion o es pailes exeio o spheial disibuions