A note on proper curvature collineations in Bianchi types VI

Transcription

1 note on roer curvture collinetions in inci tyes VI nd VII sce-times Gulm Sbbir nd mjd li Fculty o Engineering Sciences GIK Institute o Engineering Sciences nd Tecnology Toi Swbi NWFP Pkistn Emil: sbbir@gikieduk bstrct We study roer curvture collinetions in te most generl orm o te inci tyes VI nd VII sce-times using te rnk o te 6 6 iemnn mtrix nd direct integrtion tecnique It is sown tt wen te bove sce-times dmit roer curvture collinetions tey orm n ininite dimensionl vector sce INTODUTION Te im o tis er is to ind te existence o roer curvture collinetions in te inci tyes VI nd VII sce-times Since te curvture tensor is n imortnt in dierentil geometry nd generl reltivity Hence te study o its symmetries is imortnt Dierent roces [-] were doted to study curvture collinetions Trougout M reresents our dimensionl connected Husdor sce-time mniold wit Lorent metric g o signture (- ) Te curvture tensor ssocited wit g b troug te Levi-ivit connection is denoted in comonent orm by bcd Te usul covrint rtil nd Lie derivtives re denoted by semicolon comm nd te symbol L resectively ound nd squre brckets denote te usul symmetrition nd skew-symmetrition resectively Here M is ssumed non-lt in te sense tt te curvture tensor does not vnis over ny nonemty oen subset o M ny vector ield on M cn be decomosed s ; b b Gb ()

2 were ( ) L g is symmetric nd G G ) is skew symmetric tensor on M b b b b ( b I b; c is sid to be ine nd urter stisies b cg b c ten is sid to be omotetic (nd Killing i c ) Te vector ield is sid to be roer ine i it is not omotetic vector ield nd lso is sid to be roer omotetic vector ield i it is not Killing vector ield vector ield on M is clled curvture collinetion () i it stisies [] or equivlently L bcd () e e e e bcd ; e ecd ; b bed ; c bce ; d e bcd ; e Te vector ield is sid to be roer i it is not ine [] on M LSSIFITION OF THE IEMNN TENSOS In tis section we will clssiy te iemnn tensor in terms o its rnk nd bivector decomosition Te rnk o te iemnn tensor is te rnk o te 6 6 symmetric mtrix derived in well known wy [] Te rnk o te iemnn tensor t is te rnk o te liner m wic b cd ms te vector sce o ll bivectors G t to itsel nd is deined by : G b cdg Deine te subsce N o te tngent sce T M consisting o tose members k o T M wic stisy te reltion () d bcd k Ten te iemnn tensor t stisies exctly one o te ollowing lgebric conditions [] lss Te rnk is nd te rnge o is snned by te dul ir o non-null simle bivectors nd dim N Te iemnn tensor t tkes te orm bcd b cd * b * G G β G G () cd were G nd its dul G * re te (unique u to scling) simle non-null scelike nd timelike bivectors in te rnge o resectively nd β

3 lss Te rnk is or nd tere exists unique (u to scling) solution sy k o () (nd so dim N ) Te iemnn tensor t tkes te orm bcd i b j ij G G cd () i j b were ij or ll i j nd G i bk or ec o te bivectors i G wic sn te rnge o lss D Here te rnk o te curvture mtrix is Te rnge o te m is snned by single bivector G sy wic s to be simle becuse te symmetry o iemnn tensor [ bcd ] mens G G Ten it ollows rom stndrd result tt G is simle Te curvture [ b cd ] tensor dmits exctly two indeendent solutions k u o eqution () so tt dim N Te iemnn tensor t tkes te orm G G (6) bcd b cd were nd G is simle bivector wit blde ortogonl to k nd u lss O Te rnk o te curvture mtrix is (so tt ) nd dim N lss Te iemnn tensor is sid to be o clss t i it is not o clss D or O Here lwys dim N study o te S or te clsses D nd O cn be ound in [ ] bcd Min esults onsider inci tyes VI nd VII sce-times in usul coordinte system ( t x y ) (lbeled by ( x x x x ) resectively) wit line element [] ds dt ( ( ) ( )) dx ( ) ( ) ( ) ( ) ( ) dx dy ( t ) d dy ()

4 were nd re nowere ero unctions o t only For sin cos or sin cos te bove sce-time () becomes inci tye VI or VII resectively Te bove sce-time dmits tree linerly indeendent Killing vector ields wic re y x x y y x () Te non-ero comonents o te iemnn tensor re [] [ ] [ ] [ ] [ ] { } { } [ ] [ ] { } { } [ ] 6 9 Writing te curvture tensor wit comonents bcd t s 6 6 symmetric mtrix bcd (9)

5 It is imortnt to note tt we will consider te iemnn tensor comonents s bcd or clculte S Since we know rom teorem [] tt wen te rnk o te 6 6 iemnn mtrix is greter tn tree tere exists no roer Hence we will consider only tose cses were te rnk o te 6 6 iemnn mtrix is less tn or equl to tree Tere exist six iteen nd twenty cses wen te rnk o te 6 6 iemnn mtrix is one two nd tree resectively ie we ve ltogeter orty one cses wen te rnk o te 6 6 iemnn mtrix is less tn or equl to tree were roer my exist Suose te rnk o te bove 6 6 iemnn mtrix (9) is one Ten tere is only one non ero row or column in (9) I we set ive rows or columns re identiclly ero in (9) ten tere exist six ossibilities wen te rnk o te 6 6 iemnn mtrix is one In tese six ossibilities ive give te contrdiction nd only one will rise wic is given in cse (II) For exmle consider te cse wen te rnk o te 6 6 iemnn mtrix is one ie nd 6 9 Te constrints wic gives contrdiction (ere we ssume tt ) Hence tis is not ossible Now suose rnk o te bove 6 6 iemnn mtrix (9) is two Ten tere is only two non ero row or column in (9) I we set our rows or columns re identiclly ero in (9) ten tere exist iteen ossibilities wen te rnk o te 6 6 iemnn mtrix is two I one roceed urter ter some lengty clcultion one inds tt non o te bove iteen cses or rnk two exists y similr nlysis we come to te conclusion tt tere exist only two cses wen te rnk o te 6 6 iemnn mtrix is tree or less wic re (I) nk nd 9 6 (II) nk nd 6 9 We will consider ec cse in turn se (I): In tis cse we ve 6 9 te rnk o te 6 6 iemnn mtrix is tree nd tere exists unique (u to multile) no were ero time like vector ield solution o eqution () nd t ; t t ( t) e( t) From te bove constrints we ve ( t) d ( t) b d e ( d e > ) Te line element in tis cse tkes te orm nd ( t) ( t b) b were

6 ( ) ( ) e ( ) dx d dy ds dt ( t b ) () ( d ) ( ) ( ) dx dy e d Substituting te bove inormtion into te S equtions one ind tt [] () () () ( ) ( ) ( ) () () (6) () Eqution () gives K() t F ( x y) F ( x y) n rbitrry unction o t only nd ( x y) nd F F F ( x y) nd F I one roceeds urter one ind tt curvture collinetions in tis cse [] re () t c y c c x c c K were ( t) K is re unctions o integrtion () were c c c One cn write te bove eqution () subtrcting Killing vector ields s ( K() t ) (9) Proer curvture collinetion in tis cse clerly orm n ininite dimensionl vector sce It is imortnt to note tt te constnts nd b cn not be ero simultneously se (II): In tis cse we ve nd 6 9 te rnk o te 6 6 iemnn mtrix is one Here tere exist two liner indeendent solutions nd o eqution () Te vector ield t is covrintly constnt weres is t t not covrintly constnt From te bove constrints we ve ( t) ( t) ( t) ( t q) were q Te line element tkes te orm ( ) ( ) ( ) dx dy ds dt ( t q ) () ( ) ( ) dx dy d urvture collinetions in tis cse [] 6

7 were ( t ) c y c c x c c N () c c c nd N ( t ) is n rbitrry unction o t nd One cn write te bove eqution () subtrcting Killing vector ields s ( N( t ) ) () Proer curvture collinetion in tis cse clerly orm n ininite dimensionl vector sce SUMMY In tis er n ttemt is mde to exlore ll te ossibilities wen te inci tyes VI nd VII sce-times dmit roer S n roc is doted to study roer S o te bove sce-times by using te rnk o te 6 6 iemnn mtrix nd lso using te teorem given in [] wic suggested were roer curvture collinetions exist From te bove study we obtin te ollowing results: (i) We obtin te sce-time () tt dmits roer curvture collinetions wen te rnk o te 6 6 iemnn mtrix is tree nd tere exists unique nowere ero indeendent timelike vector ield wic is te solution o eqution () nd is not covrintly constnt In tis cse roer curvture collinetions orm n ininite dimensionl vector sce (or detils see cse I) (ii) Te sce-time () is obtined wic dmits roer curvture collinetions (see cse II) wen te rnk o te 6 6 iemnn mtrix is one nd tere exist two indeendent solutions o eqution () but only one indeendent covrintly constnt vector ield In tis cse roer curvture collinetions orm n ininite dimensionl vector sce eerences [] G H Ktin J Levine nd W Dvis J Mt Pys (969) 6 [] G Sbbir mjd li nd M mn Grvittion nd osmology 6 () 6 [] G Sbbir nd mjd li dv Studies Teor Pys () [] G S Hll nd J d ost J Mt Pys (99) [] G S Hll Symmetries nd urvture Structure in Generl eltivity World Scientiic

8 [6] G Sbbir Grvittion nd osmology 9 () 9 [] G Sbbir Nuovo imento () [] G S Hll nd G Sbbir lssicl Quntum Grvity () 9 [9] G S Hll Gen el Grv (9) [] H okri Qdir M S med nd M sgr J Mt Pys (99) 69 [] Tello-Llnos Gen el Grv (9) 6 [] J rot nd J d ost Gen el Grv (99) [] G S Hll lssicl Quntum Grvity (6) [] G S Hll nd Lucy McNy lssicl Quntum Grvity () 9 [] H okri M sgr M S med K sid nd G Sbbir Nuovo imento (99) 9 [6] G Sbbir H okri nd Ksi Nuovo imento () [] G S Hll nd G Sbbir Grvittion osmology 9 () [] G Sbbir Nuovo imento 9 () [9] G Sbbir Nuovo imento (6) 9 [] G Sbbir nd bu kr Memood Modern Pysics Letters () [] G Sbbir nd M mn Interntionl journl o Modern Pysics () 9 [] mjd li P D Tesis GIK Institute Pkistn 9 [] H Steni D Krmer M H Mcllum Hoenselers nd E Herlt Exct Solutions o Einstein s Field Equtions mbridge University Press