PROPER CURVATURE COLLINEATIONS IN SPECIAL NON STATIC AXIALLY SYMMETRIC SPACE-TIMES

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1 POPE CUVATUE COLLINEATIONS IN SPECIAL NON STATIC AXIALLY SYMMETIC SPACE-TIMES GHULAM SHABBI, M. AMZAN Fculty of Engineeing Sciences, GIK Institute of Engineeing Sciences nd Technology, Toi, Swbi, NWFP, Pkistn. Emil: eceived Ail, 009 We consideed the secil fom of the non sttic xilly symmetic sce-times fo studying oe cuvtue collinetions by using the nk of the 6 6 iemnn mtix, diect integtion nd lgebic techniques. Studying oe cuvtue collinetions in ech cse it is shown tht when the bove sce-times dmit oe cuvtue collinetions, they fom n infinite dimensionl vecto sce.. INTODUCTION The im of this e is to find the existence of oe cuvtue collinetions (CCS) in the secil non sttic xilly symmetic sce-times. The cuvtue collinetion which eseves the cuvtue stuctue of sce-time cies significnt infomtion nd lys n imotnt ole in Einstein s theoy of genel eltivity nd gvittion. The theoy of genel eltivity, which is ctully field theoy of gvition nd is descibed in tems of geomety, is highly nonline []. Due to this non-lineity it becomes vey hd to solve the gvittionl field equtions unless cetin symmety estictions e imosed on the scetimes. These symmety estictions my be exessed in tems of Killing vecto fields (KVF), homothetic vecto fields (HVF), icci collinetions (CS) nd cuvtue collinetions. Killing vecto fields give ise to some consevtion lws. In the Einstien sce KVF nd CS become simil, othe wise, they my be diffeent, in genel []. Ktzin et l. [, ] suggests tht iemnn cuvtue tenso my lso ovide some ext undestndings which e not ovided by (KVF) nd (HVF). It is, theefoe, imotnt to study CCS. In this e, n och which is given in [5], is doted to study CCS in secil fom of the non sttic xilly symmetic sce-times by using the nk of the 6 6 iemnn mtix nd diect integtion techniques. Thoughout M eesents fou dimensionl, connected, Husdoff scetime mnifold with Loentz metic g of signtue (, +, +, +). The cuvtue tenso om. Joun. Phys., Vol. 56, Nos., P. 5 7, Buchest, 0

2 6 Ghulm Shbbi, M. mzn ssocited with gb, though the Levi-Civit connection, is denoted in comonent c fom by bcd, nd the icci tenso comonents e b = cb. The usul covint, til nd Lie deivtives e denoted by semicolon, comm nd the symbol L, esectively. ound nd sque bckets denote the usul symmetiztion nd skew-symmetiztion, esectively. Hee, M is ssumed to be non-flt in the sense the cuvtue tenso does not vnish ove ny non-emty oen subset of M. The covint deivtive of ny vecto field X on M cn be decomosed s X b ; = hb + Fb, () whee hb ( = hb ) = LX gb nd Fb ( = Fb ) e symmetic nd skew symmetic tensos on M, esectively. If h b; c = 0, X is sid to be ffine nd futhe stisfies hb = cgb, c, in which cse X is sid to be homothetic (nd Killing if c = 0 ) [5]. The vecto field X is sid to be oe ffine if it is not homothetic vecto field nd oe homothetic if it is not Killing vecto field [5]. A vecto field X on M is sid to be CC if it stisfies [] L = 0, () o equivlently X X X X X X bcd e e e e e bcd ; e + ecd ; b + bed ; c + bce ; d bcd ; e = 0. The vecto field X is sid to be oe cuvtue collinetion if it is not ffine [5] on M. One cn exnd the bove eqution in set of couled CC equtions which cn be seen in [6].. CLASSIFICATION OF THE IEMANN TENSOS The iemnn tenso cn be clssified in tems of its nk nd bivecto decomosition. The nk of the iemnn tenso is the nk of the 6 6 symmetic mtix, deived in well known wy [5]. The nk of the iemnn tenso t is the nk of the line m f which ms the vecto sce of ll bivectos F t M to b b cd itself nd is defined by f : F cd F. Futhe, we define the subsce S of the tngent sce TM s consisting of those membes k TM which stisfy the eltion d k = 0. () bcd Then the iemnn tenso t M stisfies exctly one of the following lgebic conditions [5].

3 Cuvtue collinetions in secil non sttic sce-time 7 Clss B The nk is nd the nge of f is snned by the dul i of non-null simle bivectos nd dim S = 0. The iemnn tenso t tkes the fom * * b cd = αf F + β F F, () bcd b cd whee F nd its dul F * e the unique (u to scling) simle non-null scelike nd timelike bivectos in the nge of f, esectively nd α, β. Clss C The nk is o nd thee exists unique (u to scling) solution, sy k of () (nd so dim S = ). The iemnn tenso t tkes the fom i j bcd αij b cd i, j= = F F, (5) i b i whee αij fo ll i, j nd F bk = 0 fo ech of the bivectos F which sn the nge of f. Clss D Hee the nk of the cuvtue mtix is one. The nge of the m f is snned by single bivecto F, sy, which hs to be simle becuse the symmety of the iemnn tenso [ ] = 0 mens F bcd [ F ] = 0, which, togethe with b cd stndd esult imlies tht F is simle. The cuvtue tenso dmits exctly two indeendent solutions k, u of () so tht dim S =. The iemnn tenso t tkes the fom whee = α F F, (6) bcd b cd α nd F is simle bivecto with blde othogonl to k nd u. Clss O The nk of the cuvtue mtix is 0 (so tht bcd = 0 ) nd dim S =. Clss A The iemnn tenso is sid to be of clss A t if it is not of clss B, C, D o O. Hee lwys dim S = 0. A study of the clsses A, B, C, D, O nd CCS in the two dimensionl submnifolds cn be found in [5, 7].

4 8 Ghulm Shbbi, M. mzn. MAIN ESULTS Conside secil non sttic xilly symmetic sce-time in the usul 0 coodinte system (, t,, φ ) (lbeled by ( x, x, x, x ), esectively) with line element [8] At (,, ) Bt (,, ) ds = e dt + e ( d + d + dφ ). (7) The bove sce-time dmits only one Killing vecto field which is. φ non-zeo indeendent comonents of the iemnn tenso e The e ( A(, t, ) + A (, t, ) A(, t, ) B(, t, ) + A(, t, ) B(, t, )) At (,, ) 00 = Bt (,, ) α e ( Bt (, t, ) + Btt(, t, ) At(, t, ) Bt(, t, )) A (, t, ) A (, t, ) + A (, t, ) At (,, ) 00 = e α A(, t, ) B(, t, ) A(, t, ) (, t, ) Bt (,, ) 0 = 0 = e [ Bt ( t,, ) A( t,, ) Bt( t,, ) ] α, At (,, ) e ( A( t,, ) + A( t,, ) A( t,, ) B( t,, ) + A( t,, ) ( t,, )) 00 = Bt (,, ) α e ( Bt (, t, ) + Btt(, t, ) At(, t, ) Bt(, t, )) Bt (,, ) 0 = 0 = e [ Bt ( t,, ) A ( t,, ) Bt ( t,, ) ] α5, At (,, ) e ( A( t,, ) ( t,, ) + A( t,, ) B( t,, )) 00 = Bt (,, ) α6 e ( Bt (, t, ) + Btt(, t, ) At(, t, ) Bt(, t, )) Bt (,, ) At (,, ) At (,, ) Bt (,, ) = e ( (, t, ) + B (, t, )) e e Bt (, t, ) α7, Bt (,, ) At (,, ) At (,, ) Bt (,, ) = e ( B ( t,, ) + ( t,, )) e e Bt ( t,, ) α8, Bt (,, ) = e [ ( t,, ) ( t,, ) B( t,, ) ] α9, Bt (,, ) At (,, ) At (,, ) Bt (,, ) = e ( ( t,, ) + B ( t,, )) e e Bt ( t,, ) α0.,,,, Witing the cuvtue tenso with comonents mnifold s 6 6 symmetic mtix bcd t oint of the

5 5 Cuvtue collinetions in secil non sttic sce-time 9 α α 0 α 0 0 α α 0 α α 0 α α 6 5 bcd = α α5 0 α α5 0 α8 α α 0 α9 α0 It is imotnt to note tht we will conside iemnn tenso comonents s bcd fo clculting CCS. We know fom theoem [5, 7] tht when the nk of the 6 6 iemnn mtix is gete thn thee thee exists no oe cuvtue collinetions. Hee, we e inteested in those cses whee the nk of the 6 6 iemnn mtix is less thn o equl to thee. Thee e, ltogethe, foty-one ossibilities fo the nk of 6 6 iemnn mtix to be ( ), tht is, twenty fo nk thee, fifteen fo nk two nd six fo nk one. Suose the nk of the 6 6 iemnn mtix is thee. Then thee exist only thee non zeo ows o columns in (8). If we set thee ows o columns identiclly zeo in (8) then thee exist twenty ossibilities when the nk of the 6 6 iemnn mtix is thee. In these twenty ossibilities fifteen give contdiction nd only five will suvive. Fo exmle, conside the cse when the nk of 6 6 iemnn mtix is thee, i.e. α = α = α = α = α5 = α8 = α9 = 0, α6 0, α7 0 nd α0 0. The constints α = α = α = α = α5 = α8 = α9 = 0 imly tht (, t, φ ) = 0 nd B (, t, φ ) = 0. Substituting this infomtion in (8) we get α 7 = 0 which gives contdiction becuse we ssumed tht α7 0. So this cse is not ossible. Now conside nothe ossibility when the nk of bove mtix is gin thee, i.e. α = α = α = α = α5 = α6 = α9 = 0, α7 0, α8 0 nd α0 0. The constints α = α = α = α = α5 = α6 = α9 = 0 imly tht B (, t, φ) 0, At (, t, φ) 0, Bt (, t, φ ) = 0, A ( t,, φ) = 0, A ( t,, φ) = 0, (, t, φ) 0, (, t, φ) B (, t, φ ) ( t,, φ) = 0 nd ( t,, φ) + B ( t,, φ) 0. This is the cse C. Now suose the nk of the 6 6 iemnn mtix is. Then thee is only one non zeo ow o column in (8). If we set five ows o columns identiclly zeo in (8) then thee exist six ossibilities when the nk of the 6 6 iemnn mtix is one. In these six ossibilities two give contdiction nd only fou will suvive. Fo exmle, conside the cse when the nk of the bove 6 6 iemnn mtix is one, i.e. α = α = α = α = α5 = α7 = α8 = α9 = α0 = 0 nd α6 0. The constints α = α = α = α = α5 = α7 = α8 = α9 = α0 = 0 imly tht ( t,, φ ) nd B ( t,, φ ) must be zeo. Substituting this infomtion in (8) we get α 6 = 0 which gives contdiction becuse we ssumed tht α6 0. Hence this cse is not ossible. Now we e gin consideing the cse when the nk of the 6 6 iemnn mtix (8)

6 0 Ghulm Shbbi, M. mzn 6 one, i.e. α = α = α = α = α5 = α6 = α7 = α9 = α0 = 0 nd α8 0. These constints give, At (, t, φ) 0, A (, t, φ) = 0, A (, t, φ) = 0, Bt (, t, φ ) = 0, (, t, φ ) = 0, B (, t, φ) 0 nd B (, t, φ ) = 0. This is the cse D. By the simil nlysis we summized tht thee e ltogethe twenty fou suviving ossibilities when the nk of the 6 6 iemnn mtix is thee o less which e: (A) nk=, At (, t, ) = 0, A (, t, ) 0, Bt (, t, ) = 0, (, t, ) 0, B (, t, ) = 0, (, t, ) = 0nd A (, t, ) + A(, t, ) = 0. (A) nk=, At ( t,, ) = 0, A (, t, ) 0, Bt ( t,, ) = 0, (, t, ) 0, B (, t, ) = 0, (, t, ) = 0nd A (, t, ) + A(, t, ) 0. (A) nk=, At (, t, ) 0, A (, t, ) = 0, Bt (, t, ) = 0, (, t, ) = 0, B (, t, ) 0, B (, t, ) = 0nd A (, t, ) + A(, t, ) = 0. (A) nk=, At (, t, ) 0, A (, t, ) = 0, Bt ( t,, ) = 0, (, t, ) = 0, B (, t, ) 0, B (, t, ) = 0nd A (, t, ) + A (, t, ) 0. (C) nk=, At (, t, ) = 0, A (, t, ) = 0, Bt ( t,, ) = 0, (, t, ) 0, B (, t, ) 0, (, t, ) B(, t, ) (, t, ) = 0 nd (, t, ) + B (, t, ) 0. (C) nk=, At (, t, ) = 0, A (, t, ) = 0, Bt ( t,, ) = 0, (, t, ) 0, B ( t,, ) 0, ( t,, ) B( t,, ) ( t,, ) 0 nd (, t, ) + B (, t, ) 0. (C) nk=, At (, t, ) = 0, A (, t, ) = 0, Bt (, t, ) = 0, (, t, ) = 0, B (, t, ) 0 nd B (, t, ) 0. (C) nk=, At (, t, ) = 0, A (, t, ) = 0, Bt ( t,, ) = 0, B (, t, ) = 0, ( t,, ) 0 nd ( t,, ) 0. (C5) nk=, At (, t, ) = 0, A (, t, ) = 0, Bt (, t, ) = 0, (, t, ) 0, B (, t, ) 0, (, t, ) = 0, B (, t, ) = 0 nd (, t, ) B (, t, ) ( t,, ) 0. (C6) nk=, At (, t, ) 0, A ( t,, ) 0, Bt ( t,, ) = 0, (, t, ) = 0, B (, t, ) = 0, A (, t, ) + A(, t, ) 0, A (, t, ) + A ( t,, ) 0 nd A(, t, ) A(, t, ) + A (, t, ) 0. (C7) nk=, At (, t, ) 0, A (, t, ) 0, Bt (, t, ) = 0, (, t, ) = 0, B (, t, ) = 0, A (, t, ) + A(, t, ) 0, A (, t, ) + A ( t,, ) 0 nd A( t,, ) A( t,, ) + A ( t,, ) = 0.

7 7 Cuvtue collinetions in secil non sttic sce-time (C8) nk=, At (, t, ) 0, A (, t, ) 0, Bt (, t, ) = 0, (, t, ) = 0, B (, t, ) = 0, A (, t, ) + A(, t, ) = 0, A (, t, ) + A ( t,, ) 0 nd A( t,, ) A( t,, ) + A ( t,, ) 0. (C9) nk=, At (, t, ) 0, A (, t, ) 0, Bt (, t, ) = 0, (, t, ) = 0, B (, t, ) = 0, A (, t, ) + A(, t, ) 0, A (, t, ) + A ( t,, ) = 0 nd A( t,, ) A( t,, ) + A ( t,, ) 0. (C0) nk=, At (, t, ) 0, A (, t, ) 0, Bt (, t, ) = 0, (, t, ) = 0, A (, t, ) + A(, t, ) = 0, A (, t, ) + A(, t, ) = 0, B (, t, ) = 0nd A ( t,, ) A ( t,, ) + A ( t,, ) 0. (C) nk=, At (, t, ) = 0, A (, t, ) = 0, Bt (, t, ) 0, B t ( t,, ) + Btt( t,, ) At( t,, ) Bt( t,, ) = 0, B (, t, ) = 0nd (, t, ) = 0. (C) nk=, At ( t,, ) = 0, A (, t, ) 0, Bt ( t,, ) = 0, (, t, ) 0, B (, t, ) 0, B (, t, ) = 0 nd A (, t, ) + A(, t, ) A( t,, ) B( t,, ) = 0. (C) nk=, At (, t, ) 0, A (, t, ) 0, Bt ( t,, ) = 0, (, t, ) A(, t, ) + A (, t, ) A(, t, ) B(, t, ) = 0, B (, t, ) 0, B (, t, ) = 0nd A ( t,, ) + A( t,, ) A( t,, ) B( t,, ) = 0. (C) nk=, At (, t, ) 0, A (, t, ) = 0, Bt ( t,, ) = 0, (, t, ) 0, (, t, ) = 0, A ( t,, ) + A( t,, ) A( t,, ) ( t,, ) = 0 nd B (, t, ) = 0. (D) nk=, At (, t, ) 0, A ( t,, ) = 0, Bt ( t,, ) = 0, (, t, ) = 0, B (, t, ) = 0nd A (, t, ) + A(, t, ) 0. (D) nk=, At (, t, ) = 0, A ( t,, ) 0, Bt ( t,, ) = 0, (, t, ) = 0, B (, t, ) = 0nd A (, t, ) + A (, t, ) 0. (D) nk=, At (, t, ) = 0, A ( t,, ) = 0, Bt ( t,, ) = 0, (, t, ) = 0, B (, t, ) 0 nd B (, t, ) = 0. (D) nk=, At (, t, ) = 0, A ( t,, ) = 0, Bt ( t,, ) = 0, (, t, ) 0, B (, t, ) = 0nd (, t, ) = 0. (D5) nk=, At (, t, ) 0, Bt (, t, ) = 0, B (, t, ) = 0 (, t, ) = 0, A (, t, ) + A(, t, ) 0, A (, t, ) 0, A (, t, ) + A(, t, ) = 0, A (, t, ) 0 nd A ( t,, ) A ( t,, ) + A ( t,, ) = 0.

8 Ghulm Shbbi, M. mzn 8 (D6) nk=, At (, t, ) 0, Bt (, t, ) = 0, B (, t, ) = 0 (, t, ) = 0, A (, t, ) 0, A (, t, ) + A(, t, ) = 0, A (, t, ) + A (, t, ) 0 nd A( t,, ) A( t,, ) + A ( t,, ) = 0. We will discuss ech cse in tun. Cse A In this cse At (, t, ) = 0, A (, t, ) 0, Bt ( t,, ) = 0, (, t, ) 0, B (, t, ) = 0, (, t, ) = 0 nd A (, t, ) + A(, t, ) = 0. The bove equtions imly tht At (, ) = ln( D() t + D()) t nd B( ) = + b, whee b, ( 0) nd D () t is nowhee zeo function of integtion nd D () t is function of integtion. The nk of the 6 6 iemnn mtix is thee nd thee exists no non tivil solution of eqution (). Substituting the bove infomtion in eqution (7) nd the line element tkes the fom ds = ( D ( t) + D ( t)) dt + e ( d + d + dφ ). (9) ( + b) This cse belongs to the clss A. In this clss the nk of the 6 6 iemnn mtix my be 6, 5,, o (excluding the clss B) nd thee exists no non tivil solution of eqution (). It follows fom [5, 7] CCS in this cse e homothetic vecto fields. Hence in this cse no oe CC exists. Cses (A) to (A) e exctly the sme. Cse C In this cse we hve At ( t,, ) 0, A ( t,, ) = 0, A ( t,, ) = 0, Bt (, t, ) = 0, (, t, ) 0, (, t, ) B(, t, ) (, t, ) = 0, (, t, ) + B (, t, ) 0, B (, t, ) 0 nd the nk of the 6 6 iemnn mtix is thee. Hee, thee exists unique (u to scling) nowhee zeo vecto field t = t, such d tht t b ; = 0. Fom the icci identity bcdt = 0. The bove constints give (, ) = ln( () + q()) nd A= A( t), whee ( ) nd q( ) e nowhee zeo functions of integtion. Substituting the bove infomtion in eqution (7) nd fte suitble escling of t, the line element cn witten in the fom ds = dt + ( ( ) + q( )) ( d + d + dφ ). (0) The bove sce-time is clely + decomosble nd belongs the cuvtue clss C. CCS in this cse [5] e X = Nt () + X, () t

9 9 Cuvtue collinetions in secil non sttic sce-time whee N( t ) is n bity function of t nd X is homothetic vecto field in the induced geomety on ech of the thee dimensionl submnifolds of constnt t. The induced metic g αβ (whee α, β =,, ) with non zeo comonents is given by g = g = g = ( ( ) + q( )) () A vecto field X is clled homothetic vecto field if it stisfies L g = cg, c. () X αβ One cn exnd the bove eqution () by using () to get αβ ( ) X + q ( ) X + ( ( ) + q( )) X, = c( ( ) + q( )), () X, + X, = 0, (5) ( ) X + q ( ) X + ( ( ) + q( )) X, = c( ( ) + q( )), (6) X, + X, = 0, (7) X, + X, = 0, (8) ( ) X + q ( ) X + ( ( ) + q( )) X, = c( ( ) + q( )). (9) Diffeentiting equtions (7) nd (8) with esect to nd, esectively nd fte subtcting them we get X, X, = 0. Now diffeentiting eqution (5) with esect to φ gives X, + X, = 0. Solving the bove equtions we hve X, = X, = 0 X = A (, ) + A (, φ ) nd X = A (, ) + A ( φ, ), whee A (, ), A (, φ ), A (, ) nd A (, φ ) e functions of integtion. Fom the bove infomtion eqution (7) gives X = A (, ) 5 φ φ d+ A ( φ, ), 5 whee A (, ) bove system of equtions we need to find A (, ), A (, φ ), A ( ) 5 A (, φ ) nd (, ). φ is function of integtion. In ode to find the solution of the,, A φ To void lengthy clcultions hee we will only esent the esults. Solution of the bove equtions fom () to (9) φ X = c+ c, X = c+ c, X = c+ c, + k + k + k k + k + k whee () = c+ c, q( ) = c+ c nd c, c, c, k k c + k k c + k ( k 0, ). The sub cse when k = 0 o k = will be discussed lte. In this cse k (0)

10 Ghulm Shbbi, M. mzn 0 the induced geomety on ech of the thee dimensionl submnifolds of constnt t dmit oe homothetic vecto field. CCS in this cse e given by use of equtions (0) in () s 0 X Nt (), φ X c c, X c c, X c c. + k + k + k = = + = + = + One cn wite the bove eqution () fte subtcting oe homothetic vecto fields X = ( N( t),0,0,0). () CCS clely fom n infinite dimensionl vecto sce. Now conside the sub cse when k =. In this cse homothetic vecto field is Killing vecto field which is X X X c () = = 0, =, () whee c. Poe CCS in this ce e given in (). Now conside the sub cse when k = 0. In this cse the induced geomety on ech of the thee dimensionl submnifolds of constnt t dmit oe homothetic vecto field which is X = c + c, X = c + c, X = φc + c, () whee () = ln( c+ c ), c q( ) = ln( c+ c ) c nd c, c, c, c ( c 0). CCS in this cse e given by the use of eqution () in () s 0 X Nt (), = φ X = c + c, X = c + c, X = c + c. (5) Poe CCS in this ce e given in (). Cses (C) to (C0) e ecisely the sme. Cse C In this cse At (, t, ) = 0, A (, t, ) = 0, Bt (, t, ) 0, B t ( t,, ) + Btt( t,, ) At( t,, ) Bt( t,, ) = 0, B (, t, ) = 0, (, t, ) = 0 nd the nk of the 6 6 iemnn mtix is thee. Hee, thee exists unique (u to multile) t = t, solution of eqution () but t is not covintly constnt. Fom the bove constints, we hve A= A( t) nd B() t = ln( e dt+ b), whee b, ( 0). The line element fte suitble escling of t cn be witten s At () ds = dt + ( t + b) ( d + d + dφ ). (6)

11 Cuvtue collinetions in secil non sttic sce-time 5 The bove sce-time become secil clss of FW K=0 model. It follows fom [9] oe CCS in this cse e given in eqution (). Cses (C) to (C) e exlicitly the sme. Cse D Hee, we hve At (, t, ) 0, A (, t, ) = 0, Bt (, t, ) = 0, (, t, ) = 0, B (, t, ) = 0, A (, t, ) + A(, t, ) 0nd the nk of the 6 6 iemnn mtix is one. Thee exist two linely indeendent solutions =, nd φ = φ, of eqution () nd stisfying b ; = 0 nd φ b ; = 0. Fom the bove constints, we get A= A(, t ) nd B = m, whee m. The line element fte escling nd φ, cn be witten s At (, ) m ds = ( e dt + e d ) + d + dφ. (7) The bove sce-time (7) is clely ++ decomosble nd belongs to the cuvtue clss D. CCS in this cse e [5] X = J( φ, ) + I( φ, ) + Y, φ (8) whee J (, φ ) nd I(, φ ) e bity functions of nd φ nd Y is CC on ech of two dimensionl submnifolds of constnt nd φ. The next ste is to wok out the CCS in the induced geomety of the submnifolds of constnt nd φ. The method fo finding CCS in two dimensionl submnifolds is given in [5]. The non zeo comonents of the induced metic on ech of the two dimensionl submnifolds of constnt nd φ e given by g e At (, ) 00 = g e m. The non-zeo icci tenso comonents e ( ) A A e At (, ) m 00 = + = (9) = ( A + A ) (0) nd the icci scl is given by = ( A + A ) e m. We lso hve Gαβ gαβ, whee α, β = 0, with non-zeo comonents G ( A A ) e At (, ) m 00 = + G = ( A + A ) ()

12 6 Ghulm Shbbi, M. mzn It follows fom [5] CCS in the two dimensionl submnifolds of constnt nd φ e the solution of the eqution LG Y αβ = 0. Exnding the evious eqution nd using () to get (( A + A ) e ) X + (( A + A ) e ) X + ( A + A ) e X = 0, () At (, ) 0 At (, ) At (, ) 0,0 m A( t, ) 0 e X,0 e X, 0, = () ( A + A ) X + ( A + A ) X + ( A + A ) X = 0. () 0, The only solution of the bove system is (which is tivil solution) X 0 X 0. = = (5) Poe CCS in this cse e X = (0,0, J(, φ), I(, φ)). (6) Clely CCS in this cse lso fom n infinite dimensionl vecto sce. Cse (D) is exctly the sme. Cse D In this cse At (, t, ) = 0, A (, t, ) = 0, Bt ( t,, ) = 0, (, t, ) = 0, B (, t, ) 0, B (, t, ) = 0 nd the nk of the 6 6 iemnn mtix is one. Fom the bove constints, we hve A= A( t) nd B = + b, whee b, ( 0). Hee, thee exist two linely indeendent solutions t = t, nd =, of eqution (). The vecto field t is covintly constnt whee s is not covintly constnt. The line element, fte escling of t cn be witten s + b ds = dt + e ( d + d + dφ ). (7) The bove sce-time (7) is clely + decomosble nd belongs to the cuvtue clss D. Substituting the bove infomtion in CC equtions in [6] one finds tht CCS in this cse e = 0 X M(, t ), X = cφ + c X N(, t ),, = X = c+ c (8), whee M ( t, ) nd N( t, ) e the bity functions nd cc,, c. One cn wite the bove eqution (8) fte subtcting the Killing vecto fields s X = ( M(, t ), 0, N(, t ), 0). (9) CCS clely fom n infinite dimensionl vecto sce. Cses (D) to (D6) e ecisely the sme.

13 Cuvtue collinetions in secil non sttic sce-time 7 CONCLUSION In this e we investigted the secil non-sttic xilly symmetic sce times ccoding to thei oe CCS. An och is doted to study oe CCS the bove sce times by using the nk of the 6 6 iemnn mtix nd lso using the theoem given in [5], which suggested whee oe cuvtue collinetions exist. Fom the bove discussion we obtin the following esults: (i) The cse when the nk of the 6 6 iemnn mtix is thee nd thee exists unique nowhee zeo indeendent timelike vecto field which is solution of eqution () nd is covintly constnt. This is the sce-time (0) nd it dmits oe CCS which fom n infinite dimensionl vecto sce (see cse C). (ii) The cse when the nk of the 6 6 iemnn mtix is thee nd thee exists unique nowhee zeo indeendent timelike vecto field which is solution of eqution () nd is not covintly constnt. This is the sce-time (6) nd it dmits oe CCS which fom n infinite dimensionl vecto sce (fo detils see cse C). (iii) The cse when the nk of the 6 6 iemnn mtix is one nd thee exists two nowhee zeo indeendent vecto fields which e solutions of eqution () nd both e covintly constnt. This is the sce-time (7) nd it dmits oe CCS, which fom n infinite dimensionl vecto sce (see cse D). (iv) The cse when the nk of the 6 6 iemnn mtix is one nd thee exists two nowhee zeo indeendent vecto fields which e solutions of eqution () but only one is covintly constnt. This is the sce time (7) nd it dmits oe CCS, which fom n infinite dimensionl vecto sce (see cse D). EFEENCES. C.W. Misne, K.S. Thone nd J.A. Wheele, Gvittion, Feemn, Sn Fncisco, 97.. A. Bnes, Clss. Quntum Gv. 0 (99) 9.. G.H. Ktzin, J. Levine nd W.. Dvis, J. Mth. Physics, 0 (969) 67.. G.H. Ktzin nd J. Levine, Colloq. Mthemticum, 6 (97). 5. G.S. Hll nd J. d. Cost, J. Mth. Physics, (99) G. Shbbi, A.H. Bokhi nd A.. Kshif, Nuovo Cimento B, 8 (00) G.S. Hll, Symmeties nd cuvtue stuctue in genel eltivity, Wold Scientific, S.. oy nd V.N. Tithi, Gen. eltivity nd Gvittion, (97). 9. G.S. Hll nd G. Shbbi, Clssicl Quntum Gvity, 8 (00) 907.