Proper Projective Symmetry in some well known Conformally flat Space-Times

Transcription

1 roper rojeie Smmer in some well nown onformll fl Spe-Times Ghulm Shir Ful of Engineering Sienes GIK Insiue of Engineering Sienes nd Tehnolog Topi Swi NWF isn Emil: sr sud of onformll fl u non fl Binhi pe I nd linill smmeri si spe-imes ording o proper projeie smmer is gien using some lgeri nd dire inegrion ehniques. I is shown h he speil lss of he oe spe-imes dmi proper projeie eor fields.. INTOUTION Through ou M is represening he four dimensionl onneed husdorff spe-ime mnifold wih Loren meri g of signure -. The urure ensor ssoied wih g hrough Lei-ii onneion is denoed in omponen form d he Wel ensor omponens re d nd he ii ensor omponens re. The usul orin pril nd Lie deriies re denoed semiolon omm nd he smol L respeiel. ound nd squre res denoe he usul smmeriion nd sew-smmeriion respeiel. spe-ime is sid o e onformll fl if d eer where on M. Finll M is ssumed o e non-fl in he sense h he urure ensor does no nish oer non-emp open suse of M nd is no of onsn urure. n eor field on M n e deomposed s ; h F

2 where h h L g nd F F re smmeri nd sew smmeri ensor on M respeiel. Suh eor field is lled projeie if he lol diffeomorphisms ψ for pproprie ssoied wih mp geodesis ino geodesis. This is equilen o he ondiion h h sisfies h ; g g g for some smooh losed -form on M wih lol omponens. Thus is loll grdien nd will where pproprie e wrien s for some funion on some open suse of M. If is projeie eor field nd ; hen is lled speil projeie eor field on M. If h ; on M is from equilen o eing ero on M nd is in urn equilen o eing n ffine eor field on M so h he lol diffeomorphisms presere no onl geodesis u lso heir ffine prmeers. If is projeie u no ffine hen i is lled proper projeie []. Furher if is ffine nd h g hen is homohei oherwise proper ffine. If is homohei nd i is proper homohei while i is Killing. ψ. OJETIVE SYMMETY If e projeie eor field on M. Then from nd [] L d δ d δ L ; ; ;. lso he ii ideni on h gies h e e d h e e d g g g g ; d d ; ; d d ;. Le e projeie eor field on M so h nd holds nd le F e rel urure eigenieor p M wih eigenlue λ so h d F d λf p hen p one hs [] F F λ h ; Equion gies relion eween F nd seond order smmeri ensor p nd refles he lose onneion eween h ; nd he lgeri

3 sruure of he urure p. If F is simple hen he lde of F wo dimensionl suspe of M T p onsiss of eigeneors of wih sme eigenlue. Similrl if F is non-simple hen i hs wo well defined orhogonl imelie nd spelie ldes p eh of whih onsiss of eigeneors of wih sme eigenlue [5].. Eisene of rojeie eor field in non fl onformll fl linill smmeri si spe-imes onsider linill smmeri si spe-ime in he usul oordine ssem r leled respeiel wih line elemen [6] ds r r - e d e u d e w r d. Sine we re ineresed in hose ses when he oe spe-ime is eome onformll fl u non fl I follows from [7] here eiss onl one possiili whih is: r u r w r se In his se he oe spe-imes eomes ds r r - e d e d d. 5 The oe spe-ime 6 dmis si independen Killing eor fields whih re. These si Killing eor fields re lerl ngen o he fmil of hree dimensionl imelie hpersurfes of onsn r. onsequenl hese hpersurfes re onsn ero urure. The ii ensor Segre of he oe spe-ime is { } or { }. If he Segre is { } hen he spe-ime is of onsn urure nd he projeie eor fields re gien in []. Here i is ssumed h he spe-ime is no onsn urure. The non-ero independen omponens of he iemnn ensor re

4 . Wriing he urure ensor wih he omponens well nown w [9] d dig 6 d p s 6 6 mri in where nd re rel funions of r onl nd where he 6-dimensionl lelling is in he order wih. Here p M one m hoose erd r sisfing r r wih ll ohers inner produs ero suh h he eigenieor of he urure ensor p re ll simple wih ldes spnned he eor pirs r r r eh wih eigenlue nd eh wih eigenlue. Here p we re onsidering he open suregion where nd re nowhere equl if hn i follws from 6 he oe spe-ime 5 eomes onsn urure whih gies onrdiion o our ssumion. So nd if hn he rn of he 6 6 iemnn mri eome hree nd i follows p from [] no proper projeie will eis. So. Thus p he ensor hs eigeneors r wih sme eigenlue s δ nd h ψ ; h ψ ; hs eigeneors wih sme eigenlue s δ. Hene on M one hs fer use of he ompleeness relion h ψ ; δg h ψ ; δ g δ r r 7 where δ δ nd δ re some rel funions on M. Sine hen i follows from 7 h h g r r ψ F g F r r ; for some rel funions E nd F on M. Ne one susiues he firs equion of in nd onrs he resuling epression firs wih hen o ge ψ ψ nd ψ nd so ψ η r for some funion η.

5 The sme epression onred wih gies r. ψ Now gin he sme epression onred wih gies r r nd using he oe informion η r nd hene r. onsider he equion ψ η r nd fer ing he orin deriie we ge ψ η r η r. Ne onsider he ; ; seond equion of nd use ψ ; η r ; η r nd hen onr wih r o ge η r so h η ηr. onsider he firs equion of nd use 5 one hs he following non-ero omponens of h h e h h e nd h e. 9 Now we re ineresed in finding projeie eor fields using he following relion L g h. Using equion 9 nd 5 in nd wriing ou epliil we ge e 5 e 6 e 7 9. Equions nd gie 5

6 6 e e e where nd re funions of inegrion. In order o deermine nd we need o inegre he remining si equions. To oid deils here we will presen onl he resul. The soluion of he equions is 6 5 proided h d where. 6 5 fer suring Killing eor fields from one hs proided h. d Suppose r where r nd. r The eor field is hen projeie if i sisfies. So using he oe informion in gies nd lso. r ψ priulr soluion of is r e r

7 r where nd e. Thus he spe-ime 5 dmis proper projeie eor field for he speil hoie of s gien in.. Eisene of rojeie eor field in non fl onformll fl Binhi pe I spe-imes onsider Binhi pe- spe-ime in he usul oordine ssem leled respeiel wih line [] ds d d h d f d. 5 The oe spe-ime dmis hree linerl independen illing eor fields whih re. Sine we re ineresed in hose ses when he oe spe-ime 5 is eome onformll fl u non fl I follows from [7] here eiss onl one possiili whih is: h f se In his se he oe spe-imes eomes d d d 6 ds d nd i dmis si independen Killing eor fields whih re. These si Killing eor fields re lerl ngen o he fmil of hree dimensionl imelie hpersurfes of onsn. onsequenl hese hpersurfes re onsn ero urure. The Segre pes of he oe speime re { } or {}. If he Segre is { } hen he spe-ime is of onsn urure nd he projeie eor fields re gien in []. Here i is ssumed h he spe-ime is no of onsn urure. The proper projeie 7

8 eor fields for he oe spe-ime 6 re lso ille in [ ]. The non-ero independen omponens of he iemnn urure ensors re && & & B. Wriing he urure ensor wih he omponens well nown w [9] 7 d p s 6 6 mri in d dig B B B where nd B re rel funions of onl nd where he 6-dimensionl lelling is in he order wih. Here p M one m hoose erd sisfing wih ll oherr inner produs ero suh h he eigenieors of he urure ensor p re ll simple wih ldes spnned he eor pirs eigen lue p nd eh wih eigenlue B p eh wih. Here we re onsidering he open suregion where nd B re nowhere equl if B hn i follws from 7 he oe spe-ime 6 eomes onsn urure whih gies onrdiion o our ssumion. Hene B nd if hn he rn of he 6 6 iemnn mri eome hree nd i follows from [] no proper projeie will eis. Hene. Thus p he ensor h ; ψ hs eigeneors nd wih sme eigenlue s γ nd ψ hs eigeneors nd wih sme eigenlue s γ. Bh ; Firs onsider he equion ψ where is seond order h ; smmeri ensor wih eigeneors nd wih sme eigenlue γ. The Segre pe of is { } nd γ g. Susiuing we ge h ψ γ ; g. Now onsider Bh ; ψ where is seond order smmeri ensor wih eigeneors nd wih sme eigenlue s

9 γ. The Segre pe of is { } nd γ g. Susiuing γ we ge ψ γ γ. Hene on M one hs Bh ; g h ψ γ g Bh ψ γ g γ ; ; where γ γ nd γ re some rel funions on M. Sine from equion h h g α ; B hen i follows ψ E g F 9 for some rel funions α E nd F on M. Now one susiues he firs equion of 9 in nd onrs he resuling epression wih nd hen wih ψ nd one hs ψ ξ for some o ge ψ ψ funion ξ. The sme epression onred wih hen show h α ξ nd so α is funion of onl. Now gin onr he sme epression wih one finds ξ whih implies. Susiuing o ge α ξ nd hene α α. Now onsider he seond equion of 9 nd use ψ nd onr wih. One n esil find h ξ ξ. ; ξ ξ ; onsider he firs equion of 9 nd use 6 one hs he following non ero omponens of h h α h h nd h where α α nd α α. Now we re ineresed in finding projeie eor fields using he relion. Wriing ou equion epliil nd using 6 nd we ge α 9

10 & & 9. & Equions nd gie d d d d α where nd re funions of inegrion. In order o deermine nd we need o inegre he remining si equions. To oid lengh lulions here we will presen onl he resul. The soluion of he equions is d α proided h d & & α where fer suring Killing eor fields from one hs

11 proided h d α & α d & Suppose η η. η d α nd where The eor field is sid o e projeie if i sisfies. & So using he oe informion in gies && η & & & η && η && η & & η η & η & nd ψ & η. priulr soluions of re I F FG I η e F F Le η Ne 5 where F G I L N F. Thus he spe-ime 5 dmis proper projeie eor field for he speil hoie of s gien in nd 5.

12 eferenes []. Brnes lss. Qunum Gr [] G. S. Hll lss. Qunum Gr [] G. S. Hll Smmeries nd urure Sruure in Generl elii World Sienifi. [] G. S. Hll ifferenil Geomer nd is ppliion Msr Uniersi Brno eh epuli 996. [5] G. S. Hll nd. B. G. MInosh In. J. Theor. hs [6]. Krmer H. Sephni M.. H. Mllum nd E. Herl E Soluions of Einsein s Field Equions mridge Uniersi ress. [7] G. Shir Nuoo imeno B 7. [] G. Shir. H. Bohri nd.. Kshif Nuoo imeno B 7. [9] G. Shir lss. Qunum Gr. 9. [] G. Shir Griion nd osmolog 9 9. [] G. Shir Nuoo imeno B 9. [] G. S. Hll nd.. Lonie lss. Qunum Gr [] G. S. Hll nd M. T. el lss. Qunum Gr. 9 9.