The Dirac equation in D-dimensional spherically symmetric spacetimes

Transcription

1 The Dir equion in D-dimensionl spherilly symmeri speimes A López-Oreg Cenro de Invesigión en Cieni Aplid y Tenologí Avnzd Unidd Legri Insiuo Poliénio Nionl Clzd Legri # 694 Coloni Irrigión Delegión Miguel Hidlgo Méxio D F C P 500 Méxio E-mil: lopezo@ipnmx (Reeived June 009 eped 3 Augus 009) Asr We expound in deil mehod frequenly used o redue he Dir equion in D-dimensionl (D 4) spherilly symmeri speimes o pir of oupled pril differenil equions in wo vriles As simple ppliion of hese resuls we exly lule he qusinorml frequenies of he unhrged Dir field propging in he D- dimensionl Nrii speime Keywords: Dir field; Nrii speime Spherilly symmeri usinorml modes Resumen Exponemos on delle un méodo freuenemene usdo pr simplifir l euión de Dir en espioiempos esférimene simérios en D-dimensiones (D 4) un pr de euiones difereniles priles en dos vriles Como un pliión dire de esos resuldos lulmos ls freuenis usinormles del mpo de Dir sin rg en el espioiempo D-dimensionl de Nrii Plrs lve: Cmpo de Dir espioiempo de Nrii esférimene simério modos usinormles PACS: 0460Kz 0470-s 0470Bw ISSN I INTRODUCTION Reenly in mny reserh lines of heoreil physis he models in whih he speime hs more dimensions hn he four dimensions oservle in our dily experiene hve een sudied exensively The mos nlyzed models re hose reled o sring heory [] Also he sruiny of he properies nd soluions of higher dimensionl generl reliviy hs red lo of enion (see Ref [] nd referenes herein) In severl of hese reserh lines we need o know he lssil properies of he higher dimensionl speimes o exmine differen phenomen Therefore he invesigion of hese lssil properies is n ive reserh field To nlyze he lssil properies of given speime ommon mehod is o use field s proe [3 4] Thus in he ps severl sering phenomen of lssil fields were sudied in order o know how o lule he physil prmeers of he speime from he mesured vlues of he physil quniies orresponding o he lssil field The qusinorml modes (NMs) re soluions o he equions of moion for lssil field h sisfy he rdiion oundry ondiions h re nurl in he speime in whih he field is propging [3 4] For exmple in sympoilly fl lk holes he oundry ondiions of he NMs re h he field is purely ingoing ner he even horizon nd purely ougoing ner infiniy [3] For sympoilly ni-de Sier lk holes we impose he oundry ondiion h he field vnishes infiniy nd is ingoing ner he even horizon I hs een shown h he NMs re useful ool o lule he physil prmeers of speime [3 4] Hene if we know he qusinorml frequenies (NF) of lssil field we n infer he vlues of severl physil quniies of he speime suh s is mss hrge nd ngulr momenum [3] Furhermore i hs een proposed h he NMs enode some informion ou he qunum properies of he lk holes [5] To ompue he NF of lssil field in given speime he usul proedure is o redue he equions of moion for he field o rdil ordinry differenil equion (ssuming given dependene on he ngulr vriles nd hrmoni ime dependene) nd impose o he rdil funion he oundry ondiions of he NMs Also noie h he redued form of he equions of moion is useful (nd someimes neessry) o sudy mny oher lssil or semilssil phenomen Thus we elieve h presen ime he undersnding of he sepriliy properies of he equions of moion for lssil fields in higher dimensionl urved speimes mus e relevn pr in he eduion of physiis Moived y hese heories h ssume numer of speime dimensions greer hn four he sepriliy properies of he equions of moion for severl lssil L Am J Phys Edu Vol 3 No 3 Sep hp://wwwjournllpenorgmx

2 A López-Oreg fields were sudied in higher dimensionl kgrounds I ws found h mny of he well known resuls h re rue for four-dimensionl spherilly symmeri speimes exend o D-dimensionl (D 4) spherilly symmeri speimes For exmple he reduion of he equions of moion for Klein-Gordon eleromgnei nd grviionl perurions o ordinry differenil equions whih is rue in four-dimensionl unhrged spherilly symmeri speimes [4] lso is vlid in D-dimensionl unhrged spherilly symmeri speimes s showed in Refs [6] [7] Moreover for he oupled grviionl nd eleromgnei perurions he reduion of he equions of moion o Shrödinger ype equions whih is rue in four-dimensionl hrged spherilly symmeri kgrounds lso is vlid for D-dimensionl hrged spherilly symmeri kgrounds [8] Alhough he sudy of he lssil dynmis of fields in urved speimes (in four nd D dimensions) is foused on oson fields [ ] minly on grviionl perurions we elieve h he undersnding of he lssil dynmis of he fermion field in D-dimensionl speimes is of gre vlue euse he Dir field someimes ehves in differen wy h he oson fields For exmple i is well known f h in he fourdimensionl Kerr lk hole he fermion field does no show superrdin sering [9 0] unlike o oson fields [4] For he Dir equion is sepriliy properies in D- dimensionl spherilly symmeri speimes were previously sudied in Refs [ ] In hese ppers is shown h he Dir equion redues o pir of oupled pril differenil equions in wo vriles Owing o he ps nd fuure ppliions of he redued sysem of pril differenil equions for he Dir equion in D-dimensionl spherilly symmeri speimes we elieve h he mehod used in Refs [] o redue he Dir equion o pir of oupled pril differenil equions deserves deiled exposiion euse his oun my e pril nd useful Here we presen he mehod in more deil hn in he originl referenes h is in he presen work we expliily wrie some mhemil seps omied in Refs [ ] (see lso [5 6 7]) Noie h Seion II is no n exhusive review of he previous work on he dynmis of fermion fields in spherilly symmeri speimes Also oserve h in his pper we do no onsider in deil he mhemil properies of D-dimensionl spinors hese n e sudied in mny ooks nd ppers (see for exmple Refs [8 9 0]) We only wrie he essenil properies of he spinors neessry o mke he reduion of he Dir equion o wo oupled pril differenil equions h we shll expound in Seion II Reenly he ex ompuion of he NF for severl higher nd lower dimensionl speimes hs red lo of enion see []-[39] for some referenes in whih n ex lulion of he NF ws rried ou As mny exly solvle models in heoreil physis we elieve h hese exmples re useful models nd i is possile h hey ply relevn role in fuure reserh The Nrii speime is vuum soluion o he Einsein equions wih posiive osmologil onsn [40] This speime is simple soluion o he field equions of generl reliviy Owing o his simpliiy of he Nrii soluion i is possile o lule he vlues of severl physil quniies in ex form For exmple for his speime in Refs [39] were ompued exly he vlues of he NF for Klein-Gordon fields nd ensor ype grviionl perurions Furhermore in he D-dimensionl hrged Nrii speime [8 40] he NF for he oupled grviionl nd eleromgnei perurions were luled exly in Ref [38] To our knowledge he resul of he previous referene in he hrged Nrii speime is he only ex lulion of NF for he oupled eleromgnei nd grviionl perurions in higher dimensions As n ppliion for he redued sysem of differenil equions oined in Seion II for he Dir field moving in D-dimensionl spherilly symmeri speimes we exly lule he NF of his field in D-dimensionl Nrii speime [40] These vlues of he NF for he Dir field exend hose lredy pulished in Refs [38 39] In his pper we ssume h he reder hs working knowledge of generl reliviy nd differenil geomery Furhermore in he following seions we use Einsein s sum onvenion nd undersnd sum on repeed indies (Lin nd Greek indies) unless we expliily se h in given formul we do no undersnd sum on repeed indies The pper is orgnized s follows In Seion II we presen in deil he mehod of Refs [] (see lso [5 6 7]) h redues he Dir equion in D-dimensionl spherilly symmeri speimes o pir of oupled pril differenil equions in wo vriles Using hese resuls in Seion III we exly lule he NF of he Dir field propging in he D-dimensionl Nrii speime Finlly in Seion IV we disuss some reled fs II DIRAC S EUATION IN D-DIMENSIONAL SPHERICALLY SYMMETRIC SPACETIMES As is well known in D-dimensionl spherilly symmeri kgrounds he Dir equion iγ ψ = mψ () Noie h for he Dir field some resuls vlid in four-dimensionl roing lk holes hve een exended o roing lk holes in higher is dimensions see Refs [3] for n inomplee lis of referenes For review of he reen work on he sepriliy properies for he equions of moion Noie h Greek indies snd for he oordine indies wheres he for severl fields in higher dimensionl speimes see Ref [4] Lin indies snd for he frme indies L Am J Phys Edu Vol 3 No 3 Sep hp://wwwjournllpenorgmx redues o pir of oupled pril differenil equions in wo vriles [ 5 6 7] In his seion we desrie in deil he mehod of Refs [] frequenly used o ge his resul For differen mehod see Refs [] Here we shll onsider wo -dimensionl speimes M nd M whose meris g ν nd g ν re onforml h

3 g ν =Ω gν () where Ω is funion of he oordines We poin ou h if he symols γ ψ m nd γ ψ m denoe he Dir operor he Dir spinor nd he mss of he Dir field orresponding o he speimes M nd M respeively hen he following relions re sisfied [ 5 6 7] ( D )/ ψ =Ω ψ ( D+ )/ γ ψ =Ω γ ψ (3) m=ωm I is well known h he previous resuls n e generlized when here re guge fields [] u in his pper we do no nlyze his exension As in formule (3) in he res of he presen seion ilde snds for he quniies orresponding o he speime wih meri g ν nd similrly for oher symols o e used One wy o oin he resuls of formule (3) is he following When he meris g ν nd g ν re onformlly reled s in formul () nd we define he sis of oneforms e suh h gν = eeνη nd he sis e suh h g ν = e e νη where η = η = dig ( ) is he Minkowski meri [4 4] we find h he one-forms e nd e sisfy e =Ωe To ge he relion eween he onneion one-forms nd ω orresponding o he sis of one-forms e nd respeively we rell h he onneion one-forms deermined y he firs Crn sruure equion de = ω e ω e ω re where he symol snds for he wedge produ [4] [4] Using Semen 3 66 of Ref [4] we see h he one-forms ω nd ω re reled y ω = ω + ( e ( Ω) e e ( Ω) e ) (6) Ω The Dir equion in D-dimensionl spherilly symmeri speimes where e ( Ω ) denoes he ion of he veor e = η e on he slr funion Ω From expression (6) we ge ω ω = + ( e ( Ω ) η e ( Ω ) η ) (7) Ω Ω ω where ω = ω ( e ) nd similrly for Thus if he symol snds for he ovrin derivive of spinor h is [8 4] = e + ωγ γ (8) 4 where γ snds for he D-dimensionl gmm mries h sisfy [8 9 0] γ γ + γ γ = η (9) nd we lso oserve h γ = γ Tking ino oun he s previous resuls nd if we ke ψ =Ωψ hen i is possile o show h he D-dimensionl Dir operor rnsforms ino [ 5 6 7] γ ψ (0) γ ψ = γ Ω e + ( ωω+ e ( Ω) η 4 s e ( Ω) η) γ γ Ωψ () Ω D s = γ Ω e + ωγ γ γ γ e( Ω) Ωψ 4 s+ s+ Ω e =Ω γ ψ + ω γ γ γ ψ 4 s D +Ω γ e ( Ω) s ψ Tking s = ( D ) / in he previous formul we finlly ge he resul ( D+ )/ γ ψ =Ω γ ψ () 3 Semen 66 of Ref [4]: If ny se of -forms A is given nd {θ } is dul frme sis hen here exiss unique se of -forms χ suh h A + χ θ = 0 χ = χ ( D )/ This expression nd ψ =Ω ψ re he firs wo resuls of formule (3) The resul given in expressions (3) for he mss m immediely follows from he previous formule for The -forms χ n e expressed y he formul γ ψ nd ψ In he following prgrphs we sudy he Dir equion χ = χθ χ = ( A A A ) in he D-dimensionl spherilly symmeri speimes wih where A re he oeffiiens in he deomposiion oordines ( rφ i ) where i= ( D ) nd whose A = Aθ θ A = A line elemens we wrie in he form L Am J Phys Edu Vol 3 No3 Sep hp://wwwjournllpenorgmx

4 A López-Oreg ds = F() r d G() r dr H() r dσ (3) D where F ( r ) Gr () nd Hr () re funions only of he oordine r he symol dσ D snds for he line elemen of ( D ) -dimensionl invrin se spe whih depends only on he oordines φ i To simplify he Dir equion in speime whose meri kes he form (3) we for ou he funion H( r ) in he line elemen (3) nd define where 4 ds H( r) = (4) ds ν F G ds = g νdx dx = d dr dσ D (5) H H Nex we use he resuls (3) o find he relion mong he quniies ψ γ ψ m nd ψ γ ψ ψ m orresponding o he speimes wih line elemens ds of formul (3) nd ds of expression (5) We poin ou h in he line elemen (5) he firs wo erms depend only on he wo oordines r nd he erm dσ D depends only on he ( D ) oordines φ i As sis of one-forms for he speime wih meri (5) we hoose [ 5 6 7] e = e () r = f () rd+ f () rdr () ( r) r r e = e ( r ) = g ( rd ) + g ( rdr ) (6) () ( r) i i i j e = e ( φ ) = h ( φ ) dφ k j k where i j k = D We prefer his sis euse mny of he onneion one-forms re equl o zero for exmple he onneion one-forms ω j nd ω rj I is onvenien o oserve h in D even dimensions he gmm mries re squre mries of dimension D/ D/ wheres in D odd dimensions hese re of ( D )/ ( D )/ dimension [8 9 0] Thus if in he D- dimensionl speime wih meri (5) we use he represenion of he gmm mries [ ] γ = σ σ σ σ = σ I ( D )/ γ r = iσ σ σ σ = iσ I ( D )/ γ = iσ σ σ σ = σ ˆ γ (7) γ = iσ σ σ σ = σ ˆ γ γ 3 = iσ ˆ 3 σ3 σ σ0 = σ3 γ3 γ = = σ ˆ γ D 3 D where I ( )/ is he ideniy mrix of dimension D ( D )/ ( D )/ he symol snds for he dire produ [8 9] ˆγ ˆ γ ˆD γ re represenion of he gmm mries for ( D ) -dimensionl spe wih signure ( ) nd 0 0 σ0 = σ = i 0 σ = σ3 = i 0 0 (8) h is σ σ nd σ re he Puli mries 3 Using he sis of one-forms (6) nd he represenion for he gmm mries given in formule (7) in he speime wih line elemen ds of formul (5) we find h he D-dimensionl Dir operor γ eomes 5 [ 5 6 7] = r D γ γ γ r γ γ D ( D) r ( D) ( ˆ γ ˆ γ r ) ( D )/ ( D ) D ( D ) + σ ˆ ˆ 3 ( γ + + γ D ) = + I (9) = [ γ I iσ γ ] D ( D )/ 3 dσ ( D) ( D ) where r nd i snd for he ovrin derivives of spinor in wo nd ( D ) dimensions respeively ˆ γ nd ˆ γ r re represenion of he gmm mries in wo dimensions γ is he Dir operor D on he wo-dimensionl speime whose line elemen is F G ds = D d dr H H (0) 4 We shll wrie he funions F(r) G(r) nd H(r) simply s F G nd H respeively In generl we shll use similr onvenion for he funions h we shll define in he following prgrphs 5 In formule (9) here is no sum on he repeed indies r nd D- L Am J Phys Edu Vol 3 No 3 Sep hp://wwwjournllpenorgmx

5 nd γ is he Dir operor on he dσ ( D ) - dimensionl sumnifold wih line elemen dσ D nd wih signure ( + + ) For mny relevn speimes dσ D is he line elemen of ( D ) -dimensionl sphere u his is no he only opion lso he quoiens of hyperoli spes re possile [ ] Tking he spinor ψ of he speime wih line elemen (5) in he form ψ ( rφ) = ψ () r χ( φ) () i D i where ψ () r D is wo-spinor on he speime wih line elemen ds D of formul (0) nd he funions χ ( φ i ) sisfy γ χ = κχ () dσ h is χ nd κ denoe he eigenfunions nd eigenvlues of he Dir operor on he mnifold wih line elemen dσ D [44] From formul (9) we oin h he spinor ψ of expression () sisfies γ ψ = [ γ iσ κ] ψ χ (3) D 3 D Thus in he D-dimensionl spherilly symmeri speime wih line elemen (3) he Dir equion () redues o γ ψ = ( κiσ imhi ) ψ (4) D D 3 D The Dir equion in D-dimensionl spherilly symmeri speimes dy G = (7) dr F we find h Eq (6) eomes 6 y F ( γ + γ y) ψ D = ( i κσ3 imh I ) ψ D (8) H where γ nd γ y re represenion of he gmm mries in wo speime dimensions Here for he wo-dimensionl gmm mries γ nd y γ we use he represenion [ 5 6 7] 0 y 0 γ = γ = 0 0 (9) o find h in he D-dimensionl spherilly symmeri speime wih line elemen (3) he Dir equion () redues o ψ F F G ψ κ H = ψ r i i F ψ F F i i F G ψ κ H + = + ψ r where he funions ψ nd ψ re he omponens of he wo-spinor ψ h is D (30) ψ ψ D = (3) ψ Nex from he wo-dimensionl line elemen (0) we define he line elemen ds D y ds D of Eq F G F D = = D ds d dr ds (5) H F H nd using he resuls (3) we oin h he wo-spinor of he speime wih line elemen equion where ψ D ds D sisfies he F γ Dψ D = ( i κσ3 imh I ) ψ D (6) H γ D snds for he Dir operor on he wodimensionl speime wih line elemen formul (5) Tking he vrile y s ds D defined in Thus we ge h in he D-dimensionl spherilly symmeri speime wih line elemen (3) he Dir equion () redues o he pir of oupled pril differenil equions in he vriles nd r given in Eq (30) This sysem of wo oupled pril differenil equions in wo vriles is he min resul h we se in his seion nd i ws previously oined in Refs [ ] I is onvenien o noie h in D-dimensionl de Sier speime whose line elemen in si oordines kes he form (3) when we wrie Eqs (30) for his speime we oin Eqs () of Ref [33] h we oin y using he resuls of Refs [] o redue he Dir equion Alhough we sudy spherilly symmeri kgrounds in D 4 dimensions we hink h he resuls oined in his seion lso re vlid in hree-dimensionl speimes whose meri n ke he form (3) We nno ompre in srighforwrd wy he redued sysem of pril differenil equions presened in his seion wih h of Ref [45] euse in he previous referene differen sis of oneforms ws hosen 6 Noie h in Eq (8) here is no sum on he repeed indies nd y L Am J Phys Edu Vol 3 No3 Sep hp://wwwjournllpenorgmx

6 A López-Oreg III UASINORMAL MODES OF THE DIRAC FIELD IN THE D-DIMENSIONAL NARIAI SPACETIME As we previously menioned he NF re omplex quniies h depend on he physil prmeers of speime Thus if we know he NF we n infer he vlues of severl physilly relevn quniies of he speime For mny relevn kgrounds for exmple Shwrzshild nd Kerr lk holes i is no possile o lule he vlues of heir NF in ex form we mus use pproxime or numeril mehods [3] Neverheless presen ime we know mny higher nd lower dimensionl speimes whose NF were ompued exly in Refs []-[39] The sysems h llow ex ompuions of some physil prmeers hve he dvnge h we n nlyze in more deil heir properies nd verify in simple seing some prediions of he physil heories Hene we elieve h hese exmples deserve deiled sudy Douless hese models will e useful in fuure reserh As n elemenry ppliion for he oupled sysem of pril differenil equions (30) for he Dir field propging in he D-dimensionl spherilly symmeri speimes h we presen in he previous seion here we exly ompue he NF for his field in he D-dimensionl Nrii speime whih is simple vuum soluion of he Einsein equions wih posiive osmologil onsn The line elemen of he D-dimensionl Nrii kground is [40] where dr d s = ( σ r )d d Σ D ( σ r ) (3) σ = ( D ) Λ (33) dσ D is he line elemen of ( D ) -dimensionl uni sphere he onsn is equl o ( D 3) = ( D ) Λ (34) nd he onsn Λ is reled o he osmologil onsn If σ > 0 hen he meri (3) hs wo osmologil horizons [40] r =± (35) σ In he following we ssume h he rdil oordine r ( / σ + / σ ) We noe h he D-dimensionl Nrii speime (3) hs he following feures [40]: () i hs geomery D S S where S snds for he wo-dimensionl de D Sier speime nd S denoes he ( D ) -dimensionl sphere () i is spherilly symmeri homogeneous nd lolly si () i is geodesilly omplee Owing o he D-dimensionl Nrii speime (3) is spherilly symmeri is meri n e wrien in he form (3) wih he funions F G nd H equl o / F = = ( σr ) H = (36) G We define he NMs of he Nrii speime s he modes h re purely ougoing ner oh horizons [38] [39] We lso noie h he resuls of his seion re n exension of hose lredy pulished in he previous wo referenes for oupled grviionl nd eleromgnei perurions Klein-Gordon fields nd ensor ype grviionl perurions To ompue he NF of he unhrged Dir field h is propging in he D-dimensionl Nrii speime (3) we firs wrie in his speime he redued sysem of pril differenil equions (30) for he Dir equion in D- dimensionl spherilly symmeri speimes We ge he following sysem of pril differenil equions / iκ ψ ( σr ) rψ = ( σr ) im ψ (37) / iκ ψ + ( σr ) rψ = ( σr ) + im ψ where κ re he eigenvlues of he Dir operor on he D- dimensionl sphere h is κ =± il ( + ( D )/) where l = 0 [44] If we ke he omponens ψ nd ψ of he wo spinor ψ D of formul (3) s ()e iω ψ = R r ()e iω ψ = R r (38) hen Eqs (37) rnsforms ino he oupled sysem of ordinry differenil equions ( ) dr dr σ r + iωr / ik = ( σ r ) + im R ( σr ) dr iωr (39) dr / ik = ( σ r ) im R L Am J Phys Edu Vol 3 No 3 Sep hp://wwwjournllpenorgmx

7 where we define he quniy K = κ Moreover defining he following quniies z= σr ˆ ω = ω/ σ mˆ = m/ σ nd λ = ik / σ we find h Eqs (39) eome dr / ( z ) + i ˆ ωr ˆ = ( z ) ( λ + im) R dz dr / ( z ) i ˆ ωr ˆ = ( z ) ( λ im) R (40) dz Also noie h z ( ) Nex we define (s in Chndrsekhr ook's [4]) h is nd king mˆ θ = rn λ mˆ os( ) λ = λ + θ ˆ λ mˆ sin( θ ) (4) m = + (4) R = e R R iθ / / ei θ R we see h Eqs (40) redue o = (43) dr / ( z ) + iˆ ωr = ( z ) αnr dz dr / ( z ) i ˆ ωr = ( z ) αn R (44) dz where α ˆ N = λ + m From he previous equions we ge h he deoupled ordinry differenil equions for he funions R nd R re equl o dr z dr ( ˆ ω iˆ ωzr ) α R N + = 0 dz z dz ( z ) z dr z dr ( ˆ ω + iˆ ωzr ) α R N + = 0 dz z dz ( z ) z To solve Eqs (45) we mke he hnge of vrile (45) The Dir equion in D-dimensionl spherilly symmeri speimes wih y (0) nd he nsz where y= ( z+ ) (46) R = ( y) y S ( y) B C R = ( y) y S ( y) (47) B C i ˆ ω + B = i ˆ ω i ˆ ω + C = i ˆ ω i ˆ ω B = i ˆ ω + i ˆ ω C = i ˆ ω + (48) o find h he funions S () y nd S ( y ) mus e soluions of he hypergeomeri differenil equion [46] d f df y( y) ( ( ) y) f 0 dy dy = (49) wih prmeers (he lower indies or deermine if he prmeer orrespond o he funion S ( y ) or o he funion S ( y ) ) = B+ C + iα N = B+ C iα N = C + (50) = B + C + iα N L Am J Phys Edu Vol 3 No3 Sep hp://wwwjournllpenorgmx

8 A López-Oreg = B + C iα N = C + In he following we sudy he funion R (we oin similr resuls for he funion R ) Also we ke he quniies B C s B ˆ = iω / nd C ˆ = iω / From he previous nd resuls we see h if he prmeer is no n ineger hen we wrie he funion R s [46] R = e ( y) D y F( ; ; y) (5) iθ / i ˆ ω / i ˆ ω / / i ˆ ω/ E + y F( + + ; ; y) where D nd E re onsns A his poin we noe h he oroise oordine for he D-dimensionl Nrii speime is equl o [ ] dr x = = σ r rnh( ) σ z (5) where x ( + ) nd from expression (46) we ge σ x s x y e σ x s x y e nd + (53) Thus ner he horizon r = / σ (h is s x ) he funion R (5) ehves s R De + E e e (54) iωx iωx σx In he D-dimensionl Nrii speime o sisfy he NMs oundry ondiion ner r = / σ h is he funion R ehves s exp( iωx) if he oroise oordine goes o minus infiniy x we ke D = 0 in formul (54) nd herefore he funion R eomes e i / ( ) i ˆ/ / i ˆ/ R θ y ω = E y ω F( + + ; ; y) (55) iθ / i ˆ ω / / i ˆ ω = E e ( y) y / F( α β ; γ ; y) We rell h if he quniy γ α β is no n ineger hen he hypergeomeri funion F ( α β; γ ; u) sisfies [46] Γ( γ ) Γ( γ α β) F( α β; γ; u) = Γ ( γ α) Γ ( γ β) F( α βα ; + β+ γ; u) (56) Γ( γ ) Γ ( α+ β γ) ( u ) γ α + β Γ( α) Γ( β) F ( γ αγ βγ ; + α β; u ) Thus if he quniy γ α β is no n ineger hen using formul (56) we wrie he rdil funion (55) s iθ/ / i ˆ ω/ Γ( γ) Γ( γ α β) R = E e y Γ ( γ α) Γ ( γ β) i ˆ ω / ( y) F( α β; α+ β+ γ; y) Γ( γ ) Γ ( α+ β γ) ( ) / + i ˆ ω / + y Γ( α) Γ( β) F ( γ α γ β ; γ + α β ; y) ] (57) Therefore s x + king ino oun expressions (53) we see h he funion R is pproximely equl o R Γ( γ) Γ( γ α β) e iωx Γ ( γ α) Γ ( γ β) Γ( γ) Γ ( α+ β γ) e iωx e σ x + Γ( α ) Γ( β ) (58) The oundry ondiion for he NMs of he D-dimensionl Nrii speime imposes h he funion R ehves in he form exp( iω x) s x + Thus o sisfy he oundry ondiion of he NMs for Nrii speime ner he horizon r =+ / σ we mus nel he seond erm in formul (58) One wy is o exploi he zeros of he erms / Γ ( x) whih re loed x = n n = 0 Hene o sisfy he oundry ondiion ner he horizon r =+ / σ we mus impose he ondiion α = n or β = n (59) whih imply h he NF of he Dir field in D - dimensionl Nrii speime re deermined y he expression ˆ ω =± α N i n+ (60) A similr ompuion for he rdil funion R lso yields he NF of formul (60) Tking ino oun h L Am J Phys Edu Vol 3 No 3 Sep hp://wwwjournllpenorgmx

9 α (l+ D ) 4 σ ˆ ˆ N = λ + m = m + (6) we find h in he D-dimensionl Nrii speime (3) he NF of he Dir field re equl o (l+ D ) ˆ ω=± σ m + i σ n+ 4 σ (6) In our noion he previously luled NF for Klein- Gordon field nd ensor ype grviionl perurion re wrien s [39] ω KG D 3 ( ll D ) / ( + 3) 4 =± i σ n+ (63) We poin ou h for l = 0 he NF (63) of he Klein-Gordon field re purely imginry This f ws no noed in Ref [39] When in he D-dimensionl hrged Nrii speime sudied in Ref [38] we ke he eleri hrge of he speime equl o zero for eleromgnei nd grviionl perurions of veor ype we ge heir NF re equl o 3D 3 ( ll D ) / ( 3) ωv =± i σ n+ (64) 5D ( ll ( + D 3) ) / 4 ωv =± i σ n+ wheres for eleromgnei nd grviionl perurions of slr ype heir NF re D 3 ( ll D ) / ( 3) ωs =± i σ n+ 9( D 3) / (( ll D 3) 4 ) i σ n + ωs =± + (65) The Dir equion in D-dimensionl spherilly symmeri speimes he NMs of hese fields he dey ime does no depend on he ngulr momenum numer l For Dir field he rel pr of he NF (6) show some differenes wih respe o rel pr of NF for Klein- Gordon eleromgnei nd grviionl perurions (63) (64) nd (65) We hink h he soure of hese differenes is h in he presen seion we sudy he Dir field wheres in Refs [38 39] he fields sudied re mssless oson fields From formul (6) we oin h for he mssless Dir field (Weyl field) is NF re equl o l D ω= + i σ n+ Noie h in Refs [38 39] o lule he NF (63) (64) nd (65) of he D-dimensionl Nrii speime he resul for he NF of Pöshl-Teller poenil ws used The NF of his poenil were previously ompued in he pper y Ferrri nd Mshhoon [47] I is onvenien o oserve h for idenil vlues of D n nd Λ he imginry pr of he NF (6) for Dir field is idenil o he imginry pr of he NF for grviionl nd eleromgnei perurions of veor ype nd slr U V( x) = (67) ype (formule (64) nd (65)) nd for Klein-Gordon fields osh ( σ x) nd ensor ype grviionl perurions (formul (63)) lredy ompued in Refs [38 39] Thus he dey ime where he vlue of he onsn U depends on he τ = / I m( ω) is he sme for fermion nd oson fields perurion ype (see Refs [38 39]) sudied here nd in Refs [38 39] Also we poin ou h for L Am J Phys Edu Vol 3 No3 Sep hp://wwwjournllpenorgmx (66) Even in his se he rel pr of he NF (66) shows differen dependene on he ngulr momenum numer h he rel pr of he NF for Klein-Gordon eleromgnei nd grviionl perurions The imginry pr is equl o h of he Dir field Thus for Dir field in he D- dimensionl Nrii speime he dey imes of is NMs do no depend on he mss Tking ino oun he resuls for NF (6) (63) (64) (65) nd (66) of he D-dimensionl Nrii speime we noe h heir rel nd imginry prs show n explii dependene on he dimension D of he speime (he imginry pr hrough he prmeer σ of formul (33)) For ll he fields whose NF hve een luled nd for given mode numer n we infer h he dey ime dereses s he dimension of he speime D inreses The dependene of he osillion frequeny on he speime dimension D is more omplied i firs dereses nd hen inreses s D inreses Moreover for l 3 nd for he sme vlues of D l nd n he osillion frequenies of he mssless Dir field (66) re greer hn he osillion frequenies for Klein-Gordon eleromgnei nd grviionl perurions of formule (63) (64) nd (65) As he hrmoni ime dependene is of he form exp( iω) (see formule (38)) in order o hve sle NMs we need h I m( ω ) < 0 We noie h for NF of formul (6) I m( ω ) < 0 hus he NMs of he Dir field dey in ime Also for Klein-Gordon eleromgnei nd grviionl perurions similr resul is rue (see formule (63) (64) nd (65)) herefore we sser h he D- dimensionl Nrii speime is perurively sle under he propgion of hese lssil fields To finish his seion we noe h in Refs [38 39] is shown h he rdil differenil equions for Klein-Gordon fields ensor ype grviionl perurions nd oupled eleromgnei nd grviionl perurions redue o Shrödinger ype equions wih Pöshl-Teller poenil of he form

10 A López-Oreg Following he proedure of Chper 0 in Ref [4] we rnsform Eqs (39) ino pir of Shrödinger ype equions wih poenils equl o (l+ D ) + mˆ σ 4 σ V± ( x) = (68) osh ( σ x) / (l+ D ) + mˆ sinh( x) σ σ 4 σ ± osh ( σ x) ( D 3) ( D 3) = ( D ) Λ+ (70) ( D ) where is reled o he eleri hrge of he speime We oserve h hese poenils re of Morse ype (see Tle I of Ref [48]) For plos of he poenils (67) nd (68) see Figures nd We noe h for idenil vlues of D Λ nd l he shpe of he poenils (67) nd (68) is similr only oserve h he heigh of Pöshl-Teller poenil is smller hn he heigh of Morse poenils Noie h in Figure is ploed he Pöshl-Teller poenil orresponding o he NF ω + V u for Pöshl-Teller poenils (67) orresponding o he NF ω V ω + S ω S nd ω KG lso is rue h for idenil vlues of D Λ nd l heir heigh is smller hn he heigh of Morse poenil (68) IV CONCLUDING REMARKS The D-dimensionl Nrii speime sudied in Seion III is unhrged Noie h here is hrged generlizion of he D-dimensionl Nrii speime only we need o reple he quniy σ of formul (33) y [8 40] ( D 3) σ = ( D ) Λ (69) ( D ) FIGURE Plo of Pöshl-Teller poenil V of Eq (67) orresponding o ω + V where we ke D=5 Λ= nd l=3 nd he prmeer of expression (34) is repled in he hrged se y whih is soluion o he equion FIGURE Plo of Morse poenils V + (solid line) nd V - (dshed line) of Eq (68) where we ke m ˆ = 0 D=5 Λ = nd l=3 Suppored in our mhemil nlysis of he prolem for he unhrged Nrii speime we sser h in he hrged D-dimensionl Nrii speime he NF of he unhrged Dir field re deermined y formule (6) only we mus reple in hese formule he vlues of he quniies σ nd y σ nd respeively For idenil vlues of he prmeers D Λ nd n Morse poenils (68) nd Pöshl-Teller poenils (67) hve NF wih idenil imginry prs h is wih idenil dey imes even when for idenil vlues of D Λ nd l he heigh of Pöshl-Teller poenil is smller hn he heigh of Morse poenils (see Figures nd ) We elieve h o find he soure of his oinidene is n ineresing quesion Also in Ref [49] ws shown h for suffiienly le imes he rdil funions of Pöshl-Teller poenil (67) form omplee sis Owing o similriy of he plos for oh poenils (see gin Figures nd ) o sudy if similr resul is vlid for Morse poenil (68) deserves deiled invesigion As we previously ommen similr reduion o h of Seion II works for hrged Dir fields propging in he D-dimensionl hrged spherilly symmeri speimes [] We elieve h good exerise is o lule he NF of he hrged Dir field propging in he D-dimensionl hrged Nrii speime o exend he resuls of Refs [38 39] nd he previous seion As we oserve in Inroduion seion he resuls on he sepriliy of he Dir equion in he four-dimensionl Kerr lk hole generlize o some D-dimensionl roing lk holes [9 3] We elieve h V he exension of he resuls oined in hese referenes o he meris of Plenski-Deminski-Klemm ype [50] deserve deiled nlysis Furhermore he sudy of he sepriliy properies of he equions of moion for grviionl nd eleromgnei perurions in he D-dimensionl Myers- Perry meris of Ref [5] is relevn prolem L Am J Phys Edu Vol 3 No 3 Sep hp://wwwjournllpenorgmx

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