heuistic eview of nczos potentil és Mo nd Ruén Sánchez Resech ente on pplied Science nd dvnced Technology of Ntionl Polytechnic Institute v egí No 694 ol Iigción P 5 México ity E-mil: cmol@ipnmx (Received July 7; ccepted 6 ugust 7) stct We pesent eview of -tenso potentil c poposed y nczos fo the Weyl confoml cuvtue tenso We show the ole tht plys nczos tenso in Genel Reltivity fo theoeticl physics postgdute students In the sme wy tht the electomgnetic vecto potentil cn e used to compute the Mxwell field the nczos potentil cn lso e employed to compute the Weyl cuvtue tenso of gvittionl field Key wods: nczos potentil theoy Weyl-nczos equtions nczos coefficients Resumen Relizmos un evisión del potencil -tensoil popuesto po nczos c p el tenso de Weyl de cuvtu confome Mostmos el ppel que jueg el tenso de nczos en Reltividd Genel p estudintes de posgdo de físic teóic En l mism fom que el vecto potencil puede se usdo p clcul el cmpo de Mxwell el potencil de nczos tmién puede se empledo p clcul el tenso de cuvtu de Weyl de un cmpo gvitcionl Pls clve: Teoí de potenciles de nczos ecuciones de Weyl-nczos coeficientes de nczos PS: 4-q 49+e 95Sf I INTRODUTION In the ecent yes thee hs een enewed inteest in the -tenso potentil c poposed y nczos fo the Weyl cuvtue tenso [] Howeve in the Genel Reltivity nd gvittion most popul textooks like Misne Thone nd Wheele [] Wld [] Hwkings et l [4] Weineg [5] Penose nd Rindle [6] Stephni et l [7] etc thee is not ny tetment out nczos potentil theoy Nevetheless it is impotnt in theoeticl nd physicl spects ecuse we cknowledge tht Einstein equtions cn e witten in tems of the covint deivtive of nczos Potentil in its Jodn fom; lso thee is possiility of the existence of n nov- Bhom s gvittionl quntum equivlent effect to the tditionl one [8] In the eductionl field the impotnce of nczos potentil is cle ecuse of its nlogy with the electomgnetic 4-vecto potentil Ou im in this wok is to pesent n heuistic point of view of nczos potentil tht effims the lst mentioned nlogy etween gvity nd electomgnetism This is focused to postgdute Genel Reltivity students The ppe is ognized s follows: in Sect II we pesent some lgeic popeties of nczos potentil; in Sect III we mentioned the physicl intepettion of nczos potentil; en Sect IV we show the method fo the vcuum spce-times; in Sect V we show ief exmple fo Schwzschild spce-time Finlly in Sect VI we pesent ou conclusions ppendix I is devoted to spino fomlism II GEBRI PROPERTIES OF NZOS POTENTI In Genel Reltivity we often need to wok in given spce-time o to deive one in the fom of n exct solution to Einstein s gvity equtions In 96 nczos suggested n uxiliy potentil [9]; though it nd the covint deivtive we cn otin the confoml cuvtue cd of decomposition of Riemnn cuvtue tenso R = + E + G () cd cd cd cd whee the following evitions e followed t m J Phys Educ Vol No Sept 7 78 http://wwwjounllpenogmx
n heuistic eview of nczos potentil E ( g S + g S g S g S ) cd c d d c d c c d R R G ( g g g g ) g S cd c d d c cd R Rg R R 4 The nczos potentil is studied in eltion to Weyl cndidte ie 4-nk tenso with the following expession W = + cd c; d d ; c cd; cd; + g + g g ( d ) c ( c) d ( c) d s g + ( g g g g ( d ) c ; s c d d c ) whee we define d d ; ; d (4) The ove eqution is equivlent to the dul fom W = + cd [ c; d ] cd [ ; ] * * * * [ c; d ] cd [ ; ] (5) Initilly n ity tenso of thid ode hs 4 fee components ut futhe we hve to impose the following 4 lgeic symmeties = c c (6) the fou (guge lgeic conditions of nczos) = (7) nd the dul fou conditions * = (8) o equivlently + + c c c = (9) then his initilly sixty fou degees of feedom e educed to sixteen Futhe six diffeentil nczos guge conditions ; = () In fct the nczos potentil lso dmits wve eqution [] the tensoil fom of this wve eqution hs poved to e vey useful to find nczos potentil y some inteesting methods due to Velloso nd Novello [] The complete fom of the tensoil eqution fo nczos potentil is + R R R c c c c () s s g R + g R R = J c s c s c c III PHYSI INTERPRETTION OF NZOS POTENTI The ole of nczos potentil c with espect to the Weyl tenso W cd is the sme tht plys the vecto potentil fo the Mxwell tenso F Then in the sme () () wy tht we deive the electomgnetic tenso F fom we cn deive the electomgnetic field F s: F = W ( ) = () ; ; we cn lso deive the Weyl cuvtue tenso fom potentil; nevetheless this could not e vecto potentil s in the electomgnetic cse insted it must e -nk te nso clled the nczos potentil c with simil popeties see eqution () lso ssuming tht some spce-time M dmits n electomgnetic field F = - F then this field oeys cetin ules Fo exmple F + F + F = ; c c; c; () cd c F = J ; d Theefoe the gvittionl nczos potentil c physiclly coesponds with the electomgnetic vecto potentil In cetin point of view could e deived in covint wy to get the electomgnetic tenso F (s is suggested y ()) lso the nczos potentil could e deived in covint wy to get the Weyl gvittionl field s it is coespondingly suggested y () The souce of the electomgnetic eqution (second eqution of ()) coesponds to () nczos eqution with souces IV METHOD FOR THE VUUM SPE- TIMES If we hve spce-time with glol Killing vecto field [] it is sometimes possile to genete nczos p otentil with the following method: If is non-null Killing vecto tht stisfies Killing eqution + = (4) ; ; nd lso is hype sufce othogonl vecto (ie stisfies the following mthemticl eltion) = (5) [ ; c] Then it is possile to tke n unit vecto u fom the goup of motions such tht u = = s > s = (6) u Thus the Killing eqution (4) guntees tht u is expnsion-less nd she-fee ie u ( δ u u )( δ u u ) = (7) ( ; ) m m n n nd y mens of the hype sufce othogonllity condition of we hve u u = (8) [ ; c] theefoe u = s u (9) ; whee we hve defined the fist cuvtue vecto of the goup of conguence (lso clled nd known s t m J Phys Educ Vol No Sept 7 79 http://wwwjounllpenogmx
és Mo nd Ruén Sánchez cceletion) to e =u ; u which fo ll goup of motions even those not stisfying (8) this is gdient =(ln(/)) Then cndidte of nczos potentil is given y = ( u u ) u c c () sg ( g ) c c which stisfies (6) (7) nd (9) Then veifying the nczos guge () nd the () condition of the potentil we cn fix completely ou cndidte s full-fledged nczos potentil We cn tce the spinoil nlog of () to fix ou cndidte In the following section we show n exmple fo the Schwzschild spce-time V N EXMPE FOR THE SHRZSHID SPE-TIME Now we show ief exmple of the method [ ] using the Schwzschild line element in the coodintes (t θ ø) s follows ds M d = dt M d sin( ) θ θ φ d () The time-like Killing vecto = δ hs squed nom M = g = () clely this vecto fields e hypesufce-othogonl If we intoduce the coesponding unit vecto field u = u u = (4) then u = u with = u u (5) ; ; now with the velocity vecto u nd the cceletion it is possile to wite nczos potentil in the following fom = ( u u ) u ( g g ) (6) c c c c th is is the -nk tenso tht Novello nd Velloso [] pov e to e nczos potentil We stt to mnge the exmple y using pefeed null tetd in the Newmnn- Penose convention sy { e } = ( m m l k ) m m g = m m k l = (7) ( ) ( ) = k l = we continue the pesent exmple employing the null tetd povided y Kinnesley which e l = ( Δ ) n = ( Δ) Δ Δ M (8) m = ( i csc( θ )) m = ( i csc( θ )) Hee the non-vnishing spin coefficients nd Weyl scl components e cot( θ ) M ρ = β = α = μ = (9) = M M Ψ = then the intinsic deivtives e D = l = + Δ t D' n Δ = = t () i δ = m = + θ sin( θ) φ i δ ' = m = θ sin( θ) φ fom these set of eltions we hve tht the only non- components of nczos tenso e vnishing M M M = = () 6 6 whee we used the following eltions etween the null tetd nd the Weyl scls c d c d Ψ = l m l m Ψ = l m l n cd c d c d Ψ = lmmn Ψ = lnmn () cd cd cd c d Ψ = m n m n 4 cd nd nczos scls c c = l m l = l m m c c c = m n l = m n m c c c = l m m = l m n 4 c 5 c c () c c = m n m = m n n 6 c 7 c This is n exmple of nczos potentil tht hs een computed fom the Novello nd Velloso s nczos gvittionl potentil -nk tenso tht illusttes the use of the null tetd nd spinoil coefficients fom the Newmnn- Penose fomlism [] VI ONUSIONS It hs een found tht the method of Novello nd Velloso [] descied in section IV could e used to compute nczos gvittionl potentil nd it hs een pointed out c t m J Phys Educ Vol No Sept 7 8 http://wwwjounllpenogmx
n heuistic eview of nczos potentil tht we cn use it in the deivtion of the Weyl cuvtue tenso fo the simple cse of Schwzschild spheiclly symmetic vcuum spce-time The nczos tenso is impotnt in the deivtion of the Einstein field equtions in his Jodn fom We hve used the spin-coefficients nd the Newmnn-Penose fomlism in this method to chieve ou ojective of showing n esy nd convenient wy to get this impotnt quntity KNOWEDGEMENTS This wok ws suppoted y EDI OF-IPN SNI- ONyT gnts nd Resech Poject SIP-748 REFERENES [] Begqvist G nczos potentil in Ke geomety J Mth Phys 8 4-54 (997); Edg S B nd Hoglund The nczos potentil fo the Weyl cuvtue tenso: existence wve eqution nd lgoithms Poc Roy Soc ond 45 85-85 (997); ópez-bonill J Ovndo G nd Peñ J J nczos potentil fo plne gvittionl wves Found Phys ett 4-45 (999); tin D The nczos potentil s spin- field hep-th/85v (); O Donnell P nczos tenso potentil fo confomlly flt spce-times 9 4-45 (4); Men F nd Tod P nczos potentils nd definition of gvittionl entopy fo petued FRW spce-times g-qc/757v (7) [] Misne h W Thone K S nd Wheele J (Gvittion Feemn New Yok 995) [] Wld Roet M (Genel Reltivity The Univesity of hicgo Pess hicgo nd ondon 984) [4] Hwkings S W Ellis G F R (The ge Scle Stuctue of Spce-Time midge Univesty Pess midge 97) [5] Weineg S (Gvittion nd osmology: Pinciples nd plictions of the Genel Theoy of Reltivity John Wiley New Yok 97) [6] Penose R nd Rindle W (Spinos nd spce-time vol Two-Spino lculus nd Reltivistic Fields midge Univesity Pess midge 984) [7] Stephni H Kme D Mccllum M Hoenseles Helt E (Exct Solutions of Einstein s Field Equtions midge Univesity Pess midge 6) [8] Holgesson D (nczos potentils fo pefect fluid cosmologies inköping Univesity Thesis 4); on-line t: http://unkse/esolve?un=u n:nn:se:liu:div-58 [9] nczos The splitting of the Riemnn tenso Rev Mod Phys 4 79-89 (96) [] Novello M nd Velloso The connection etween genel oseves nd nczos potentil Gen Rel Gv 9 5-65 (987) [] Doln P nd Kim W Some solutions of the nczos vcuum wve eqution Poc R Soc ond 447 577-585 (994) I NEWMNN-PENROSE ONVENTION FOR THE SPIN-OEFFIIENTS In this section we show the symology employed y Newmnn nd Penose to denote sevel spin-coefficients tht we hve ledy computed fo the vcuum spce-time This nottion hs een poved to e vey useful The histoffel symols of second kind cn e witten in tems of spinoil coefficients s c ε B ε B B B Γ = + (4) whee we followed the convention of decomposition of the null tetd in tems of the cnonicl spinoil fme m = o ι m = ι o (5) l = ιι k = o o the sis spinos o nd ι fom dyd in the sense tht stisfies the nomlized eltion of inne poduct: B o ι = ε o ι = B nd ε is the locl metic of the spin spce S() B which hs the components B ε = ε = B The spino indices e ised nd loweed ccoding to the ules B B α = ε α α = ε α (6) B B The spin inne poduct is ntisymmetic in the sense tht the spinoil indices B do not commute ie B B αε ε α B (7) ε α β = α β = α β B The digms of Newmnn-Penose e the following TBE I B Symology of Newmnn-Penose fo the spino \ B ε κ τ ' ' α ρ σ ' β ' β σ ρ ' α ' τ κ ' ε TBE II Symology of Newmnn-Penose fo the spino B \ B = κ ε = ' π = τ ' ρ α = β ' λ= σ ' σ β = α ' μ = ρ ' τ = ε ' ν = κ ' t m J Phys Educ Vol No Sept 7 8 http://wwwjounllpenogmx
és Mo nd Ruén Sánchez Thee e lso two moe tles fo the complex conjugte spin-coefficients symols TBE III Symology of Newmnn-Penose fo the spino B \ B ε κ τ ' ' α ρ σ ' β ' β σ ρ ' α ' τ κ ' ε ' TBE IV Symology of Newmnn-Penose fo the spino B \ B = κ ε = ' π = τ ' ρ α= β ' λ = σ ' σ β = α ' μ = ρ ' τ = ε ' ν = κ ' Notice tht in these tles the second nd thid ows of the complex conjugte of spin-coefficient nd B B e intechnged Becuse of the ovious eltions nd coespondingly = = B B B B = = B B (8) (9) (We lso cn think tht the cpitl ltin spinoil indices with dot B cn not see the spinoil indices without dot B nd then cn pemute with this lst set without intefeence But the cpitl indices without the dot B cn not commute etween them ecuse in doing so they could intefee with the vlue of the spinoil quntity) t m J Phys Educ Vol No Sept 7 8 http://wwwjounllpenogmx