Angular momentum in general relativity: The definition at null infinity includes the spatial definition as a special case

Transcription

1 PHYSICL REVIEW D, VOLUME 63, ngula momenum in geneal elaiviy: The definiion a null infiniy includes he spaial definiion as a special case nhony Rizzi Califonia Insiue of Technology, LIGO, Livingson, Louisiana Received 7 Febuay 2000; published 4 pil 200 I show ha he definiion of angula momenum a null infiniy given by. Rizzi, Phys. Rev. Le. 8, , educes o he nowi-dese-misne definiion a spaial infiniy in he appopiae i. This idenificaion einfoces he gauge condiion of he null definiion as well as he null definiion iself. Insigh ino he place of he null and spaial definiions is also gained. The mehods used in his pape ae vey useful fo undesanding fa-field poblems in geneal elaiviy; a se of ppendixes gives deails of he mehods. DOI: 0.03/PhysRevD PCS numbes: Cv, Ha I. INTRODUCTION Unil ecenly, hee was no definiion of angula momenum in geneal elaiviy ha allowed fo a nonivial consevaion law. The only definiion of angula momenum ha was consisen wihin geneal elaiviy was he nowi- Dese-Misne DM spaial definiion,2,3; his definiion gave an angula momenum fo he enie space-ime. The DM definiion was only maginally useful, because i allowed no way of speaking of he exchange of angula momenum. Tha is, one could no even make simple consevaion saemens, such as a sysem wih angula momenum L gave up L and was heeby lef wih LL. Pevious aemps a defining his basic quaniy in a saisfacoy way failed fo vaious easons; pime among hese easons was he so-called supeanslaion ambiguiy. These poblems wee solved in Ref. 4. In he pesen pape, I will show ha he definiion of angula momenum a null infiniy given in 4 educes o he DM definiion in he appopiae i. The ageemen beween he wo definiions senghens ou oveall undesanding of angula momenum in geneal elaiviy. Since he DM definiion is well esablished i lends anohe successful es o he definiion given in 4. Fuhemoe, i gives conex o he DM definiion. To avoid wodiness, hencefoh he definiion given in 4 will be efeed o as he null-m definiion o simply he null definiion. The mos inuiive way of seeing ha hee is a poin whee he definiion a null infiniy mees he spaial definiion is o make use of he confomal picue. In he confomal picue one maps infiniy o a finie locaion. The confomal picue of an asympoically fla space-ime is shown in Fig.. The null definiion of angula momenum exiss along he axis labeled I, Sci, spaial infiniy is shown as a small cicle maked i 0. Mahemaically, one says L null ul spaial u whee u paameizes I. To make his mahemaically pecise, we ecas he spaial and null definiions in ems of a space-ime foliaion adaped o he appopiae is. We will wok in songly asympoically fla SF space-imes cf. ppendix. We use he opical-maximal foliaion. This is my name fo he foliaion used in Chisodoulou and Klaineman s book 8 on he nonlinea sabiliy of he Minkowski space. s he name suggess, he opical-maximal foliaion is he S 2 foliaion an S 2 foliaion is one wih leaves ha ae opologically S 2 ha esuls fom he inesecion of an opical foliaion and a maximal foliaion. maximal slice of space-ime, designaed by ime,, is a spaial slice which has maximal volume wih espec o he embedded space; mahemaically his means (k ij )0 whee k ij is exinsic cuvaue of. One mus impose wo By DM definiion, I mean wha is canonically aken o be he DM definiion. The DM definiion is only clea and unambiguous when given in he cene of mass fame (P CM 0) 5,6,7. Theefoe, in his pape, DM definiion efes o he DM equaion aken in his case. FIG.. The null definiion of angula momenum is valid all along I which should include spaial infiniy, i 0. Hence, if he i is aken along I o i 0, one should ge he spaial definiion also know as he DM definiion /200/630/040023/$ The meican Physical Sociey

2 NTHONY RIZZI PHYSICL REVIEW D FIG. 2. The opical foliaion is shown saing wih one maximal slice. This final slice sas ou a finie ime,, and hen a i is aken o he infinie fuue. The opical foliaion is a null 2-suface folaion. FIG. 3. The opical-maximal foliaion is a foliaion of spaceime wih opologically S 2 sufaces as leaves. I is consuced by aking he inesecion of he null cones of he opical foliaion, he C u s, and he spaial hypesufaces, s, which ae he leaves of he maximal foliaion. moe consains o make he slicing of space-ime unique: pick an DM linea momenum fo he slice and 2 choose he meic such ha he lapse funcion, 2, is one a spaial infiniy i.e., becomes pope ime a spaial infiniy. We need he zeo DM momenum cene of mass fame SF condiions imply daa se in P DM 0. ssigning such a maximal hypesuface, fo each geneaes wha will be called he maximal foliaion. The opical foliaion cf. Chaps. 9 and 4 of 8 is a null 2-suface foliaion. I is defined eveywhee fa fom he egion of mass so ha geodesics ae, in a suiable sense, wellbehaved. Tha is, he foliaion is defined eveywhee exeio o a ime-like cylinde enclosing he mass. The foliaion is shown in Fig. 2. To ceae he foliaion we do he following. a Sa wih a maximal hypesuface, final, of zeo DM momenum and consuc a -paamee family of S 2 sufaces i.e. sufaces which ae diffeomophic o S 2 on final by solving a given equaion of moion cf. 8, p. 4 on final. This pocess gives us he funcion u f defined on final whose level sufaces will be labeled by S f,u f. b Fo a given S f,u f of final send ligh ays ino he pas hus ceaing a null cone called C u. If one does his fo each S f,u f, he exends he opical funcion u f on final o an opical funcion u in he space-ime. c To make he soluion valid fo all space-ime ouside he imelike cylinde one les final. 3 This gives an opical funcion u defined in he enie egion of space-ime ha is fa fom all mass. Thus, u is he uniquely defined funcion ha saisfies he so-called Eikonal equaion: u u g x x 0. d Finally, o ceae he opical-maximal foliaion we ake he inesecion of a given C u wih a maximal hypesuface o ge a opological S 2 suface which we name S,u. This foliaion is shown in Fig. 3. We can always choose as big as needed o ge he desied closeness o null infiniy. Taking his inesecion is he same as using l,l 4 o popagae he S,u foliaion along each C u. In his way, one could hus vay he foliaion by picking ohe null pais cf. ppendixes C and D; we shall have occasion in his pape o do his. In ode o undesand he eason fo picking his foliaion we fis inoduce he DM definiion. The DM definiion of angula momenum on a leaf has he fom L s i 8 S iab x a k bj g bj kn j d 2 2 One can wie he space-ime meic as ds 2 2 d 2 g ij dx i dx j whee is called he lapse funcion. 3 In aking his foliaion o ime-like infiniy, one is making he physically easonable assumpion ha he enie space-ime exiss global exisence. Howeve, since he poof given in his pape akes place nea spaial infiniy, infiniely fa away fom imelike infiniy, one does no, even in a echnical sense, need he poof of global exisence hee. In a wod, he foliaion is exended o ime-like infiniy only o give a complee inuiive picue of how he space-ime is sliced. whee S coodinae sphee of adius i x i 2, x i s defined in neighbohood of spaial infiniy, k ij is he exinsic cuvaue of he leaf, N is exeio uni nomal o S, and d is an aea elemen of S. Equaion 2 can be e-wien, given a songly asympoically fla daa se cf. ppendix : 4 If we use 9.4c on p. 264 of 8 noe and a ae given in consucion of maximal foliaion, we obain l al

3 NGULR MOMENTUM IN GENERL RELTIVITY:... PHYSICL REVIEW D L i s 8 iab x a k bj N j iab x a N b kd S 3 8 S iab x a k bj N j d 8 k N i d S 8 i d cons S u whee k N 2 D l,l, ltn, ltn, wih la l, lal W ln a, L i null u ucons s 8 Se,u W i d Z 2 8 Z H i d 0. Su Hee S u S,u is he u elemen of he foliaion of null infiniy, and d 0 efes o he aea elemen of S u. The definiion of he ohe quaniies ae given in ppendixes B and C. Equaion 9 is he same as he definiion in 4, because he geodesic-null pai manifesly cf. ppendix D educes o he affine foliaion 4 nea null infiniy. ppendix B, in un, shows ha he i using he geodesic-null pai o foliae he null cones is he same as using he -null pai cf. ppendixes C and D l,l. Thus, we have L i null u W ucons 8 i d. 9 W 2 D l,l, also ls, lu2, a D T uu 7 Now, o make he above somewha fuzzy posulae,, moe pecise we ake L i,u u 8 i d means covaian deivaive on he S 2 suface,2, i (i) is oaion veco field aound he î axis. ae such ha (i), ( j) ijk (k). They ae oaion veco fields hey ae defined picoially in 4,9 defined on pulled back sphee s cf. ppendix F. The oaion veco fields coespond, in a Euclidean space, o oaions aound he x, y and z coodinae axes. We will popagae hese fields fom one pulled back sphee o anohe using Lie dagging in he mos naual way. We sa fom null infiniy on he appopiae C u whee he sphees ae acual wo-sphees wihou need of pull-back and so (i) ae defined vey inuiively and dag hem ino he space-ime using l which o a sufficien ode, nea null infiniy, is he same as popagaing using l al see below. l is he mos naual because i is a angen o a null geodesic which is he pah aken by gaviy waves. In sho, in ems of he paamees of he above opicalmaximal foliaion, he DM angula momenum is using he suface 0 i L DM L spaial u 8 i d. S0,u Hee d is he diffeenial aea elemen of S 0,u and is he meic on S 0,u. The null-m definiion akes he following fom using he geodesic-null pai cf. ppendixes C and D l,l in he opical-maximal foliaion: 8 whee eveyhing is again in he opical-maximal foliaion using he -null pai. Fo example, he aea elemen is ha of a small piece of S,u called d. If we can show L i i null u L,u u u hen we can ewie Eq. which we seek o pove: u L spaial L i null u u becomes i L 0,u u L i,u. 0 2 Hee, o be concee, we se 0; Eq. 2 is of couse ue fo evey consan slice. Noe using he fac he DM angula momenum is he same on evey ime slice, one can wie Eq. 2 as: L i,u u u L i,u. Tha is, unde he saed condiions, one can swap he u and is. Thee ae hus wo equaions ha mus be poved: Eq. 0 and Eq. 2. We now poceed o hose poofs; we pove Eq. 2 in Secs. II and III, and Eq. 0 is poved in Sec. IV

4 NTHONY RIZZI PHYSICL REVIEW D he RHS of Eq. 4, we need o ewie d in ems of d. Fo a given S,u, one defines he aeal adius,, in he following way: 4 2 d whee d is he aea elemen of he S 2 suface of consan and on a given consan-u cone FIG. 4. Picoial of LHS of inequaliy 4. Fo each suface S,u one does he L inegal shown. One hen inegaes L ove C u on he egion fom 0 o. Finally, one akes he i of such inegals as u. II. PROOF THT u\à L i 0,u Ä u\à \ L i,u WHEN L ËCÕ FOR Ì Because L (i) 0,u is independen of, one can inse he i i.e. L (i) 0,u L (i) 0,u on he LHS of Eq. 2, and ewie i as u L i 0,u L i,u 0 8 d 2, whee he oveba means aveaged ove solid angle. Using he decay law fo given in ppendix D 2 H 2 L i 0,u L i,u 0. u Hence, we wan o show ha 0. Using he fundamenal heoem of calculus, one obains We fuhe define L,u d. u 0 u I is easy o show ha L,u d. We will now show ha 0 if L,u L C 3 Theefoe we have H 2 d O d. C C u 0 d u 0 O d u 0 u 0 C d 0 C u C d 0 C 0 0 O d do 0 5 fo all and whee C is a consan C u L d 0 u 0 d. 4 The i on he LHS of Eq. 4 is illusaed in Fig. 4. The inegand, L, in his i is an inegal on an S,u,an S 2 suface afe pull back. One inegaes L along he null cone C u fom 0 ou o null infiniy, and hen akes he i of such inegals owads spaial infiniy i 0. To evaluae whee we use he fac ha 0 as u. Now, he RHS of he las equaliy is zeo as long as. Since 0 0, we have L i i 0,u L,u u u Q.E.D. We mus now show ha he condiion given in Eq. 3 holds

5 NGULR MOMENTUM IN GENERL RELTIVITY:... PHYSICL REVIEW D III. FLL OFF OF L FOR LRGE We now poceed o show ha L falls off like C/ fo 0 whee C is a consan. In moe specific ems, a a given poin of space-ime L will be associaed wih a unique leaf of he foliaion, say S,u, which has aeal adius. s long as is sufficienly lage i.e. one is sufficienly fa fom he souce, we will see ha L decays like / independen of o u: L L,u d l d l d l d d l d d. 6 The second equaliy, makes use of ( l d ) d, and he hid equaliy uilizes he fac ha he oaion veco fields ae moved fowad accoding o he ule l l, 0. We will need he following: l V l,v,e D l V,e D V l,e D l V V B D B l,e D l V V B B whee V means any veco. Recalling l al, we obain B l V D l V V B ad l V av B a l V. B In he las line, use is made of he fis line wih l subsiued fo l. Hence Eq. 6 becomes a l d a l W ln a W ln a d. 2 To show he decay law of he inegal above, we now need he decay laws fo he each of he quaniies in he inegand. The decay laws fo he Ricci Coefficiens ae given in ppendix D; we also need ln a/ and ln / all fom 8. These decay laws ae valid fo on a consan u cone. Howeve, i can be shown ha he decay laws in ems of he luminosiy adius,, fall off he same fo, cons confe ppendix E. Hence, subsiuing he decay laws ino Eq. 2 above we obain 2 H d 0 l Z Z Z Z Z 2 2 Z l 2 Z 2 2 Z Z 2 2 Z Z 2 2 Z H 2 2H d 0 HZ 3 HZ 3 0 d 0 d

6 NTHONY RIZZI PHYSICL REVIEW D Z 2 2 Z 2 HZ 3 2 Hk 0 d Hk Z H Z O 0 d O L O Q.E.D. In he above, 5 use is made of l (/s)a (/)()/(H /2)cons/. Z d 00 and d 00 poved in ppendix B H H2 22 wih ()consk using p u L i,u 8 IV. PROOF THT u\à L i i null u Ä u\à \ L,u u 8 u 8 i d W ln a i u Z 8 u Z d Z Z 8 Z 2 Z HH i d 0 u u 8 d 0 u L i null H i 2 H i d 0 2 H i d 0 whee H consh. In he penuae equaliy, use is made of S,u Z (i) d 00. In he las equaliy, one uses Eq. 9. Finally, in ode o complee he poof, we mus show ha H i d 00 u o 5 I is ineesing o noe ha he final decay law given above would also apply in he moe geneal case: ln a/ O( () ). H i d We will designae he LHS of Eq. 23 by. To evaluae he inegal we mus find expessions fo H and ;we will find he expessions in he given ode.. Expession fo H À Using an equaion on p. 508 of 8 and ZE noe ha he u he eaded ime and null ingoing shee ha I use below ae elaed o he u and of Chisodoulou and Klaineman CK 8 in he following way: u CK u and CK /2; he minus supescip on a quaniy means u

7 NGULR MOMENTUM IN GENERL RELTIVITY:... PHYSICL REVIEW D Z 2 2 PP. Now using (PP ) 0, 0, and Z /2 H one obains 2 H. In he above, he oveba means aveaged ove S 2. Nex we use he gauge condiion fom 4 ha he odd pa of he shee is zeo; mahemaically, one can he wie B B B B even 0. Hence, ecalling ha cons: 24 2 H H 2 25 f. 26 Now, expanding H and f in spheical hamonics whee S 2 e.g.,, e.g. H l,m H l,m Y l,m (), implies ha Eq. 26 becomes llh l,m Y l,m l,m f lm Y l,m l,m H l,m ll f l,my,m. 27 Now, we mus find he hamonic coefficiens f l,m ha chaaceize 2(). Rewiing he soluion fo hese quaniies on p. 504 of 8, one ges 6 du S 2 2 u,x 2 u d xx x /2 du sgnu 2 u, 2u 8 whee he oveba means aveage ove S 2, and d (x ) is he diffeenial aea elemen of S 2 in ems of he pimed coodinaes, x. Noe ha sgn(u) and M(u)/u/32 2 (u)d 0 gives 2 (u)du8(m f M i )8M whee M f M() and M()M i. Noe also ha we have picked abiay z axis and defined he coodinaes, wien abbeviaed as in usual way. We also need S 2 xx l0 ml ml 4 2l Y lm *,Y lm, using xx xx x d 4Y 00, d xx x 4Y 00, Y 00 *,4Y 00,d 4 whee dsin dd and using F ()8S du(u,) 2 du(u,,) 2 S/4he oal enegy adiaed in he pope sense pe uni solid angle emied in he diecion, 0 we ge F 2 x 6 S 2xx d x2m 8 F 8M F x 2 6 S 2xx d x 8 F 2 6 l,m F l,m Y lm l,m 2 4 l,m 8 l,m F l,m 2l Y lm 8 l,m 2l 2l F l,my l,m 4 2l Y * Y lm l,m d 8 l,m F l,m Y l,m F l,m Y l,m

8 NTHONY RIZZI PHYSICL REVIEW D whee Finally, using Eqs we ge F l0 ml F l,m Y l,m,. ml H 2 l,m ll l 2l F l,my l,m. 28 In he manne above one ges B. Expession fo À 4 l,m F l,m 2l Y lm2m. 29 Because of he fom of Eq. 23, we can dop he 2M and, hus, fo ou puposes 4 l,m F l,m 2l Y lm. 30 Since, he LHS of Eq. 23, is an inegal on a sphee we can ewie ha inegal as H i d 0. C. Symmey makes Ä0 Theefoe, if we pick, as an example, (i) (z) hen H L z H wih L z / and L z Y lm imy lm and using Eqs. 28 and 30 one obains L z H d 0 4 l,m F l,m 2l Y lml z 2 l0 ml ml ll l 2l F l,m Y l,m d 0 i 8 l,m F l,m 2ll Y lm l0 ml ml mf l,m ll2l Y lm d 0. Taking m m and using Y l,m Y l,m *, one ges i 8 l,m i 8 l0 i 8 l0 ml ml F l,m 2ll Y lm l0 mf l,m F l,m 2l 2 ll 2l 2 ll m ml ml l mf l,m 2lll Y * d lm 0 l mf l,m F l,m m mf l,m F l,m

9 NGULR MOMENTUM IN GENERL RELTIVITY:... PHYSICL REVIEW D Since z was picked abiaily, he same logic follows fo x and y as well. Hence, H i d 0 0. Si,u Q.E.D. Wha ae he easons his inegal is zeo? Fis, i would no, in geneal be zeo wihou he gauge condiion given fo he null definiion. I also elies on he invaiance of he laplacian boh in wo and hee dimensions unde oaions and on he pesevaion of a ceain symmey due o he conen of he inegand; i conains H and, boh of which ae inegals of 2. Specifically, he invaiance unde oaion of he laplacian makes he kenels of he inegals of 2 oaion invaian funcion of only disance. In ems of he spheical hamonic decomposiion, one mus have F l,m in boh he H em and he ems wih no ohe m dependence; ohewise, he cancelaion o give zeo would no occu. V. CONCLUSION We have jus shown ha he definiion of angula momenum given in 4, valid fo all eaded ime a null infiniy, educes o he DM spaial angula momenum in he i of eaded imes infiniely fa in he pas. Mahemaically, (i) his is wien L spaial DM u L null (u). The DM angula momenum can be hough of as, in some sense, he oal angula momenum of he space-ime. The null-m definiion should yield his oal befoe any adiaion has had a chance o leave: ha is, in he infinie pas. Hence, he poof fuhe solidifies he coecness of he new definiion. Wihou he use of he gauge condiion given in 4, and saed in Eq. 24 of his pape, he poof would fail. One may now ask: why does he null definiion equie he gauge condiion, while he spaial definiion does no? null infiniy gaviy waves ae consanly changing and mixing he choices fo S 2 sufaces of inegaion; one has o pick among he many possible choices. The gauge condiion makes his choice. By conas, he spaial definiion equies no such choice, because i only gives he angula momenum in he infinie pas 6 and has no changing sufaces o deal wih. Fuhemoe, in he case of he canonical DM momenum P DM 0 he cene of mass case consideed in his pape, one does no have o pick ou he S 2 sufaces ha coespond o posiion and heeby, fo example, sepaae ou he spin pa fom he obial pa. Since he P0 case does involve such a selecion, i does equie a gauge condiion, bu appaenly fo a diffeen eason. To conside he P DM 0 is an ineesing and wohy pojec. I would involve deciding how o define he cene of mass S 2 sufaces and would even fuhe elucidae he issue of angula momenum in geneal elaiviy. Seveal ohe aspecs of he poof given in his pape ae wohy of noe. Fis, he physical impo of Eq. 23, specifically he symmey issues and he funcional dependence of F is sill o be exploed and may hold significan insigh. In a simila vein, i is ineesing o simply noe ha he definiion of angula momenum wih he inegand and he definiion using he inegand W do no educe o he same esul a null infiniy; his eveals somehing of he subley of angula momenum a null infiniy. Lasly, he mahemaics used o show he educion is also ineesing in is own igh, because i is a poweful ool in woking in he egion fa fom all mass whee gaviy wave deecos opeae. CKNOWLEDGMENTS I would like o hank Demeios Chisodoulou fo helpful convesaion and fo eading his manuscip fo conen. This wok was suppoed in pa by NSF Gans PHY and PHY PPENDIX : STRONGLY SYMPTOTICLLY FLT SPCE-TIMES Biefly, a songly asympoically fla space-ime is one ha can be consuced fom an iniial daa se of he ype descibed below. n iniial daa se saisfies he condiion of songly asympoically flaness cf. 8 if g and k he meic and exinsic cuvaue ae sufficienly smooh and hee is a coodinae sysem x i such ha as he adius i (x i ) 2 g ij 2M ij o 4 3/2 k ij o 3 5/2. Noe ha a funcion is said o be o m ( k ) as if l f (x)o( kl )l0,,...m whee l denoe all he paial deivaives of ode l elaive o he coodinaes x,x 2,x 3. The above condiions imply ha he daa se is aken in he cene of mass fame (P DM 0). PPENDIX B: NULL DEFINITION CN USE S,u FOLITION Fis, ecall he null definiion given in ems of he geodesic-null pai o use he affine foliaion 4, diecly: L i null u W ucons 8 i d. Ss,u s Using W Z /Z 2 / 2 6 Effecively, his means he spaial definiion gives he in some sense oal angula momenum fo he space-ime. This unfounaely means i can give no useful consevaion law. Tha is, one canno, using i, make he simple saemen: a sysem had iniial angula momenum L, and L escaped leaving he sysem wih L L. One can do his wih he definiion a null infiniy. 2/H/ 2 d 2 HH d 0 whee supescip 0 means aea elemen on S 2 slice of null infiniy gives

10 NTHONY RIZZI PHYSICL REVIEW D W 8 i d Ss,u Z 2 8 Z H i d 0 Su ucons s whee use has been made of Su Z (i) d 00. This is poven as follows: Z i d S,u 0 S,u B B 2 H i d 0 H B B k B ZZ ln a Y 0 2 N ln V k N TBLE I. Sandad null pai (l,l). H B B k B ZZ ln Y ln a 2 N ln S,u B B i 2 H i 0 d 0 because (i) is killing and B is symmeic. Noe: fom he same agumen, i is clea ha S,u (i) d 00. Now, using he -null pai, in a pecisely paallel way one ges ucons W 8 i d 8 Su Z 2 Z H i d 0. Bu, as given in Eq. 24, H Hk whee k is a consan so one ges L i null u W ucons 8 i d. If one uses he sucue equaions h and k given in oiginally fom 8 one can show he exisence of his i follows in same manne as fo affine foliaion. Fo he sake of compleeness, he definiion in ems of paamees ha ae in pinciple measuable is 4 L i null u 8 B C CB I i d 0. B PPENDIX C: RICCI ROTTION COEFFICIENTS The Ricci oaion coefficiens ae defined below wih espec o a null pai, e 3,e 4 whee e 3, e 4 2 and, in he case used mos fequenly in his pape, e 3 l, e 4 l, and he spaial vecos, e,e 2, ohogonal o hese wo vecos and angen o he opologically S 2 sphees which make up he foliaion. The e ae called he null eads. H B D e 4,e B, ZZ 2 D 3e 4,e, H B D e 3,e B, ZZ 2 D 4e 3,e, Y 2 D 4e 4,e, Y 2 D 3e 3,e, C 4 D 4e 4,e 3, V 2 D e 4,e 3. 4 D 3e 3,e 4, Noe ha he quaniy is no elaed o he quaniy labeled in Eq. 25. The unsupesciped null pai, l,l efe o he geodesic pai so called because l is angen o a null geodesic; unsupesciped Ricci oaion coefficiens efe o hose wih espec o he geodesic null pais l,l. Supescip on he null pai efes o l l, l l he S 2 foliaion is popagaed o be on a maximal hypesuface. The pimed null pai efes o he sandad null pai: ltnal, l TNa l whee T is he uni nomal o he maximal spaial hypesuface, and N is he uni nomal o S,u in he maximal hypesuface. Ricci oaion coefficiens associaed wih hese null eads ae disinguished especively by supescip and pimes. Fo he sandad null pai one ges eaanging 8, p. 7 he esuls in Table I, whee a he lapse funcion of he foliaion induced by u u B he exinsic cuvaue of he sufaces S,u k ij exinsic cuvaue of he maximal slice elaive o on each 2 spaial componens Tg ij 2 g ij Decomposiion of k : B k B, k N, k NN

11 NGULR MOMENTUM IN GENERL RELTIVITY:... PHYSICL REVIEW D TBLE II. Geodesic null pai (l,l). H B a ( B k B ) H B a( B k B ) ZZ ln a ZZ ln Y 0 a a 0 V W ln a One can ansfoms o a diffeen null pai; his is called a lapse ansfomaion 7 and has he fom l Tans a l, l Tans al whee a is called he lapse funcion and is any funcion on he S 2 suface. Noe ha he nomalizaion l l2 is peseved unde a lapse ansfomaion. Unde he lapse ansfomaion above, he Ricci coefficiens ansfom as H Tans a H, ZZ Tans ZZ, H Tans ah, ZZ Tans ZZ, Y Tans a 2 Y, Y Tans a 2 Y, C2 Tans 2 a2 D l aa, H B ( B k B ) ZZ ln a Y D l 2 N ln Y a 2 ln a V Tans V ln a. a 2 N ln 2 D l a TBLE III. -null pai (l,l ). V ln W ln(a) Tans a 2 D la, H B ( B k B ) ZZ ln Y 2 ln a 2 N ln 2 D l Hence, in ems of he geodesic null pai, la l, lal one obains he esuls in Table II, whee use is made of D l ad a l a2 D la2a deived using elaions fom 8, p In ems of he -null pai, l l, l l, one ges he esuls in Table III. PPENDIX D: DECY LW FOR RICCI COEFFICIENTS This appendix gives he decays laws fo he Ricci oaion coefficiens using esuls fom 8 and 0. We will need he following: fo any 2-symmeic covaian enso, H B,ona opologically S 2 suface p. 40 of 8: H B Ĥ B 2 B H whee he ha means he aceless pa. In he opical-maximal foliaion using he geodesic null pai (l,l) in a coodinae basis, he Ricci oaion coefficiens cf. ppendix B nea null infiniy ae whee 0 seen in Table IV. Noe ha he ode of can be found by using a / and he fac ha is independen of u. Using he sandad null pai (l,l) one has he esuls in Table V. Z E and Z 2 E 2 2 Using he -null pai (l,l ) one has he esuls in Table VI. PPENDIX E: DECY T NULL INFINITY GIVES DECY S FUNCTION OF LONE given geomeic quaniy, say Q, defined on space-ime migh decay like a null infiniy on a fixed C u as : QQ 0 u, Q u, Q2 u, 2. Hee S 2, and u, ae as defined in he ex. ssume all Q (n) finie u. Noe hee ha he aeal adius of S,u can be wien (,u) and Q can be wien Q(,u). To see ha i is he same law on a fixed obseve ha fo lage enough u which also means lage Q 0 u,qu,q u,, 7 The physical meaning of his lapse funcion is discussed in. In geneal, he lapse funcion, ogehe wih is counepa, he shif veco, may be descibed as he nondynamical vaiable ha ells one how o move fowad in ime cf. e.g. 2,. whee I use he fac ha each poin of is on one of he C u cones and ha hee mus be a unique i a spaial infiniy. In wods, on, fo lage enough, Q is popoional o / o leading ode wih popoionaliy Q () (u,). This is ue

12 NTHONY RIZZI PHYSICL REVIEW D TBLE IV. Geodesic null pai (l,l). Ĥ B B O( ); Ĥ B B O( ); H 2 H 2 O 2 H 2 O TBLE V. Sandad null pai (l,l). Ĥ B B O( ) H B B O( ); H 2 H2 2 H 2 O ZZ W Z z 2 2 O 2 ZZ Z O O 2 ZZ Z Z 2 2 O 2 ZZ Z O Y 0 Y 2 O 0 O( 2 ) V W Z Z 2 2 O 2 Y 0 Y 2 2 O O( 2 ) O( 2 ) V E 2 E 2 O 2 fo evey such quaniy Q, and i is no necessay ha he decay law a null infiniy have inege powes of /. If one wans o add o muliply many such quaniies as one does in he inegals in quesion in his pape, i is obvious ha he esulan of such addiion and muliplicaion will also obey hese ules. n inuiive, simplified, way of hinking abou his is o ecall ha he choice of 0 in he ex was abiay. One can pick a saing as lage as one wishes, and, in his way, ge a lage enough on he S 2 suface on he maximal slice,, as one would like and hus eveywhee ou o null infiniy. The above agumens show ha he decay laws fo apply also in geneal fo wih cons. One can go fuhe and noe ha fo nonsingula Q (), one can wie a decay law ha is a funcion of only. Fo example, in he above case, le Cmax ove u and of Q () (u,) and ge Qu,Q 0 C whee C is a consan. PPENDIX F: PULLED BCK QUNTITIES In ode fo all of he wok done hee o ake place on acual sphees of uni adius a null infiniy, in ems of pulled back quaniies, I will always use he appopiae diffeomophism, divide by he appopiae powe of he luminosiy aeal adius defined by suface aea/4 and ake he i as s in he affine foliaion. Fo example, he meic on he opological sphee S,u will have a diffeomophism u, ; S 2 S,u ; i akes one fom an acual S 2 suface o he opological sphee S,u. One can use he pull back and show s u, * 0, he sandad meic on he uni sphee. Moe deails on his can be found in 8,0,4,. TBLE VI. -null pai (l,l ). Ĥ B B O( ) Ĥ B B O( ) H 2 H2 2 O 2 H 2 O ZZ Z Z 2 2 O 2 ZZ Z O Y 0 O( 2 ) O( 2 ) Y 2 O V Z Z 2 2 f whee f is some paicula funcion on S

13 NGULR MOMENTUM IN GENERL RELTIVITY:... PHYSICL REVIEW D Robe M. Wald, Geneal Relaiviy Univesiy of Chicago Pess, Chicago, S. Dese, R. nowi, and C. W. Misne, in Gaviaion: n Inoducion o Cuen Reseach, edied by L. Wien Wiley, New Yok, S. Dese, R. nowi, and C. W. Misne, Phys. Rev. 7, ; 8, ; 22, Rizzi, Phys. Rev. Le. 8, James W. Yok, J., in Essays in Geneal Relaiviy, edied by Fank J. Tiple cademic, New Yok, 980, Chap. 4, pp Geneal Relaiviy and Gaviaion: One Hunded Yeas fe he Bih of lbe Einsein, edied by. Held Plenum, New Yok, 980, Vol Niall Ó Muchadha and James W. Yok, J., Phys. Rev. D 0, D. Chisodoulou and S. Klaineman, The Global Nonlinea Sabiliy of he Minkowski Space Pinceon Univesiy Pess, Pinceon, NJ, nhony Rizzi, in Poceedings of he Eighh Macel Gossmann Meeing on Geneal Relaiviy, edied by Tsvi Pian Wold Scienific, Singapoe, 999, pp D. Chisodoulou, Phys. Rev. Le. 67, Rizzi, Ph.D. hesis, Pinceon Univesiy, C. W. Misne, K. S. Thone, and J.. Wheele, Gaviaion Feeman, San Fancisco,