aa r X i v : . [ g r- q c ] S e p General Relativity and Gravitation manus ript No.(will be inserted by the editor)Adam D. HelferAngular momentum of isolated systemsRe eived: date / A epted: dateAbstra t Penrose's twistorial approa h to the de(cid:28)nition of angular momen-tum at null in(cid:28)nity is developed so that angular momenta at di(cid:27)erent uts anbe meaningfully ompared. This is done by showing that the twistor spa esasso iated with di(cid:27)erent uts of I + an be identi(cid:28)ed as manifolds (but not asve tor spa es). The result is a well-de(cid:28)ned, Bondi(cid:21)Metzner(cid:21)Sa hs-invariantnotion of angular momentum in a radiating spa e(cid:21)time; the di(cid:30) ulties andambiguities previously en ountered are atta hed to attempts to express thisin spe ial-relativisti terms, and in parti ular to attempts to identify a singleMinkowski spa e of origins. Unlike the spe ial-relativisti ase, the angularmomentum annot be represented by a purely j = 1 quantity M ab , but hashigher- j ontributions as well. Applying standard kinemati pres riptions,these higher- j ontributions are shown to orrespond pre isely to the shear.Thus it appears that shear and angular momentum should be regarded asdi(cid:27)erent aspe ts of a single uni(cid:28)ed on ept.Keywords angular momentum · gravitational radiation · asymptoti stru ture of spa e(cid:21)time1 Introdu tionEnergy(cid:21)momentum and angular momentum (throughout this paper, `angu-lar momentum' will refer to the relativisti quantity) are of prime importan ephysi ally. Pra ti ally, they are key onserved quantities in the analysis ofspe i(cid:28) systems; theoreti ally, they are of fundamental signi(cid:28) an e, both las-si ally and in quantum physi s.Department of Mathemati s, University of Missouri, Columbia, MO 65211, U.S.A.Tel.: 573-882-7283Fax: 573-882-1869E-Mail: adammath.missouri.edu Yet while these quantities are well-understood in spe ial relativity, theirextension to general-relativisti situations is still in development. This is be- ause the foundation of the usual analysis (cid:22) the existen e of the Poin aréisometry group, generating the onserved quantities (cid:22) is absent: a generalspa e(cid:21)time will have no isometries. We must expe t, therefore, that in gen-eralizing momentum and angular momentum to urved spa e(cid:21)time, we willbe for ed to give up at least some of their familiar features, and perhaps toreassess what their roots are.For systems whi h are isolated in the sense of Bondi and Sa hs, that is,isolated in that one has a well-de(cid:28)ned on ept of what it means for radiationto es ape from them, the Bondi(cid:21)Sa hs energy(cid:21)momentum is now a epted as orre t. The su ess here has been losely onne ted with the existen e of asatisfa tory four-dimensional group of asymptoti translations. For the angu-lar momentum of radiating systems, however, there have remained problems.The asymptoti symmetry group, the Bondi(cid:21)Metzner(cid:21)Sa hs (BMS) group,(even though it has the anoni al four-dimensional translation subgroup)does not have any physi ally preferred Poin aré subgroup, but rather anin(cid:28)nite-dimensional family of them, apparently on equal footings. This is-sue often surfa es as an ambiguity in de(cid:28)ning a satisfa tory spa e of originswith respe t to whi h angular momentum is to be measured. One an in fa tmake a good argument that it is impossible to identify a preferred physi allynatural spa e of su h origins.1What properties should we look for in a de(cid:28)nition of angular momentumfor isolated general-relativisti systems? Certainly the de(cid:28)nition itself shouldbe natural, whi h is to say it should respe t the BMS symmetries. (It neednot be made in terms of the BMS generators, however.) And of ourse itshould be a plausible extension of the spe ial-relativisti de(cid:28)nition (althoughperhaps from an unorthodox perspe tive). Finally, I think it is essential thatangular momenta at di(cid:27)erent uts be unambiguously omparable, for thatis what is ne essary for a on ept of onservation, and a useful measure ofex hange between systems.In this paper, I shall show that it is possible to a hieve these goals by mak-ing a de(cid:28)nition of angular momentum dire tly on a twistor spa e. The twistorspa e, and the measure of angular momentum on it, have natural, invari-ant, and unproblemati existen es, whi h extend the de(cid:28)nitions in spe ial-relativisti twistor theory. (It is the attempts to pass from this twistor spa eto various andidate `spa es of origins' whi h introdu e di(cid:30) ulties and am-biguities.) But the most important feature of the twistor spa e is that it isuniversal, in that it depends only on those aspe ts of the asymptoti stru turewhi h are ommon to all Bondi(cid:21)Sa hs systems.The onstru tion is a development of Penrose's quasilo al twistors appliedat future null in(cid:28)nity I + [5,6℄. Penrose showed that for ea h ut S of I + , thereexists a natural twistor spa e T ( S ) with familiar spe ial-relativisti proper-1 Suppose the gravitational radiation is on(cid:28)ned to several (cid:28)nite intervals of re-tarded time. Then in the intervening, quies ent, intervals, one has andidate spa esof origins, but there are in general gauge mismat hes (distortions by `supertrans-lations') between these quies ent periods whi h prevent the identi(cid:28) ation of theirdi(cid:27)erent spa es of origins in any natural and onsistent way. Cf. [3,1,6℄.ties, most notably being a four- omplex-dimensional ve tor spa e. What Ishall show here is that there is a natural identi(cid:28) ation of all these twistorspa es as real manifolds. This means that we may say we have single twistorspa e T with many di(cid:27)erent possible omplex-linear stru tures, one for ea h ut. The spa e T retains enough stru ture that there is a meaningful twistorialde(cid:28)nition of angular momentum on it. As the origin-dependen e of angularmomentum be omes, in twistor terms, a dependen e on the hoi e of twistor,it is the existen e of a well-de(cid:28)ned twistor spa e whi h is the solution of theproblem.While the twistor spa e T has no anoni al linear stru ture, it does anon-i ally have a (cid:28)bre-bundle stru ture, a bundle of a(cid:30)ne omplex planes overa two- omplex dimensional ve tor spa e. This base spa e may be identi(cid:28)edwith asymptoti spin spa e S A ′ . The elements of S A ′ an be used to spe -ify the omponents of the angular momentum in question, and the hoi e ofpoint in the (cid:28)bre orresponds to a hoi e of origin.This means that if we think of measuring the angular momentum byspe ifying both a omponent and an origin, then (cid:28)xing the omponent onegets a sensible de(cid:28)nition (with a onventional origin-dependen e); it is thebehavior of this onstru tion as the omponent varies (in T , as we pass fromone (cid:28)bre to another) whi h is ompli ated. We shall see that the omponentsof the angular momentum no longer onstitute a pure j = 1 quantity M ab ,but a more ompli ated obje t with higher- j ontributions.For instan e, we will derive this formula for the spin: spin(ˆ r ) = s a v ˆ r a + M ℑ λ (ˆ r ) , (1)where ˆ r is a unit spatial ve tor representing the dire tion in whi h the spin isto be measured, the ve torial part of the spin is s a v , the Bondi(cid:21)Sa hs mass is M , and ℑ λ is an angular potential for the magneti part of the Bondi shear,with omponents for j ≥ . In other words, the spin an be measured in dif-ferent dire tions, but these measurements do not `integrate up' to give simplya spin-ve tor; they give a more ompli ated angular dependen e. Note thathere (as throughout this paper) the quantity j refers, not to the values thatangular momentum takes, but to the fun tional dependen e of the angularmomentum on dire tion.The formula (1) is interesting for a number of reasons. First, it givesan expli it and transparent role to the magneti part of the shear. (Investi-gations have repeatedly turned up a sensitivity of angular-momentum on-stru tions to the magneti shear, but the pre ise role has been di(cid:30) ult to pindown.) Se ond, the remarkable simpli ity of this formula an fairly be takenas eviden e in favor of the present approa h. Third, this result is intuitivelyplausible, be ause in spe ial relativity it is well-known that M − s a v an beinterpreted as a displa ement of the enter of mass into the omplex and intwistor theory ℑ λ has a similar interpretation as a displa ement of the utinto the omplex.One an also apply standard kinemati formulas to identify the enterof mass, and it has a non-ve torial part whi h may be identi(cid:28)ed with theangular potential ℜ λ for the ele tri part of the Bondi shear. Thus it isthe Bondi shear whi h ontrols the non-Poin aré hara ter of the angularmomentum, the ele tri and magneti parts of the shear ontributing non-Poin aré behavior to the enter-of-mass and spin, respe tively.These results suggest that the shear forms a sort of angular momentum,essentially the j ≥ parts of the angular momentum, and that, in passingfrom spe ial to general relativity, the mixing of gravitational radiation andmatter mixes `ordinary' ( j = 1 ) angular momentum and radiative modes ofthe (cid:28)eld, so that the ` orre t' understanding of angular momentum embra esthe two on epts.The most important onsequen e of the lose identi(cid:28) ation of angularmomentum with shear is that gravitational radiation may arry o(cid:27) angularmomentum (in luding j = 1 parts) as a (cid:28)rst-order e(cid:27)e t (in the Bondi news),whereas energy(cid:21)momentum is radiated as a se ond-order e(cid:27)e t. Thus theradiation of angular momentum may be a more important e(cid:27)e t for, and amore signi(cid:28) ant onstraint on, many systems than is the radiation of energy(cid:21)momentum.Still, the e(cid:27)e ts are not generally expe ted to be large ex ept in highlyrelativisti ir umstan es (or perhaps umulatively over long times or largevolumes). They are typi ally hara terized by the length s ale set by theBondi shear (cid:22) more pre isely, by the magneti part of the shear, and by hanges in the ele tri part of the shear. This s ale is typi ally of order ∼ ( GM/c )( v/c ) n , where M is a hara teristi mass and v a hara teristi velo ity and n ≥ .2 Even for relativisti velo ities, this is only of orderthe S hwarzs hild radius of the mass. Thus while it is to be hoped thateventually astrophysi al measurements will be re(cid:28)ned enough to warrant theuse of general-relativisti ally- orre ted angular momenta, the main interestat present is theoreti al.One would also expe t the ideas here to give quantum-gravitational or-re tions to angular momentum and spin. These e(cid:27)e ts do not appear tobe signi(cid:28) ant for individual mi ros opi systems, sin e the orre tions aretypi ally of the order ( E/E Pl ) ~ , where E is the energy of the system and E Pl is the Plan k energy. However, the development of a quantum theory ofspin in orporating the ideas here ould very well have impli ations for the(cid:28)nal stages of bla k hole evaporation, be ause it ould onstrain the possibletransitions.1.1 Real twistors, magneti shear and originsThis subse tion deals with the reality stru ture on twistor spa e. While thiswill (cid:28)gure essentially in the paper, the reader not spe ially interested intwistor theory an safely skip this part, referring ba k to it later if ne essary.While the ideas sket hed above present an a urate outline of the mainpoints in onventional spa e(cid:21)time terms, they do leave out one ru ial ele-ment of the twistor stru ture, its reality properties. It turns out that these2 This is of ourse a very rude statement and is only useful at the oarsest level.Given any spe i(cid:28) system, one needs to think about what the appropriate hoi esare for M and v . For example, in ases where there is a hange in ele tri shearthe orre t hara teristi velo ity v may be something like the square root of the hange in the squared velo ity of a omponent of the system.are losely tied to the magneti part of the shear, to the failure of a modelMinkowski spa e of origins to exist, and to di(cid:27)eren es between the stru turesof T and T ( S ) .In order to explain this, a qui k outline of the spe ial-relativisti ase isin order. There, twistor spa e has a Hermitian form of signature + + − − ,and the twistors Z α whi h are null ( Z α Z α = 0 ) with respe t to it are alled`real'. The points in Minkowski spa e may be identi(cid:28)ed with the totally real(that is, totally null) omplex two-planes, and thus real twistors are involved entrally in de(cid:28)nition of origins for angular momentum.In the general-relativisti ase, we shall (cid:28)nd that there is a natural de(cid:28)ni-tion of a quantity Φ on T analogous to the form Z α Z α on spe ial-relativisti twistor spa e, and one an de(cid:28)ne real twistors with respe t to Φ and thusdevelop a theory of angular momentum and spin. Of ourse, sin e T hasno preferred linear stru ture, one annot properly say that Φ is a (squared)norm. However, the failure is not just a semanti ni ety.Given any ut S , there is a natural identi(cid:28) ation of the twistor spa e T with Penrose's T ( S ) , whi h ould then be used to give T an S -dependent omplex-linear stru ture. However, in general the fun tion Φ does not respe tthat linear stru ture (cid:22) that is, in general, the fun tion Φ is not the restri tionof any Hermitian form on T ( S ) , but is more strongly nonlinear. This meansthat in general one annot (cid:28)nd totally Φ -real two-planes in T ( S ) , and thus one annot re onstru t from T ( S ) a Minkowski spa e of origins. In fa t, we shallsee that the ondition for Φ to arise from a Hermitian form on T ( S ) is pre iselythat S should have no magneti shear. Thus it is pre isely the presen e ofmagneti shear on S whi h prevents the identi(cid:28) ation of a Minkowski spa eof origins from T .1.2 OrganizationThe plan of the paper is this. The next se tion reviews twistorial kinemati sand Penrose's onstru tion. Se tion 3 is the heart of the paper; it explainshow the twistor spa es at di(cid:27)erent uts may be identi(cid:28)ed, and what thestru ture of the resulting nonlinear twistor spa e is. Se tion 4 explains howthis spa e may be used to ompute angular momentum. There the formulasfor the spin and enter of mass are dedu ed, as well as a (cid:29)ux law. Se tion 5explains how the present (non-Poin aré) angular momentum is onne tedto the familiar (Poin aré) spe ial-relativisti one, by dis ussing the ase oflinearized gravity. An a ount of the number of onstants of motion produ edby this framework, and the relation of the relation of angular momentumto gravitational degrees of freedom, is given in se tion 6. The (cid:28)nal se tionre apitulates the paper's main on lusions.Notation, onventions and ba kground. The notation and onventions arethose of Penrose and Rindler [6℄, in whi h also all on epts not explained here an be found. This paper assumes a familiarity with two-spinors and spin- oe(cid:30) ients, as well as Penrose's onformal treatment of I + . All omputationsat I + are in terms of the onformally res aled quantities. Fa tors of G aregiven expli itly, but some fa tors of c are suppressed. In this paper, we work at future null in(cid:28)nity I + . However, parallel results,applying to in oming radiation, ould be obtained at past null in(cid:28)nity I − ,by reversing the sense of time.For a review of the problems of de(cid:28)ning angular momentum and relatedissues, see the arti le by Szabados [7℄. Another sort of urved twistor spa ewas introdu ed by Penrose [4℄; this was essentially equivalent to the H -spa eof Newman [2℄, whi h had roots in work on angular momentum as well.2 Twistorial kinemati s and Penrose's onstru tionThis se tion summarizes the main points of twistor theory whi h will beused. Se tion 2.1 an be safely skipped by those familiar with twistor theory.The same is true for most of se tion 2.3, although the (cid:28)nal two paragraphsdo ontain some new omments about the ase of uts with magneti shear.Se tion 2.2 should be read, be ause the material is presented from a slightlyun onventional vantage adapted to later arguments.2.1 Twistors in Minkowski spa eIn Minkowski spa e, twistors an be regarded as solutions of the twistorequation ∇ A ′ ( A ω B ) = 0 . (2)It an be shown that there is a four- omplex-dimensional family T of solu-tions, ea h of the form ω A ( x ) = ω A − i x AA ′ π A ′ , where ω A and π A ′ are (cid:28)xedspinors. A twistor, when it is thought of as an element of this spa e (that is,when its hara ter as a spinor (cid:28)eld is not emphasized) is generally denoted by Z α , and one sometimes writes Z α = ( ω A , π A ′ ) . Under Lorentz motions, thespinors transform onventionally. Under a translation x a x a + k a , we have ω A ( x + k ) = ω A − i( x AA ′ + k AA ′ ) π A ′ , so ( ω A , π A ′ ) ( ω A − i k AA ′ π A ′ , π A ′ ) .This means that the proje tion Z α π A ′ is Poin aré- ovariant. Complemen-tarily, the twistors with π A ′ = 0 have ω A translation-invariant and are saidto lie at in(cid:28)nity.The following twistorial stru tures are also important. There is an alter-nating twistor ǫ αβγδ and a norm Z α Z α = ω A π A + ω A ′ π A ′ . (These are both onformally invariant.) A twistor with Z α Z α = 0 is said to be null or real.Additionally, there is a (Poin aré-invariant) in(cid:28)nity twistor I αβ = (cid:20) ǫ AB
00 0 (cid:21) . (3)For any skew bitwistor X αβ = − X βα , its dual is de(cid:28)ned by X αβ = (1 / ǫ αβγδ X γδ . A skew bitwistor is real i(cid:27) X αβ = X αβ . In parti ular, the in(cid:28)nitytwistor is real.Any point x in Minkowski spa e determines a omplex two-plane oftwistors, those for whi h ω A ( x ) = 0 , and all twistors on this plane are null.Conversely, any omplex two-plane in T all of whose elements are null de-termines a point in Minkowski spa e (or a point at in(cid:28)nity). The points atin(cid:28)nity in Minkowski spa e are pre isely those whose twistor planes in ludeat least one twistor at in(cid:28)nity. Correspondingly, a null twistor vanishes ex-a tly on a null geodesi in Minkowski spa e, along whi h its spinor π A ′ istangent (or the twistor orresponds to a limit at in(cid:28)nity of this situation).2.2 Kinemati s twistoriallyIf P a and M ab = µ A ′ B ′ ǫ AB + µ AB ǫ A ′ B ′ are the momentum and angularmomentum of a spe ial-relativisti system, they de(cid:28)ne a kinemati twistor,whose spinor omponents are given by A αβ = (cid:20) P AB ′ P A ′ B µ A ′ B ′ (cid:21) . (4)That is, the angular momentum has exa tly the orre t origin-dependen efor A αβ to transform as a (dual bi-)twistor.For any twistor Z α , then, the ombination A ( Z ) = A αβ Z α Z β = 2i µ A ′ B ′ π A ′ π B ′ + 2 P AA ′ ω A π A ′ (5)is invariant, that is, the origin-dependen es of µ A ′ B ′ and ω A an el out. Infa t, this quantity is simply times the π A ′ - omponent (or, more properly,the π A ′ π B ′ - omponent) of the spinor form of the relativisti angular momen-tum at any point x where ω A ( x ) = 0 . We may thus regard A ( Z ) as measuringthe angular momentum, by hoosing Z α = (i x AA ′ π A ′ , π A ′ ) , where x is thepoint at whi h to evaluate the angular momentum and π A ′ determines the omponent in question. (By polarization, or equivalently, by di(cid:27)erentiatingwith respe t to π A ′ , one an get all omponents of the angular momentumthis way.) Thus a knowledge of A ( Z ) on null twistors is equivalent to a knowl-edge of angular momentum, the hoi e of twistor oding both the origin withrespe t to whi h one is measuring and the omponent measured.We shall have also to onsider the polarized form A ( Z, ´ Z ) = A αβ Z α ´ Z β .For us, it is most natural to think of this as a sort of two-point fun tion ontwistor spa e. It is possible, of ourse, to interpret this in more onventionalspe ial-relativisti terms; a brief al ulation shows that A ( Z, ´ Z ) = 2i µ A ′ B ′ ( x av ) π A ′ ´ π B ′ − (i / P a x a diff π B ′ ´ π B ′ , (6)where x a av = (1 / x a + ´ x a ) and x a diff = x a − ´ x a . In this formula, the points x a , ´ x a are arbitrary points on the twistors Z , ´ Z . They are thus subje tto the freedoms x a → x a + ξπ A π A ′ , ´ x a → ´ x a + ´ ξ ´ π A ´ π A ′ . There is a one-parameter family of these for whi h the fa tor P a x a diff = 0 , and it de(cid:28)nes atimelike world-line of values of x a av with tangent P a . Thus one an interpret A ( Z, ´ Z ) as a omponent of the angular momentum at any point on thisworld-line. However, this interpretation will not be available in the generi general-relativisti setting, be ause we will not really have a model Minkowskispa e in whi h to envision world-lines.We are led to think of the angular momentum as a tually being thefun tions A ( Z ) , A ( Z, ´ Z ) on twistor spa e. Thus we think of evaluating theangular momentum, not by taking parti ular omponents at a parti ularspa e(cid:21)time point, but by evaluating A ( Z ) , A ( Z, ´ Z ) on twistor spa e (with adistinguished role played by the values at twistors with Z α Z α = 0 ). This willbe our point of view in what follows. This twistorial des ription of angularmomentum remains valid even when it is di(cid:30) ult to de(cid:28)ne a spa e of origins.Sin e ea h null twistor orresponds to a null geodesi with tangent spinor, thefun tion A ( Z ) is just the omponent of the angular momentum determinedby π A ′ evaluated along the geodesi . The fun tion A ( Z, ´ Z ) an be also beinterpreted as a sum of the angular momentum at the average of any twopoints on the geodesi s plus a orre tion due to the di(cid:27)eren e between thetwo points ( f. the previous paragraph), but it is really most natural to thinkof it as simply a two-point fun tion on the spa e of real twistors.Twistor expressions for the spin are a little involved. There are severalways of doing it; we give one whi h will be well-suited to later generaliza-tion. Suppose that Z α , ´ Z α are both real and also satisfy A αβ Z α Z β = 0 , A αβ ´ Z α ´ Z β = 0 . Then by dire t al ulation, one an verify the following iden-tity: ℜ A αβ Z α ´ Z β /I γδ Z γ ´ Z δ = (2 P AA ′ π A π A ′ P BB ′ ´ π B ´ π B ′ ) − · (cid:16) S AA ′ π A π A ′ P BB ′ ´ π B ´ π B ′ − S AA ′ ´ π A ´ π A ′ P BB ′ π B π B ′ (cid:17) , (7)where S a = (1 / ǫ abcd P b M cd is the Pauli(cid:21)Luba«ski spin ove tor. This iden-tity allows one to re over the spin in terms of real twistors. (It is not quiteobvious that given π A ′ and ´ π A ′ there do exist real twistors Z α , ´ Z α with A αβ Z α Z β = A αβ ´ Z α ´ Z β = 0 , but this an be dedu ed from (5).)2.3 Twistors on uts of I + In a urved spa e(cid:21)time, the twistor equation is in general over-determinedand has only the trivial solution. However, in favorable ir umstan es one anobtain twistor spa es by onsidering just ertain omponents of the equation.If S is a ut of I + , the omponents of the twistor equation involving onlyderivatives tangent to I + are ð ′ ω = 0 , ð ω = σω , (8)where we use the standard spin- oe(cid:30) ient formalism for the geometry ex-pressed in onformally res aled terms ( f. [6℄). This system of equations de-(cid:28)nes a four- omplex-dimensional ve tor spa e T ( S ) , whi h turns out to havean in(cid:28)nity twistor and alternating twistor obeying the same algebra as thosein Minkowski spa e.The kinemati twistor, dedu ed by orresponden e with linear theory, is A ( Z ) = A αβ Z α Z β = − i4 πG I (cid:8) ψ ( ω ) + 2( ψ + σ ˙ σ ) ω ω (cid:9) d S , (9)where the overdot represents di(cid:27)erentiation with respe t to u and an inte-gration by parts has been used to simplify the expression somewhat.To omplete the treatment of kinemati s, one also needs a norm (andto verify that the other quantities have the appropriate reality propertieswith respe t to it). The natural thing to do is to express Z α Z α in terms ofthe spinor (cid:28)elds ω , ω . This an be done, but one (cid:28)nds that the resultingexpression, while onstant in Minkowski spa e, or on a ut with purely ele tri shear, is not onstant if the shear has a magneti omponent. (Re all that theshear σ , being a spin-weight two quantity, an always be expressed as σ = ð λ for an ordinary fun tion λ . The ele tri and magneti parts of the shear are σ el = ð ℜ λ and σ mag = ð i ℑ λ .) Penrose suggested, with some reservations,averaging over the ut with respe t to the metri de(cid:28)ned by the Bondi(cid:21)Sa hsenergy(cid:21)momentum. Then one does get the proper spe ial-relativisti twistoralgebra.There are two on erns about this norm (in the ases where there is amagneti omponent to the shear) whi h should be raised here. The (cid:28)rst is itsstrong ut-dependen e, whi h makes it hard to understand how to ompareangular momenta at di(cid:27)erent retarded times. The se ond is that, were oneto adopt it, one would have to regard the physi al spa e(cid:21)time as displa edinto the omplex relative to the reality stru ture on the twistor spa e [6℄, andthis would mean (for example) that a massless test parti le on a light ray,whi h by lo al measurements had zero heli ity, would have to be as ribed anon-zero heli ity a ording to the twistorial de(cid:28)nition.3 Identifying the twistor spa esI show here how the twistor spa es at the di(cid:27)erent uts of I + may be identi-(cid:28)ed. While the ideas are not di(cid:30) ult, it is perhaps well to begin by aution-ing against a possible pre on eption. On ea h ut the twistors are de(cid:28)ned asspinor (cid:28)elds satisfying ertain equations, so one might be tempted to thinkthat the twistors ould be realized as spinor (cid:28)elds over all of I + , whi h simplyrestri t to the orre t values on any ut. However, the twistors have a moredelo alized existen e, taking well-de(cid:28)ned values as spinor (cid:28)elds only whenan entire ut (and not simply a point on the ut) is spe i(cid:28)ed.3.1 The (cid:28)eld ω The omponents ω , ω , and the two omponents of the two-surfa e twistorequation, are not on the same footing as far as the delo alization just dis- ussed goes. It turns out that the spa e of ω s satisfying ð ′ ω = 0 on any utdoes have a well-de(cid:28)ned existen e as a (cid:28)eld on all of I + . This feature is a well-known part of Penrose's onstru tion, the spa e of su h ω being identi(cid:28)ed0with S A ′ , the spa e of primed dual twistors `at in(cid:28)nity', and the proje tion ( ω , ω ) ω providing a (cid:28)bration T ( S ) → S A ′ of the twistor spa e on agiven ut to this spa e. The ω spinors in this ase satisfy Þ ′ ω = 0 (as wellas ð ′ ω = 0 ), whi h an be thought of as providing a propagation equationfrom one ut to another. These two equations may be expressed jointly as ι A ι B ∇ AA ′ ω B = 0 , whi h makes the ut-independen e of the system lear,sin e ι A is (apart from s aling) independent of the Bondi system.There will be a onsequen e of this whi h will (cid:28)gure entrally in whatfollows. This is that for any admissible ω (cid:28)eld whi h is not identi ally zero,there is a unique generator γ ( ω ) of I + on whi h ω vanishes. (This is awell-known onsequen e of the equation ð ′ ω = 0 , given that ω has spin-weight − / , together with the equation Þ ′ ω = 0 , whi h transports ω upthe generators.) The generator on whi h ω vanishes is also independent ofthe hoi e of Bondi system, as an be seen by writing the vanishing equationin the form ω A ι A = 0 .3.2 The (cid:28)eld ω We have seen that, for ea h twistor, the (cid:28)eld ω has a well-de(cid:28)ned ut-independent existen e. However, the situation for ω is more ompli ated.In order to treat twistors without having in ea h instan e to take up theirglobal behavior over a ut, we take advantage of the fa t that ea h twistor anbe spe i(cid:28)ed, at any point of the ut, by suitable data. These data are most ommonly taken to be the values ( ω , ω , π ′ , π ′ ) , where π ′ = i ð ′ ω − i ρω , π ′ = i ð ω .3 We may thus, in order to ask how to transport a twistor fromone ut to another, simply ask how to identify its data at one point on the(cid:28)rst ut with data at another point on the se ond ut.Our pres ription will essentially onsist of transporting the twistor alongthe generator γ ( ω ) on whi h ω vanishes; this generator must interse t ea h ut in a single point. (The ase of identi ally vanishing ω will be seen laterto be determined by ontinuity, and also to agree with Penrose's de(cid:28)nitionof the spin spa e S A .)In order to determine the orre t transport equations, we re all the for-mula for the lo al twistor onne tion: D BB ′ (cid:0) ω A , π A ′ (cid:1) = (cid:0) ∇ BB ′ ω A + i ǫ BA π B ′ , ∇ BB ′ π A ′ + i P ABA ′ B ′ ω A (cid:1) , (10)where P ab = Φ ab − Λg ab = (1 / Rg ab − (1 / R ab . The equation D BB ′ (cid:0) ω A , π A ′ (cid:1) = 0 has, in a onformally (cid:29)at spa e(cid:21)time, lo ally a four- omplex-dimensionalfamily of solutions whi h omprise twistor spa e. At I + when gravitationalradiation is present, this equation is over-determined and does not have afull four-dimensional family of solutions. However, we shall only use ompo-nents where the derivative is tangent to a generator of I + and we shall onlyenfor e it on the single generator γ ( ω ) . Thus we have a system of ordinarydi(cid:27)erential equations on a line, and there are no integrability restri tions.3 For onvenien e, we hoose ρ ′ = 0 .1The omponents of (10) in question are n b D b (cid:0) ω A , π A ′ (cid:1) = 0 on γ ( ω ) . (11)In this form, they are manifestly independent of the hoi e of Bondi system.In spin- oe(cid:30) ient notation, they be omeÞ ′ ω = 0 , Þ ′ ω = ð ω , Þ ′ ð ′ ω = 0 , Þ ′ ð ω = 0 on γ ( ω ) . (12)We shall use these now to derive an expli it transformation for twistors fromone ut to another. For simpli ity, we take these here to be two u = const uts of the same Bondi system, although this is not ne essary.The twistors on the uts may be onveniently given. We let λ be anyfun tion satisfying ð λ = σ ; then the ω (cid:28)elds on the ut have the form ω = ω ð λ − λ ð ω + ξ , (13)where ξ is any solution to ð ξ = 0 . (This observation is due to K. P. Tod.)The spa e of su h ξ is spanned by ð ω and ω ′ , and so ω = ω ð λ − λ ð ω + α ð ω + βω ′ , (14)where α = α ( u ) and β = β ( u ) are to be determined. Substituting (14) into(12) and integrating, we (cid:28)nd α = u − u + λ ( u, γ ) − λ ( u , γ ) + α (15) β = ð ω ( γ ) ð ′ ω ′ ( γ ) ( ð ′ λ ( u, γ ) − ð ′ λ ( u , γ )) + β , (16)where α ( u ) = α , β ( u ) = β , and we have abused notation slightly bywriting the generator γ as the argument of fun tions (rather than the angularvariables determining γ ).These formulas allow us to identify the twistor spa es T ( S ) at di(cid:27)erent uts, so we may say that we have a single twistor spa e T . For any (cid:28)xed u ,we may view α ( u ) and β ( u ) (together with a oordinatization of the ω (cid:28)elds) as providing oordinates on T (and identifying it with T ( S ) , where S orresponds to u ). As u varies, the transport formulas give us transitionfun tions to other oordinate systems ( α ( u ) , β ( u )) . The formulas are gener-ally nonlinear, be ause of the nonlinear dependen e of the terms on γ , thegenerator on whi h ω vanishes.So far we have ex luded the ase of twistors with identi ally vanishing ω . To obtain oordinate harts in luding su h twistors we hoose some o-ordinatization of the spa e of ω (cid:28)elds and also of the ξ (cid:28)elds; together these oordinatize the twistors a ording to (14). Whatever oordinatization ofthese we hoose, we will have ξ ( u ) − ξ ( u ) = ( u − u + λ ( u, γ ) − λ ( u , γ )) ð ω + ð ω ( γ ) ð ′ ω ′ ( γ ) ( ð ′ λ ( u, γ ) − ð ′ λ ( u , γ )) ω ′ , (17)by (14,15,16), and this will determine the transition fun tions. Sin e this hasa well-de(cid:28)ned limit (zero) as ω is taken identi ally to zero, the transitionfun tions extend to be C there.23.3 Stru ture of T We have just seen that T is a smooth manifold ex ept where ω vanishesidenti ally, where it is of lass C . The transition fun tions respe t the (cid:28)bra-tion over S A ′ , so the spa e T (cid:28)bres over S A ′ . Ea h (cid:28)bre has the stru ture ofan a(cid:30)ne omplex two-plane, for the transition fun tions only a t by trans-lations. Ea h hoi e of ut provides a trivialization of the bundle, identify-ing T with T ( S ) . There is a well-de(cid:28)ned s aling a tion ( ω , ω , π ′ , π ′ ) k ( ω , ω , π ′ , π ′ ) .We shall take as the in(cid:28)nity twistor I ( Z, ´ Z ) = ´ ω ð ω − ω ð ´ ω (whi h is onstant over I + ), in agreement with Penrose. There is a well-de(cid:28)ned realfun tion on T whi h naturally extends the de(cid:28)nition of the (squared) norm,that is the quantity Φ = ω A π A + π A ′ ω A ′ evaluated at γ ( ω ) (and equal tozero if ω vanishes identi ally). Even on a spe i(cid:28) ut S , where T ( S ) has a omplex-linear stru ture, the form Φ is not in general Hermitian, be ause theexpression ω A π A + π A ′ ω A ′ must be evaluated at the point γ ( ω ) depending onthe twistor in question; this introdu es a signi(cid:28) ant additional nonlinearity.In fa t (as Penrose noted), the ondition that this expression be onstantover S is pre isely that S be purely ele tri . Our fun tion Φ thus redu es toa Hermitian one (and agrees with Penrose's) on a ut pre isely if the ut ispurely ele tri ; otherwise it is a more ompli ated obje t.This strongly nonlinear Φ has two advantages. First, it is independentof the hoi e of ut, and this means that we have a universal notion of realtwistors, and thus a way of omparing angular momenta on di(cid:27)erent utsof I + . Se ond, it provides a dire t link to spa e(cid:21)time geometry, for withthis de(cid:28)nition of Φ we may identify the real twistors with the null geodesi s(together with their parallel-propagated tangent spinors) meeting I + , in theusual way. The fun tion Φ does restri t to a Hermitian form on those twistorswith the (cid:28)eld ω (cid:28)xed up to proportionality, be ause in this ase one doesnot need to vary γ ( ω ) . If we (cid:28)x also the s ale of ω (and require ω not iden-ti ally zero), then the set of all null twistors with this ω (cid:28)eld is evidently areal three-dimensional a(cid:30)ne spa e. In the spe ial-relativisti ase, this spa ewould be identi(cid:28)ed with Minkowski spa e modulo the translations by π A π A ′ ,where π A ′ stands for ω regarded as an element of S A ′ . We thus get, for ea h π A ′ = 0 , a spa e whi h an be regarded as a spa e of origins modulo trans-lations by multiples of π A π A ′ . As we expe t the omponent of the angularmomentum in the dire tion π A ′ to be independent of su h translations, wewill have, for this omponent, as mu h of a well-de(cid:28)ned spa e of origins as isne essary for the de(cid:28)nition of angular momentum.4 Kinemati s on T As our onstru tion allows us to identify T with Penrose's T ( S ) on a ut S ,we may take over Penrose's de(cid:28)nition of the kinemati twistor dire tly to T , A S ( Z ) = − i4 πG I (cid:8) ψ ( ω ) + 2( ψ + σ ˙ σ ) ω ω (cid:9) d S . (18)3This de(cid:28)nes the `unpolarized' form, orresponding to the Minkowski quan-tity A αβ Z α Z β . A polarized form, orresponding to A αβ Z α ´ Z β , an also bede(cid:28)ned. While these forms are quadrati and bilinear, respe tively, as fun -tions on T ( S ) , one annot say they have these properties as fun tions on T ,be ause of the la k of a linear stru ture on that spa e.Even restri ting our attention to one ut, however, the linear stru tureon T ( S ) is in generi ir umstan es mu h less signi(cid:28) ant than it is in spe- ial relativity, for the spe ial-relativisti formalism for angular momentum isre overed only when the shear is purely ele tri . If the shear is not purelyele tri , one annot re over a suitable Minkowski spa e of origins with respe tto whi h one might de(cid:28)ne angular momentum in familiar Poin aré terms. Weare thus for ed to onfront, even on a single ut, a non-Poin aré hara ter ofthe angular momentum: what does A S ( Z ) mean?Brie(cid:29)y, we may interpret A S ( Z ) , for real Z , as the π A ′ - omponent of theangular momentum about the geodesi de(cid:28)ned by Z (or really about thepoint where Z meets I + , sin e the onstru tion is really de(cid:28)ned at I + ). Thisinterpretation is valid whether Z meets I + in S or not, and whether Z , ifit does meet S , is a member of the ongruen e of null geodesi s meeting S orthogonally. A fuller understanding will be developed by investigating thespin and enter of mass.4.1 Magneti shears and spinAs noted above, when a magneti omponent of the shear is present, thede(cid:28)nition of the norm adopted here in general pre ludes the identi(cid:28) ationof a Minkowski spa e of origins. It is natural to ask what happens to thespin in this situation, for in spe ial relativity the spin is origin-independent.Thus we may hope to get an origin-independent spin in general relativity.It is not at all obvious, however, that it is possible to do so. The most ommon spe ial-relativisti approa hes to spin have no obvious analogs here,be ause these approa hes begin from the angular momentum evaluated atan origin. Remarkably, however, there is a suitable su h de(cid:28)nition. We sawin se tion 2.2 that in spe ial relativity there is an expression for the spinin twistorial terms, and it turns out that the same expression results in anorigin-independent de(cid:28)nition of spin here. The de(cid:28)nition is more ompli atedthan in spe ial relativity, though, in that the spin is not (if magneti shearis present) purely ve torial ( j = 1 ), but has higher- j omponents as well.In order to derive it, let us begin by working out A S ( Z, ´ Z ) in some detail.For simpli ity we take the time axis of the Bondi system aligned with theBondi(cid:21)Sa hs energy(cid:21)momentum, and assume that λ has no j = 0 or j = 1 parts with respe t to this hoi e. Then substituting ω = ω ð λ − λ ð ω + α ð ω + βω ′ (and similarly for ´ ω ), we (cid:28)nd A S ( Z, ´ Z ) = − i4 πG I n(cid:2) ψ + 2( ψ + σ ˙ σ ) ð λ + ð ( λ ( ψ + σ ˙ σ )) (cid:3) ω ´ ω +( ψ + σ ˙ σ )( α ´ ω ð ω + ´ αω ð ´ ω )+( ψ + σ ˙ σ )( ´ βω ´ ω ′ + β ´ ω ω ′ ) o d S . (19)4The middle term an be written as one proportional to ( α + ´ α ) ð ( ω ´ ω ) (butthis has pure j = 1 and so ontributes nothing to the integral in our hoi e offrame) and one proportional to ( α − ´ α )(´ ω ð ω − ω ð ´ ω ) = ( α − ´ α ) I αβ Z α ´ Z β ,whi h is onstant over the ut. Letting π A ′ and ´ π A ′ denote the spinor (cid:28)elds ω and ´ ω , we may write in parallel to (4) and (5) A S ( Z, ´ Z ) = 2i µ A ′ B ′ π A ′ ´ π B ′ + (i / M ( α − ´ α ) ǫ R ′ S ′ π R ′ ´ π S ′ +i P AA ′ ( ´ βπ A ′ ´ π A + βπ A ´ π A ′ ) , (20)where the three terms orrespond to the three lines of (19). (The notationhere is onventional, but there is one point where it an be onfusing. This isthat the symbol π A ′ now stands for the spinor (cid:28)eld ω as an abstra t elementof S A ′ , and not for the spinor (cid:28)eld π ′ o A ′ − π ′ ι A ′ representing omponentsof the lo al twistor, as it did previously. From this point on, we shall onlyuse π A ′ to stand for an element of S A ′ , and we shall write out the spinor (cid:28)eld π ′ o A ′ − π ′ ι A ′ when we need it.)The formula (7) for the spin requires ertain restri tions on the twistors.First, we require Z and ´ Z to be real. We have Φ ( Z ) = ( ω π + ω π + π ′ ω ′ + π ′ ω ′ ) (cid:12)(cid:12)(cid:12) γ = ( − i ω ð ′ ω ′ + i ω ′ ð ω ) (cid:12)(cid:12)(cid:12) γ = i( λ − α − λ + α ) ð ω ð ′ ω ′ (cid:12)(cid:12)(cid:12) γ . Thus Z is real i(cid:27) α − λ ( γ ) is real. (And similarly ´ Z is real i(cid:27) ´ α − λ (´ γ ) is real.)The spe ial-relativisti formula for the spin also requires A S ( Z ) = A S ( Z, Z )= 0 and A S ( ´ Z ) = A S ( ´ Z, ´ Z ) = 0 . The for e of these onditions an be dedu edfrom (20), however, by letting the twistors oin ide. We (cid:28)nd µ A ′ B ′ π A ′ π B ′ + βP AA ′ π A π A ′ = 0 , whi h we regard as an equation for β , and similarly for ´ β .With these formulas in hand, it is straightforward to show that ℜ A S ( Z, ´ Z ) /I ( Z, ´ Z ) = (2 P AA ′ π A π A ′ P BB ′ ´ π B ´ π B ′ ) − · h ( S AA ′ v + M P AA ′ ℑ λ ( γ )) π A π A ′ P BB ′ ´ π B ´ π B ′ − ( S AA ′ v + M P AA ′ ℑ λ (´ γ ))´ π A ´ π A ′ P BB ′ π B π B ′ i , (21)where the freedom in α , ´ α has an eled out and S AA ′ v = 2 ℜ i µ A ′ B ′ P AB ′ (22)is the usual formula for the Pauli(cid:21)Luba«ski spin-ve tor. Remembering thatin spe ial relativity this is M times the spin, we may in our ase de(cid:28)ne thespin by spin( π A ′ ) = M − S AA ′ v π A π A ′ + P AA ′ π A π A ′ ℑ λ ( γ ) , (23)where γ is the generator of I + orresponding to π A ′ .The ve torial part S a v is orthogonal to the time axis here. It is naturaltherefore to repla e the null ve tor π A π A ′ by its spatial proje tion. The term λ ( γ ) depends only on this proje tion as well. Thus with a slight abuse ofnotation, the spin in the unit ˆ r dire tion is spin(ˆ r ) = M − S a v ˆ r a + M ℑ λ (ˆ r ) . (24)5The simpli ity of this formula, and the lean onne tion of spin and mag-neti shear it gives, have already been noted. That this formula should existat all (cid:22) that the quantity within square bra kets in (21) should allow aseparation of the terms in Z and ´ Z (cid:22) is also remarkable.4.2 Mass-moments and enter-of-massIf P a and µ A ′ B ′ are the energy(cid:21)momentum and angular momentum of aspe ial-relativisti system, and P a is timelike (whi h we will always assume),then we may write µ A ′ B ′ = (cid:16) k AA ′ − i M − S AA ′ (cid:17) P AB ′ , (25)where M is the mass, and k a , S a , ea h orthogonal to P a , represent the enterof mass (in the plane P a x a = 0 ) and the Pauli(cid:21)Luba«ski spin ve tor. Thusthe enter of mass and spin appear as the real and imaginary parts of asingle quantity. In spe ial relativity, in order to identify the enter of massof a system, one looks for points at whi h the ve tor k a vanishes. If we wishto do this by examining µ A ′ B ′ , we must (cid:28)rst subtra t the spin ontribution.We will follow the same strategy in general relativity.In the general-relativisti ase, we may again interpret a real twistor Z asa null geodesi γ together with a tangent spinor π A ′ , and we may interpret (2i) − A S ( Z ) as the π A ′ - omponent of the angular momentum about γ . (This γ is then not the same as the generator γ ( ω ) , but γ does meet I + at γ ( ω ) .)The natural way to subtra t the spin omponent of the angular momentumis to form (2i) − A S ( Z ) + i M − P AA ′ π A ′ ∂∂π A spin = (cid:16) µ A ′ B ′ + i M − S AA ′ v P AB ′ (cid:17) π A ′ π B ′ + βP AA ′ π A π A ′ + i M − P AA ′ π A ′ ∂∂π A P BB ′ π B π B ′ ℑ λ . (26)The operator M − P AA ′ π A ′ ∂/∂π A (cid:28)guring here is essentially the ð ′ operatora ting on fun tions of π A , but this notation will be avoided here, be ause,while it is `morally the same' as the ð ′ operator appearing elsewhere in thispaper, they di(cid:27)er by a fa tor, as will now be shown. (Note that this operatorpasses through P BB ′ π B π B ′ .)We are using the symbol π A ′ to represent the (cid:28)eld ω , thought of asan element of S A ′ . Remembering that we have hosen the time-axis of theBondi system aligned with the Bondi energy(cid:21)momentum, we have P a = M t a (where t a is the unit timelike ve tor determining the frame), and thus, bybasi spin- oe(cid:30) ient results, we have t AA ′ π A ′ orresponding to ð ω , and sowe must di(cid:27)erentiate ℑ λ by perturbing ω ′ by ð ω . However, the potential ℑ λ is given, not as a fun tion of ω (or ω ′ ), but as a fun tion of γ , thegenerator on whi h ω vanishes. Perturbing the de(cid:28)ning equation ω ( γ ) = 0 ,we (cid:28)nd − ( δγ a ) m a ð ′ ω − ( δγ a ) m a ð ω + δω = 0 , (27)6where δγ a is the in(cid:28)nitesimal perturbation of γ as ω ′ is perturbed by δω ′ = ð ω . However, we have ð ′ ω = 0 , and so δγ a m a ð ω = δω . Applying thisto our perturbation δω ′ = ð ω , we have δγ a m a = ð ω / ð ′ ω ′ . Therefore t AA ′ π A ′ ∂ ℑ λ/∂π A = − ( ð ω / ð ′ ω ′ ) ð ′ ℑ λ and (2i) − A S ( Z )+ i M − P AA ′ π A ′ ∂∂π A spin = (cid:16) µ A ′ B ′ + i M − S AA ′ v P AB ′ (cid:17) π A ′ π B ′ + βP AA ′ π A π A ′ − i P BB ′ π B π B ′ ( ð ω / ð ′ ω ′ ) ð ′ ℑ λ . (28)This quantity should be interpreted as the π A ′ - omponent of the mass-moment of the system at the geodesi γ .The enter of mass is given by the vanishing of the mass-moments, and soby the vanishing of (28). The vanishing of this equation gives us a formula for β in terms of π A ′ , or, equivalently, in terms of the generator of I + . Thus forany generator of I + we get a spinor π A ′ , and this spinor determines β fromthe vanishing of (28). This hoi e of β , together with an admissible hoi e of α (re all we must have α − λ ( γ ) real for Z to be a real twistor) determinesa twistor from (14).In these formulas, the twistor Z is one meeting I + in the generator labeledby π A ′ (or equivalently, determined by the vanishing of ω ). The freedom in hoosing α simply orresponds to translating the twistor up or down thegenerator (as follows from (15)), and is here not very signi(cid:28) ant, as thetime axis is hosen to oin ide with the Bondi(cid:21)Sa hs energy(cid:21)momentumand this means the entire system is invariant under su h time translations.The freedom in β is more interesting, and varying β orresponds to makingdi(cid:27)erent hoi es of real twistor (that is, of null geodesi ), through the samepoint of I + , the omplex parameter β representing the two real degrees offreedom in this hoi e. Thus a determination of β by the vanishing of (28)is pre isely a sele tion of a null geodesi inwards from the point of I + inquestion, this geodesi being interpretable as the one, through that point,dire ted towards the enter of mass of system.The family of su h geodesi s determines a ongruen e whi h has an in-tuitive geometri signi(cid:28) an e: it is the ongruen e one would obtain by su-pertranslating the Bondi system to remove the ele tri part of the shear (asmeasured at S ). To see this, we examine the tangent spinor to the ongruen e,whi h is (not the abstra t π A ′ ∈ S A ′ but) the (cid:28)eld π ′ o A ′ − π ′ ι A ′ , evaluatedat γ : i( ð ω ) o A ′ − i( ð ′ ω ) ι A ′ = i( ð ω ) o A ′ − i( − ð ′ λ ð ω + β ð ′ ω ′ ) ι A ′ = i ð ω o A ′ + ð ′ λ − ð ′ ω ′ ð ω β ! ι A ′ ! . (29)Comparing this with the vanishing of (28), we see that β is hosen pre iselyto remove the ontribution of the imaginary part of λ , leaving only the realpart, whi h is the angular potential for the ele tri part of the shear.This result seems very reasonable. In the (cid:28)rst pla e, it would have thee(cid:27)e t of removing any shear whi h was purely gauge (cid:22) that is, arose entirelyfrom a supertranslation. In the se ond pla e, that the magneti portion of the7shear has an eled out means that the ongruen e meets I + orthogonally ina well-de(cid:28)ned family of uts (given by supertranslations by ℜ λ ). These maybe interpreted as instants of retarded time in the rest-frame of the system,as de(cid:28)ned instantaneously at S .The results here and in the previous subse tion have an overall parallelismdespite a onsiderable di(cid:27)eren e in detail. The spin and the enter of massare very di(cid:27)erent sorts of quantities in general relativity, with the spin simplya dire tion-dependent fun tion, but the enter of mass a null ongruen e. Yetthe magneti part of the shear is essentially the j ≥ part of the spin, andthe ele tri part of the shear is essentially the j ≥ part of the enter ofmass. Thus one may say that the shear itself odes the j ≥ part of theangular momentum.4.3 FluxIt is straightforward to ompute the evolution of the angular momentum;one simply uses the formula (19) for the kinemati twistor together withthe formulas (14,15,16) for the twistors and their evolution. (Su h a dire tapproa h is possible pre isely be ause the angular momentum is de(cid:28)ned on aspa e T whi h is independent of the ut.) We shall only ompute the evolutionin retarded time for a (cid:28)xed Bondi system, for simpli ity, although this paper'sapproa h would allow the omputation between two arbitrary uts.We have A S ( Z, ´ Z ) = − i4 πG I (cid:8) ψ ω ´ ω + ( ψ + σ ˙ σ )( ω ´ ω + ´ ω ω ) (cid:9) d S . (30)Di(cid:27)erentiating this, and using the relations ˙ ψ = ð ψ + 2 σψ − πGT (01)1 ′ ′ (31) ˙ ψ = ð ψ + σψ − πGT ′ ′ , (32)we (cid:28)nd, after a little work, ∂∂u A ( Z, ´ Z ) = F matter + F Bondi + F shift , (33)where the three terms on the right will be dis ussed separately.The (cid:28)rst term in (33), F matter = − i4 πG I (cid:8) − πGT (01)1 ′ ′ ω ´ ω − πGT ′ ′ [ ω ´ ω + ´ ω ω ] (cid:9) d S , (34)is the (cid:29)ux of energy(cid:21)momentum and angular momentum arried away bymatter. This term is exa tly what one would expe t on formal grounds.The se ond term in (33), F Bondi = − i4 πG I (cid:8) [ ψ + σ ˙ σ ][2 ω ´ ω ð ˙ λ − ˙ λ ð ( ω ´ ω )] + ˙ σ ˙ σ [ ω ´ ω + ´ ω ω ] (cid:9) d S , (35)8is a generalization of the Bondi energy-loss term. It is T ( S ) -linear. Its on-tribution to the energy(cid:21)momentum is through the term ˙ σ ˙ σ [ ω ´ ω + ´ ω ω ] ,and this is exa tly what one expe ts and is se ond-order in the gravitationalradiation. Its ontribution to the angular momentum is through the term [ ψ + σ ˙ σ ][2 ω ´ ω ð ˙ λ − ˙ λ ð ( ω ´ ω )] (and energy(cid:21)momentum parts an mix in aswell, of ourse, be ause of the origin-dependen e). This portion will in generalhave ontributions whi h are (cid:28)rst-order in the gravitational radiation, sin e ˙ λ is (cid:28)rst-order. What is ne essary for su h (cid:28)rst-order ontributions is thatthe angular dependen es of ˙ λ and the mass aspe t [ ψ + σ ˙ σ ] should ombineto produ e j = 0 or j = 1 terms. This means that the angular dependen e ofthe gravitational radiation should be suitably orrelated with the anisotropyof the system.The (cid:28)nal term in (33), F shift = − i4 πG I n [ ψ + σ ˙ σ ][ ˙ λ (´ γ ) ω ð ´ ω + ˙ λ ( γ )´ ω ð ω +( ð ω ð ′ ˙ λ/ ð ′ ω ′ ) (cid:12)(cid:12)(cid:12) ´ γ ω ´ ω ′ + ( ð ω ð ′ ˙ λ/ ð ′ ω ′ ) (cid:12)(cid:12)(cid:12) γ ´ ω ω ′ ] o d S = ( − i / M (cid:16) ˙ λ (´ γ ) − ˙ λ ( γ ) (cid:17) ǫ A ′ B ′ ´ π A ′ π B ′ + i´ π A ′ t AA ′ ∂∂ ´ π A ˙ λ (´ γ ) P AA ′ ´ π A π A ′ + i π A ′ t AA ′ ∂∂π A ˙ λ ( γ ) P AA ′ π A ´ π A ′ (36)(where the simpli(cid:28) ations in the last step make use of the results of the pre- eding subse tion), ontains the terms re(cid:29)e ting the shift in linear stru tureon T ( S ) as S is evolved. These terms are in general (cid:28)rst-order in the radia-tion. Note that this term is not origin-dependent: it depends only on the ω , ´ ω parts of the twistors, not on ω , ´ ω . Thus these `shift' ontributions tothe (cid:29)ux only displa e one (cid:28)bre of T relative to another; they do not alter thea(cid:30)ne stru ture within ea h (cid:28)bre whi h odes the origin-dependen e.It is of some interest to ompare the last formula for F shift with our earlierformula (6) for the spe ial-relativisti kinemati twistor: we had A ( Z, ´ Z ) = 2i µ A ′ B ′ ( x av ) π A ′ ´ π B ′ − (i / P a x a diff π B ′ ´ π B ′ , (37)where x a av = ( x a + ´ x a ) / , x a diff = x a − ´ x a , and x a , ´ x a were any two pointsin Minkowski spa e on the real twistors Z , ´ Z . The general-relativisti for-mula for the (cid:29)ux has a parallel stru ture, with M ( ˙ λ ( γ ) − ˙ λ (´ γ )) ǫ A ′ B ′ ´ π A ′ π B ′ orresponding to the term involving x a diff , and the other terms in (36) to anaverage of the `shift' ontributions to the (cid:29)ux of angular momentum at thetwo points.4.4 Nearly stationary systemsIt is worthwhile examining the evolution of the angular momentum in the ase of nearly stationary systems, partly be ause these a ount for a lass ofsubstantial interest, and partly in order to larify some of the ideas involved.9We must (cid:28)rst make pre ise what we mean by a nearly stationary system.Sin e we are interested in the asymptoti behavior of the (cid:28)eld only, what wehave in mind is a system whi h departs from stationarity only to (cid:28)rst ordernear I + . (The system may have a strong time-dependen e in the interior ofspa e(cid:21)time.)We take the Bondi system to be aligned with the (approximate) station-arity, so that ψ = − M to zeroth order, where M is the mass. We assumeas well that the Bondi frame has been hosen so that the shear itself (andnot just its u -derivative) is a (cid:28)rst-order quantity. It is no loss in generality toassume that λ is hosen to have only j ≥ omponents, as before. And weshall assume, for simpli ity, no matter (cid:28)elds are present near in(cid:28)nity. Thenthe (cid:29)ux F matter due to matter vanishes.The Bondi (cid:29)ux term F Bondi also vanishes. First, the terms proportionalto ˙ σ ˙ σ and σ ˙ σ are se ond-order, and so will be negle ted here. This leavesonly the terms proportional to ψ . However, for the parti ular spheri al har-moni s onsidered here, these must vanish, be ause the only (cid:28)rst-order termsrepresent produ ts of M (pure j = 0 ) with ˙ λ (only j ≥ ) and ω ´ ω (only j = 0 , j = 1 ).We are left with only the `shift' (cid:29)ux term F shift . The π A ′ - omponent ofthe total emitted angular momentum will be − (2i) − ∆A ( Z ) = − (2i) − Z u u F shift (cid:12)(cid:12)(cid:12) Z = ´ Z d u = − P AA ′ π A π A ′ π A ′ t AA ′ ∂∂π A ∆λ . (38)Comparing this with the dis ussions of the enter of mass and spin in thepre eding subse tions, we see that if a system departs only to (cid:28)rst orderfrom stationarity near I + , then the energy(cid:21)momentum and the j = 1 partsof the angular momentum are un hanged, but the j ≥ parts of the angularmomentum hange with the shear, the ele tri and magneti parts of theshear ontributing to the hange in enter of mass and spin, respe tively.This implies that if in addition the system is stationary initially and (cid:28)nally,then the hange in angular momentum is a pure j ≥ hange in the enterof mass, with no hange in the spin, be ause the magneti omponent of theshear vanishes in any stationary regime.5 Linearized theory and spe ial relativityIn this se tion, I dis uss some elements of the linearized (that is, post-Minkowskian) theory. Of ourse, this linearized theory would be adequatefor many physi al situations. But there is another reason for treating it, animportant on eptual one going to the liaison between spe ial and generalrelativity, and that is the main on ern here.In fa t, we seem to be onfronted with a paradox. We have been for edby general relativity to introdu e a on ept of angular momentum with anentirely new hara ter, a non-Poin aré dependen e on the omponent beingmeasured. On one hand, one an understand that general relativity, whose0asymptoti symmetry group is not the Poin aré group, might lead to su ha stru ture. But (on the other) it seems that this new, non-Poin aré, an-gular momentum is something whi h never ould be ex hanged with non-gravitational systems, sin e their angular momenta are Poin aré- ovariant! Ifthis really were the ase, the de(cid:28)nition put forward here of general-relativisti angular momentum would be very questionable.However, this paradox is only apparent. While it is true that stri tlyspe ial-relativisti angular momenta do not have the non-Poin aré hara terdis overed here, we shall see that as soon as even linearized generi gravita-tional e(cid:27)e ts are onsidered, onsisten y requires the use of the non-Poin aréangular momenta. Thus as soon as one admits any sorts of general-relativisti orre tions one is for ed to non-Poin aré angular momenta. We shall see thatthe onsistent appli ation of these ideas, far from ausing paradoxes, resolvesa onundrum that had existed.The situation might be ompared to the transition from lassi al to quan-tum me hani s. In lassi al me hani s, the angular momentum is a -numberwhi h has a de(cid:28)nite value at any time. However, as soon as one treats anangular momentum quantum-me hani ally, one is for ed to an entirely newsort of obje t, an operator. And as soon as one angular momentum is treatedquantum-me hani ally, one must in general (unless spe ial simpli(cid:28) ationsapply) treat all angular momenta whi h ouple to it quantum-me hani ally.5.1 Two sorts of linearizationThere are in fa t two di(cid:27)erent sorts of linearization one might speak of, andit will be important to distinguish them. In ea h ase, the gravitational (cid:28)eldis treated as a perturbation of Minkowski spa e. However, in one ase thetwistors are simply taken to be those of Minkowski spa e, whereas in the otherthe twistors are also perturbed to (cid:28)rst order. Roughly speaking, the (cid:28)rst ase orresponds to treating gravity as a heli ity-two (cid:28)eld on Minkowski spa e,whereas the se ond orresponds to the linearization of the full, nonlineartheory of angular momentum. It is the se ond ase whi h is relevant here.5.2 First-order orre tions to spe ial relativityLet us onsider a system with no in oming gravitational radiation. If T ab represents the linearized stress(cid:21)energy tensor, then the retarded solution tothe linearized Einstein equation has ψ = − / Gm b m d ∂ ∂u Z δ ( l a ( ut a − y a )) T bd ( y ) d y . (39)Sin e ψ = − ¨ σ , we expe t, then, σ = 2 / Gm b m d Z δ ( l a ( ut a − y a )) T bd ( y ) d y . (40)1(This is not quite obvious, be ause of the integrations involved, but it an beveri(cid:28)ed by dire t al ulations.) Thus the presen e of matter generates shearat I + , and in general this shear evolves.As an illustration, onsider perhaps the simplest ase, the s attering ofparti les due to onta t for es. Then (40) gives the ontribution of a parti lewith mass µ to σ as / Gµ ( m a ˙ γ a ) / ( l b ˙ γ b ) , where ˙ γ a is the four-velo ity ofthe parti le. Thus the shear is determined by the velo ities of the parti lesand their masses. If we onsider the ase of parti les whi h s atter o(cid:27) ea hother by onta t, the shears before and after are in general di(cid:27)erent. Thuseven for very simple intera tions, the (cid:28)rst-order general-relativisti orre tionwill result in a net hange in shear in a s attering pro ess. This will result innon-Poin aré ontributions to the angular momentum.5.3 Resolving a di(cid:30) ultyThe treatment of angular momentum given here resolves a onundrum havingto do with the extension of angular momentum to linearized gravity.Suppose we try to extend the spe ial-relativisti treatment of momentumand angular momentum to take into a ount (cid:28)rst-order gravitational e(cid:27)e ts.We must then deal with the fa t that in general (ex ept for very simple sys-tems with no intera tions), the (cid:28)rst-order orre tions to a spe ial-relativisti system will result in the emission of gravitational waves, and also hanges inthe shear. This hange in the shear breaks the Poin aré symmetry at I + andintrodu es a (cid:28)rst-order ambiguity in the sele tion of a Poin aré subgroup ofthe BMS group and hen e in the de(cid:28)nition of angular momentum.It is at this point that attempts to in orporate (cid:28)rst-order general-relativist-i orre tions to theories of angular momentum run into di(cid:30) ulties. The am-biguity in the hoi e of Poin aré subgroup leads to all the familiar di(cid:30) ultieswith de(cid:28)ning angular momentum in the full theory (although at a linearizedlevel). In other words, taking into a ount general relativity to (cid:28)rst order, theangular momenta of several billiard balls before and after a ollision are notrelated by a Poin aré motion, be ause the gravitational waves emitted in the ollision distort the geometry of spa e(cid:21)time after the ollision relative to thatbefore it. (Of ourse, the error in negle ting this is very small, amounting toa positional ambiguity ∼ GM ∆v /c ∼ − m!)The present treatment resolves this di(cid:30) ulty, for the angular momentumis de(cid:28)ned twistorially and the shift in shear is a ommodated. We see that a onventional me hani al system would be expe ted to emit small amounts ofgravitational radiation, and that radiation would give a small non-Poin aré orre tion to the angular momentum. It is only when we ompletely negle tgeneral-relativisti e(cid:27)e ts, and model the system spe ial-relativisti ally, thatthe usual Poin aré stru ture is really re overed.6 Number of independent onstants of motionIn spe ial relativity, the angular momentum omprises six independent on-stants of motion (assuming the energy(cid:21)momentum is given). The de(cid:28)nition2given here of angular momentum in general relativity seems to omprisein(cid:28)nitely many onstants of motion, be ause it onsists of apparently inde-pendent ontributions for ea h j ≥ . Have we in fa t found that somehowgeneral relativity, whi h one would expe t to have fewer symmetries thanspe ial relativity, really has in(cid:28)nitely many more onstants of motion?Surprisingly, there is an important sense in whi h the answer to thisquestion is Yes. However, the situation requires some dis ussion.First, we are not onsidering absolute onstants of motion (those, in fa t,would be less interesting to us), be ause we wish to treat a system fromwhi h material and gravitational radiation may be es aping. One should infa t think of a hoi e of ut as partitioning the system into two portions,an interior one (determined by data on a partial Cau hy surfa e meetingthe ut), and the portion of I + to the past of the ut. We are also espe iallyinterested in those onstants of motion whi h an be omputed from (cid:28)elds onthe ut, for physi ally of ourse those represent data a essible asymptoti allyat a parti ular retarded time.Se ond, for the systems we onsider, there are in(cid:28)nitely many onstantsof motion (for general relativity is a (cid:28)eld theory). Of all these onstants ofmotion, the most natural and interesting for the radiative modes are thoseforming the shear at a ut. (So there is a sphere's worth of onstants ofmotion, being the values of the shear at the di(cid:27)erent generators.) These arelo ally measurable at the ut, and they are physi ally interesting also as thetime-integrals of the wave pro(cid:28)les.If we ount the number of onstants of motion, we have six for the j = 1 ,` onventional', angular momenta, and j + 1) for the j ≥ degrees offreedom in the shear. What is new is really not the degrees of freedom in theshear, but our interpretation of these as representing angular momenta.That said, how is it that we have a quired in(cid:28)nitely many more onstantsof motion than there were in spe ial relativity? The answer is that we havetaken into a ount the gravitational degrees of freedom, and the ouplingbetween gravity in matter, and these are negle ted in spe ial relativity, and sowe now have available to us in(cid:28)nitely many degrees of freedom, and onstantsof motion, whi h were not ontemplated in spe ial relativity.Suppose, for example, we onsidered some system in spe ial relativity, and omputed the integral (40) that we know, from linearized general relativity,would lead to the shear. In spe ial relativity we would not all this a onstantof motion, and we would have no parti ular reason to be interested in it.However, in linearized gravity it gives us the shear, and we interpret its hange over time as giving us information about the gravitational radiation.It is the fa t that we keep tra k of this gravitational radiation, with itsfun tional degrees of freedom, whi h makes the di(cid:27)eren e: the hange in shearis a ounted for as a transfer of a quantity of these onstants of motion fromthe internal portion of the spa e(cid:21)time to the emitted gravitational radiation.In short, the ount of onstants of motion omes out as expe ted; it israther the uni(cid:28) ation of the on epts of shear and angular momentum whi his new.37 Con lusionsThe main on lusion of this paper is that it is possible to treat the angu-lar momentum of an isolated gravitational system by introdu ing a suitabletwistor spa e. This spa e is naturally de(cid:28)ned in terms of the universal stru -ture of null in(cid:28)nity I + . It la ks the full linear stru ture spe ial-relativisti twistor spa e has, but it does possess a (cid:28)ber-bundle stru ture. The base spa ein question is the spa e of spinors at in(cid:28)nity, or, essentially equivalently, thespa e of generators of I + .The nonlinearities as we pass from one (cid:28)bre to another in twistor spa ebe ome, in spa e(cid:21)time terms, angular dependen es for the angular momen-tum whi h are more omplex than the essentially ve torial ( j = 1 ) behaviorin spe ial relativity. The two parts of the relativisti angular momentum,the spin and the enter of mass, both have angular dependen es in ludingterms for all j ≥ . The terms for j ≥ are pre isely due to the shear, theele tri part of the shear determining the j ≥ parts of the enter of massand the magneti part of the shear determining the j ≥ parts of the spin.Additionally, the enter of mass has a dire t physi al interpretation as a null ongruen e inwards from I + , whi h one thinks of as dire ted towards the enter of the system.Angular momentum, unlike energy(cid:21)momentum, an be emitted at (cid:28)rstorder by gravitational waves, and this applies even to the ` onventional' ( j =1 ) parts. However, su h (cid:28)rst-order emission does require a orrelation betweenthe anisotropy of the waves and of the mass-aspe t of the system, and this isnot expe ted for systems departing only to (cid:28)rst order from stationarity. Fornearly stationary systems, angular momentum an be emitted by (cid:28)rst-ordergravitational radiation, but it has a purely j ≥ hara ter.The appearan e of higher- jj