Total angular momentum from Dirac eigenspinors
aa r X i v : . [ g r- q c ] D ec Total angular momentum from Dira eigenspinorsLászló B SzabadosResear h Institute for Parti le and Nu lear Physi sH-1525 Budapest 114, P. O. Box 49, Hungarye-mail: lbszabrmki.kfki.huO tober 30, 2018Abstra tThe eigenvalue problem for Dira operators, onstru ted from two on-ne tions on the spinor bundle over losed spa elike 2-surfa es, is investi-gated. A lass of divergen e free ve tor (cid:28)elds, built from the eigenspinors,are found, whi h, for the lowest eigenvalue, reprodu e the rotation Killingve tors of metri spheres, and provide rotation BMS ve tor (cid:28)elds at fu-ture null in(cid:28)nity. This makes it possible to introdu e a well de(cid:28)ned, gaugeinvariant spatial angular momentum at null in(cid:28)nity, whi h redu es to thestandard expression in stationary spa etimes. The general formula for theangular momentum (cid:29)ux arried away be the gravitational radiation is alsoderived.1 Introdu tionAngular momentum is one of the basi onserved quantities in physi s, and ingeneral relativity there is, indeed, a well de(cid:28)ned notion of total (ADM) angularmomentum of an isolated system `measured' at spatial in(cid:28)nity. On the otherhand, if we are interested in the angular momentum arried away by the gravi-tational radiation, then we should be able to de(cid:28)ne total angular momentum atfuture null in(cid:28)nity, too. Unfortunately, however, there is no generally a eptednotion of angular momentum at null in(cid:28)nity of a radiative spa etime. (For areview of lassi al results see [1℄, and for the re ent ones see e.g. [2, 3, 4, 5, 6, 7℄.)Another di(cid:30) ulty is that even if we have an ambiguity-free notion of angularmomentum, it is not guaranteed that the angular momenta measured at di(cid:27)er-ent retarded times (i.e. when they are asso iated with di(cid:27)erent uts of futurenull in(cid:28)nity) an be ompared, and hen e there is no unambiguous way of om-puting the angular momentum radiated away in a given time interval by thelo alized sour e (see e.g. [1, 6℄).The situation at the quasi-lo al level (i.e. when we are onsidering onlysubsystems of the whole universe, or, mathemati ally, we intend to asso iate1hysi al quantities with losed spa elike 2-surfa es S ) is even worse, be ause wedo not have even the asymptoti (BMS) symmetries. The best that we an do isto try to systemati ally `quasi-lo alize' the anoni al analysis of GR. Althoughthis program is not yet ompleted, the (cid:28)rst few steps towards the quasi-lo al anoni al GR have been made [8℄, yielding a lass of well de(cid:28)ned, 2+2 ovariant,gauge invariant observables O [ N a ] . These are based on divergen e free ve tor(cid:28)elds N a that are tangent to the 2-surfa e S . Unfortunately, however, withoutadditional restri tion on the ve tor (cid:28)elds these observables re(cid:29)e t properties ofthe 2-surfa e as a submanifold in the spa etime, rather than the properties ofthe gravitational `(cid:28)eld' itself: O [ N a ] may be non-zero for 2-surfa es even inMinkowski spa etime. Thus to obtain physi ally interesting observables in theform of O [ N a ] , new ideas are needed how to hoose the ve tor (cid:28)elds N a .On the other hand, and remarkably enough, the requirement of the (cid:28)nitenessof O [ N a ] at spatial in(cid:28)nity of asymptoti ally (cid:29)at spa etimes already restri tsthe asymptoti stru ture of N a su h that the orresponding observable willbe just the familiar spatial (ADM) angular momentum. Similarly, at futurenull in(cid:28)nity of a stationary asymptoti ally (cid:29)at spa etime O [ N a ] reprodu es thestandard angular momentum expression. Nevertheless, in radiative spa etimesat future null in(cid:28)nity O [ N a ] is still ambiguous. Thus to obtain a well-de(cid:28)nednotion of angular momentum at future null in(cid:28)nity a more detailed pres riptionof the divergen e-free ve tor (cid:28)elds should be given.A potentially viable onstru tion might be based on the re ent idea of ap-proximate Killing ve tors on topologi al 2-spheres [9℄. These are the divergen e-free ve tor (cid:28)elds that solve a variational problem, whose a tion is built form thenorm of the Killing operator. Sin e these ve tor (cid:28)elds are divergen e free by onstru tion, they an be used in O [ N a ] to get well-de(cid:28)ned, gauge invariantobservables.Another promising approa h of onstru ting su h observables ould be basedon the spe tral analysis of the Lapla e, or of the Dira operators on losed spa e-like 2-surfa es. Clearly, the eigenvalues of these operators are gauge invariant,and they should re(cid:29)e t, maybe in a rather impli it way, the geometri al proper-ties of the surfa e. (For example, the mass and angular momentum parameters an be re overed from the eigenvalues of the Lapla ian on the event horizon ofa Kerr(cid:21)Newman bla k hole [10℄.) Indeed, in Riemannian geometry mathemati- ians already proved the existen e of sharp lower bounds to the (cid:28)rst eigenvalueof the Dira operator in terms of the s alar urvature [11, 12, 13, 14℄ or thevolume [15, 14, 16℄. Similar results exist for hypersurfa e Dira operators whenthe lower bounds are given in terms of the urvature of the intrinsi geometryand the extrinsi urvature [17℄.In the present paper we also investigate the eigenvalue problem of the Dira operators. However, instead of the eigenvalues, we on entrate on the (almostalways) overlooked eigenspinors. A further di(cid:27)eren e between the former andthe present investigations is that the base manifold on whi h the Dira oper-ators are de(cid:28)ned is not simply a two-dimensional Riemannian manifold, but aspa elike 2-surfa e in a general Lorentzian spa etime. Moreover, the two Dira operators that we study here are built from both the intrinsi and extrinsi ge-2metri al obje ts of the surfa e. This yields a number of on eptual di(cid:30) ulties(e.g. no natural onstant Hermitian metri exists on the spinor bundle), andhen e we an de(cid:28)ne the eigenvalue problem only for the spa etime (rather thanthe 2-surfa e) Dira spinors. Moreover, in la k of any natural Hermitian metri the reality of the eigenvalues is not guaranteed. Nevertheless, we show how theNester(cid:21)Witten integral an be used to give riteria for the reality of the eigen-values. We onstru t three divergen e-free ve tor (cid:28)elds from the eigenspinors.In parti ular, we show that on round spheres with radius r two of these ve tor(cid:28)elds, built from the eigenspinors with the lowest eigenvalue of one of the twoDira operators, are proportional to the /r times, and the third to the /r times the rotation Killing ve tors of the metri sphere.Based on this observation we show that on the large spheres u = const , r = const in a Bondi type oordinate system near the future null in(cid:28)nity inan asymptoti ally (cid:29)at spa etime the divergen e-free ve tor (cid:28)elds analogous tothe (cid:28)rst two above determine the rotation BMS ve tor (cid:28)elds tangent to the ut u = const of I + . As we already mentioned, we expe t that the gravitationalenergy-momentum and angular momentum be onne ted with the Hamiltonianformulation of Einstein's theory, i.e. these observables are expe ted to be thevalue of the orre t Hamiltonian of the theory with appropriately hosen gen-erators. Thus it seems natural to de(cid:28)ne the angular momentum at the futurenull in(cid:28)nity by the observable O [ N a ] in whi h the ve tor (cid:28)elds N a are hosen tobe the divergen e-free ve tor (cid:28)elds built from the eigenspinors with the lowesteigenvalue of the Dira operator. This strategy gives two observables: the (cid:28)rst an be interpreted as spatial angular momentum. This is gauge invariant, freeof ambiguities, and trivially redu es to the standard expressions in the station-ary and in the axi-symmetri spa etimes. Moreover, there is a natural way of omparing the angular momentum measured at di(cid:27)erent retarded times, andhen e we an ompute the angular momentum (cid:29)ux arried away by the grav-itational radiation. The other observable is a non-negative expression of themagneti part of the asymptoti shear, and it is vanishing pre isely when theshear is purely ele tri . Sin e the shear is known to be purely ele tri in station-ary spa etimes, this is a measure of dynami s of the gravitational `(cid:28)eld' nearthe future null in(cid:28)nity. Its signi(cid:28) an e is, however, not yet lear. Remarkablyenough, though the two BMS ve tor (cid:28)elds are di(cid:27)erent, they de(cid:28)ne the samepair of observables.The organization of the paper follows the logi of the results above: in these ond se tion we re all the basi notions, dis uss the general aspe ts of theeigenvalue problem for the Dira operator that is based on a Sen-type deriva-tive operator, onstru t the divergen e-free ve tor (cid:28)elds and dis uss the realityproperties of the eigenvalues. Se tion 3 is devoted to the dis ussion of the spe- ial properties of another Dira operator, whi h is a redu tion of the previousone and is built only from the intrinsi geometry and the onne tion 1-form ofthe normal bundle of the 2-surfa e. The related non-existen e result for the onstant positive de(cid:28)nite Hermitian s alar produ t in given in the appendix.Then, in se tion 4, we apply these ideas to round spheres, where we al ulatethe spe trum of the Dira operators and onstru t the divergen e-free ve tor3elds expli itly. Se tion 5 is devoted to the analogous al ulations on largespheres near the future null in(cid:28)nity in asymptoti ally (cid:29)at spa etimes. We also al ulate the orresponding angular momentum and angular momentum (cid:29)ux aswell.The signature of g ab is (+ , − , − , − ) , the urvature and Ri i tensors and the urvature s alar are de(cid:28)ned by R abcd X b := − ( ∇ c ∇ d − ∇ d ∇ c ) X a , R bd := R abad and R := R ab g ab , respe tively. Then Einstein's equations take the form G ab = − πGT ab , where G is Newton's gravitational onstant.2 The ∆ AA ′ -Dira operator2.1 GeneralitiesLet S A ( S ) denote the bundle of 2- omponent (i.e. Weyl) spinors over the losed,orientable spa elike 2-surfa e S in the spa e and time orientable spa etime,and we denote the omplex onjugate bundle by ¯ S A ′ ( S ) . From the spa etimestru ture two metri s are inherited: the symple ti ε AB and the symmetri γ AB .The former is just the spinor form of the spa etime metri , g ab = ε AB ε A ′ B ′ , whilethe latter is built from the timelike and spa elike unit normals of S , t AA ′ and v AA ′ , respe tively, as γ AB := 2 t AA ′ v BA ′ . These normals de(cid:28)ne the proje tion Π ab := δ ab − t a t b + v a v b to the 2-surfa e, by means of whi h the restri tion to S of the spa etime tangent bundle de omposes in a unique way to the g ab -orthogonal dire t sum of the tangent bundle T S and the normal bundle N S .Though the a tual normals t a and v a are not uniquely determined, both theproje tion Π ab and the spinor γ AB are well de(cid:28)ned. This γ AB de(cid:28)nes a hiralityon the spinor bundle S A ( S ) . Thus the spinor bundle is endowed in a naturalway by the symple ti metri and the hirality, and in this ase the elements of ( S A ( S ) , ε AB , γ AB ) are alled 2-surfa e spinors.On the spinor bundle, two onne tions an be introdu ed in a natural way,and the orresponding derivative operators will be denoted by δ e and ∆ e , re-spe tively. The (cid:28)rst is built from the intrinsi 2-metri and the onne tion1-form A e := Π fe ( ∇ f t a ) v a on the normal bundle of S , while the other on-tains the spinor form Q AeB of the extrinsi urvature tensor Q aeb of S too,where Q AeB = Π fe ( ∇ f γ AE ) γ EB . This Q AEE ′ B has the reality property Q AAE ′ B = ¯ Q A ′ A ′ BE ′ , expressing the hypersurfa e orthogonality of the twonull normals of S (or, in other words, the reality of the two individual on-vergen es ρ and ρ ′ orresponding to the outgoing and in oming null normals,respe tively). The a tion of the ovariant derivative ∆ e is given expli itly by ∆ e λ A = δ e λ A + Q AeB λ B . This is nothing but the proje tion to S of the spa e-time Levi-Civita ovariant derivative: ∆ a := Π ba ∇ b . Both ∆ e and δ e annihilate ε AB , but γ AB is annihilated only by δ e . The urvatures f ABcd and F ABcd orresponding to δ e and ∆ e , respe tively, are given by4 ABcd = − i4 f γ AB ε cd , (2.1) F ABcd = f ABcd − (cid:0) δ c Q AdB − δ d Q AcB + Q AcE Q EdB − Q AdE Q EcB (cid:1) ; (2.2)where f := f abcd ( ε ab − i ⊥ ε ab ) ε cd = R − ε ab δ a A b , the (cid:16)s alar urvature(cid:17) ofthe urvature f abcd of the derivative δ e on the Lorentzian ve tor bundle over S .Here R is the s alar urvature of the intrinsi geometry of S , and ε ab and ⊥ ε ab are the volume 2-forms on the tangent and normal 2-spa es of S , respe tively.(For the details see [18, 5℄.)The ∆ AA ′ -Dira operator on S is de(cid:28)ned in the de omposition of ∆ A ′ A λ B inits unprimed indi es to its anti-symmetri and γ AB -tra e-free symmetri parts:The former, ∆ A ′ A λ A , gives the Dira operator, while the latter the 2-surfa etwistor operator of Penrose. (The γ AB tra e gives essentially the Dira operatortoo: γ AB ∆ A ′ A λ B = ¯ γ A ′ B ′ ∆ B ′ B λ B .) The square of the ∆ AA ′ -Dira operator is ∆ AA ′ ∆ A ′ B λ B = −
12 ∆ e ∆ e λ A − ε A ′ B ′ ε BC Q e [ ab ] ∆ e λ C −− ε A ′ B ′ ε BC F CDAA ′ BB ′ λ D . (2.3)Taking into a ount (2.1)-(2.2) and that the anti-symmetri part of the extrinsi urvature tensor is Q e [ ab ] = − ( ε A ′ B ′ Q AEE ′ B + ε AB ¯ Q A ′ EE ′ B ′ ) , we (cid:28)nd that − AA ′ ∆ A ′ B λ B = ∆ e ∆ e λ A + 14 f λ A − Q AeB ∆ e λ B −− ε A ′ B ′ (cid:16) δ AA ′ Q BBB ′ C − δ BB ′ Q BAA ′ C ++ Q BAA ′ E Q EBB ′ C − Q BBB ′ E Q EAA ′ C (cid:17) λ C . (2.4)This equation is analogous to the Li hnerowi z identity [19℄: the square of theDira operator is expressed in terms of the Lapla ian and the urvature, buthere ∆ e ∆ e is not the intrinsi Lapla ian and the (cid:28)rst derivative of the spinor(cid:28)eld also appears on the right.An analogous analysis an be arried out with the δ AA ′ -Dira operator too,but all the results an be re overed from those for the ∆ AA ′ -Dira operator bythe formal substitution Q AeB = 0 too. In parti ular, the identity (2.4) redu esto − δ AA ′ δ A ′ B λ B = δ e δ e λ A + 14 f λ A . (2.5)Note that, stri tly speaking, this is still not the Li hnerowi z identity, be ause δ AA ′ is not only an intrinsi derivative operator: it ontains extrinsi quantitiesin the form of A e as well, while the Dira operator in the genuine Li hnerowi zidentity is built ex lusively from the intrinsi geometry of S .5.2 The eigenvalue problem for the 2-surfa e Dira oper-atorsIf we had a naturally de(cid:28)ned Hermitian metri G AA ′ on S A ( S ) by means ofwhi h the bundles S A ( S ) and ¯ S A ′ ( S ) ould be identi(cid:28)ed (i.e. the primed indi es ould be onverted to unprimed ones), then the eigenvalue equation for the δ AA ′ -Dira operator ould be de(cid:28)ned as i G AA ′ δ A ′ B λ B = − α √ λ A . (2.6)(The hoi e for the apparently ad ho oe(cid:30) ient − / √ in front of the eigen-value α yields the ompatibility both with the subsequent more general analysisand the known standard results in spe ial ases.) However, it is desirable that,in addition, su h a Hermitian metri be ompatible with the onne tion in thesense that δ e G AA ′ = 0 . Nevertheless, in the appendix we show that the exis-ten e of su h a Hermitian metri is equivalent to the vanishing of the holonomyof δ e on the normal bundle (and, in parti ular, its urvature, Im f , must bezero). Therefore, on a general 2-surfa e in a general, urved spa etime we donot have any su h natural Hermitian stru ture on S A ( S ) . Thus to motivate howthe eigenvalue problem should be de(cid:28)ned for the δ AA ′ (or, more generally, forthe ∆ AA ′ )-Dira operator, let us onsider the eigenvalue problem for the spa e-time Dira (rather than the Weyl) spinors, where the primed and unprimedindi es are treated on equal footing.Re all that a Dira spinor Ψ α is a pair of Weyl spinors λ A and ¯ µ A ′ , writtenthem as a olumn ve tor Ψ α = (cid:18) λ A ¯ µ A ′ (cid:19) (2.7)and adopting the onvention α = A ⊕ A ′ , β = B ⊕ B ′ et . Its derivative ∆ e Ψ α is the olumn ve tor onsisting of ∆ e λ A and ∆ e ¯ µ A ′ . If Dira 's γ -`matri es' aredenoted by γ αeβ , then one an onsider the eigenvalue problem i γ αeβ ∆ e Ψ β = α Ψ α . (2.8)Expli itly, with the representation γ αeβ = √ (cid:18) ε E ′ B ′ δ AE ε EB δ A ′ E ′ (cid:19) (2.9)(see e.g. [20℄, pp 221), this is just the pair of equations i∆ A ′ A λ A = − α √ µ A ′ , i∆ AA ′ ¯ µ A ′ = − α √ λ A . (2.10)By (2.10) Ψ α = ( λ A , ¯ µ A ′ ) (as a olumn ve tor) is a Dira eigenspinor witheigenvalue α pre isely when ( λ A , − ¯ µ A ′ ) is a Dira eigenspinor with eigenvalue − α . In the language of Dira spinors this is formulated in terms of the hirality,represented by the so- alled ` γ -matrix', denoted here by6 αβ := 14! ε abcd γ αaµ γ µbν γ νcρ γ ρdβ = i (cid:18) δ AB − δ A ′ B ′ (cid:19) (2.11)(see appendix II. of [6℄). Sin e this is anti- ommuting with γ αeβ , from (2.8) weobtain that i γ αeµ ∆ e ( η µβ Ψ β ) = − α ( η αβ Ψ β ) . Thus if Ψ α is a Dira eigenspinorwith eigenvalue α , then, in fa t, η αβ Ψ β is a Dira eigenspinor with eigenvalue − α .On the other hand, the Dira eigenspinors with de(cid:28)nite hirality belong tothe kernel of the Dira operator. Indeed, Dira spinors with de(cid:28)nite hiral-ity have the stru ture either ( λ A , or (0 , ¯ µ A ′ ) , whi h, by (2.10), yield that ∆ A ′ A λ A = 0 or ∆ AA ′ ¯ µ A ′ = 0 , respe tively. Therefore, this notion of hirality annot be used to de ompose the spa e of the eigenspinors with given eigen-value. Its role is simply to take a Dira eigenspinor with eigenvalue α to a Dira eigenspinor with eigenvalue − α .The equations of (2.10) imply that − ∆ AA ′ ∆ A ′ B λ B = 12 α λ A , − ∆ A ′ A ∆ AB ′ ¯ µ B ′ = 12 α ¯ µ A ′ . (2.12)Thus if Ψ α = ( λ A , ¯ µ A ′ ) is a Dira eigenspinor with eigenvalue α , then itsWeyl spinor parts λ A and ¯ µ A ′ are eigenspinors of the se ond order operator − AA ′ ∆ A ′ B and − A ′ A ∆ AB ′ , respe tively, with the same eigenvalue α .Conversely, if λ A is a Weyl eigenspinor of − AA ′ ∆ A ′ B with non-zero eigen-value α , then Ψ α ± , built from λ A and ¯ µ A ′ := ∓ ( √ /α )i∆ A ′ A λ A , are Dira eigenspinors with eigenvalue ± α , respe tively, for whi h η αβ Ψ β ± = iΨ α ∓ . There-fore, there is a natural isomorphism between the spa e W α of the Weyl eigen-spinors of − AA ′ ∆ A ′ B with eigenvalue α and the dire t sum D α ⊕ D − α ,where D α is the spa e of the Dira eigenspinors of ∆ e with eigenvalue α . The hirality operator η αβ maps D ± α to D ∓ α . For zero eigenvalue, α = 0 , the Weyleigenspinors de(cid:28)ne the kernel of the ∆ AA ′ -Dira operator, whi h, apart fromex eptional 2-surfa es, is empty (see e.g. [18, 2℄). Thus for generi 2-surfa es,the eigenvalues of the ∆ AA ′ -Dira operator are nonzero.Finally, let us translate the general equations into the language of the GHPformalism [21, 20℄. Thus let us (cid:28)x a normalized spinor dyad { o A , ι A } adaptedto the 2-surfa e S (i.e. the omplex null ve tors m a := o A ¯ ι A ′ and ¯ m a := ι A ¯ o A ′ are tangent to S ). Then the GHP form of the ∆ AA ′ -Dira operator and thesquare of the ∆ AA ′ -Dira operator are ¯ o A ′ ∆ A ′ A λ A = ð ′ λ + ρλ , (2.13) − ¯ ι A ′ ∆ A ′ A λ A = ð λ + ρ ′ λ , (2.14) o A ∆ AA ′ ∆ A ′ B λ B = ðð ′ λ + (cid:0) ð ρ (cid:1) λ − ρρ ′ λ , (2.15) ι A ∆ AA ′ ∆ A ′ B λ B = ð ′ ð λ + (cid:0) ð ′ ρ ′ (cid:1) λ − ρρ ′ λ . (2.16)7ere we de(cid:28)ned the spinor omponents by the onventions λ := λ A o A and λ := λ A ι A ; ð and ð ′ are the standard edth operators and ρ and ρ ′ are thestandard GHP onvergen es orresponding to the outgoing and in oming nullnormals o A ¯ o A ′ and ι A ¯ ι A ′ of S , respe tively. The dimension of the kernel of theedth operators depends on the genus of the 2-surfa e S [22℄: let p be any realnumber. Then on topologi al 2-spheres dim ker ð ( p,p + n ) = dim ker ð ′ ( p + n,p ) = 0 for any n ∈ N , while dim ker ð ( p + n,p ) = dim ker ð ′ ( p,p + n ) = 1 + n for any n =0 , , , ... . On tori dim ker ð ( p,p + n ) = dim ker ð ′ ( p,p + n ) = 1 for any integer n . Onsurfa es with genus g ≥ one has dim ker ð ( p,p + n ) = dim ker ð ′ ( p + n,p ) = 0 if − n ∈ N , it is 1 if n = 0 , it is g − if n = 1 , it is g if n = 2 and it is ( n − g − if n > .2.3 Spe ial ve tor (cid:28)elds built from the eigenspinorsThe Weyl eigenspinors de(cid:28)ne several ∆ e -divergen e-free omplex Lorentzianve tor (cid:28)elds on S . In fa t, let λ A and ¯ µ A ′ be the unprimed and primed Weylspinor parts of a Dira eigenspinor Ψ α , respe tively. Then ontra ting the (cid:28)rstequation of (2.10) with ¯ µ A ′ and the se ond equation of (2.10) with λ A , andadding them together we obtain ∆ AA ′ ( λ A ¯ µ A ′ ) = 0; (2.17)i.e. K a := λ A ¯ µ A ′ is a ∆ e -divergen e-free omplex ve tor (cid:28)eld on S . Similarly,the ontra tion of the (cid:28)rst equation of (2.10) with ¯ λ A ′ gives ¯ λ A ′ ∆ A ′ A λ A = − i √ α ¯Φ , and the ontra tion of the se ond equation of (2.10) with µ A gives µ A ∆ AA ′ ¯ µ A ′ = i √ α Φ , where Φ := λ A µ A . These imply that ∆ AA ′ (cid:0) λ A ¯ λ A ′ (cid:1) = i √ (cid:0) ¯ α Φ − α ¯Φ (cid:1) , ∆ AA ′ (cid:0) µ A ¯ µ A ′ (cid:1) = i √ (cid:0) α Φ − ¯ α ¯Φ (cid:1) ; and hen e, for any a, b ∈ C , that ∆ AA ′ (cid:16) aλ A ¯ λ A ′ + bµ A ¯ µ A ′ (cid:17) = i √ (cid:16)(cid:0) a ¯ α + bα (cid:1) Φ − (cid:0) aα + b ¯ α (cid:1) ¯Φ (cid:17) . (2.18)Therefore, the real ve tor (cid:28)eld Z a ± := λ A ¯ λ A ′ ± µ A ¯ µ A ′ is ∆ e -divergen e free forpurely imaginary/real eigenvalue α .These ve tor (cid:28)elds an be re overed as spe ial ases of V e := Φ α γ αeβ Ψ β , builtfrom the Dira eigenspinors Φ α and Ψ α . In fa t, if i γ αeβ ∆ e Ψ β = α Ψ α , i γ αeβ ∆ e Φ β = β Φ α , then i∆ e (Φ α γ αeβ Ψ β ) = ( α − β )Φ α Ψ α . Therefore, if Φ α = ( φ A , ¯ ω A ′ ) and Ψ α =( λ A , ¯ µ A ′ ) are eigenspinors with the same eigenvalue, say α , then the ve tor (cid:28)eld V e := Φ α γ αeβ Ψ β = −√ λ E ¯ ω E ′ + φ E ¯ µ E ′ ) is ∆ e -divergen e free. In parti ular, (1)8f Φ α = Ψ α , then V e = − √ λ E ¯ µ E ′ = − √ K e ; (2) if Φ + α = Ψ α (i.e. ω A = λ A and φ A = µ A ), and hen e Ψ α is an eigenspinor with the eigenvalue − ¯ α too and α is imaginary, then V e = −√ λ E ¯ λ E ′ + µ E ¯ µ E ′ ) = −√ Z e + ; (3) if Φ + α η αβ = Ψ β (i.e. i ω A = λ A and i φ A = µ A ), and hen e Ψ α is an eigenspinor with theeigenvalue ¯ α too and α is real, then V e = − i √ λ E ¯ λ E ′ − µ E ¯ µ E ′ ) = − i √ Z e − .Another lass of spe ial ve tor (cid:28)elds an be onstru ted purely from the un-primed Weyl spinor parts of one or two Dira eigenspinors and their derivatives.Next we onsider these. Contra ting the identity (2.4) with an arbitrary spinor(cid:28)eld φ A , using the relationship between the two derivative operators ∆ EE ′ and δ EE ′ and the reality property Q AAE ′ B = ¯ Q A ′ A ′ BE ′ , a straightforward al ula-tion gives − φ A ∆ AA ′ ∆ A ′ B λ B = φ A ∆ e ∆ e λ A + 14 f φ A λ A − Q AeB φ A ∆ e λ B −− (cid:0) ∆ e Q AeB (cid:1) φ A λ B + 2 (cid:0) ∆ AA ′ Q DDA ′ B (cid:1) λ A φ B + 12 λ A φ A Q BeD Q BeD . Inter hanging the spinor (cid:28)elds λ A and φ A , adding the resulting formula to theold one and using φ A ∆ e ∆ e λ A = ∆ e ( φ A ∆ e λ A ) − (∆ e φ A )(∆ e λ A ) , we obtain thegeometri identity δ e (cid:16) φ A δ e λ A + λ A δ e φ A (cid:17) = ∆ e (cid:16) φ A ∆ e λ A + λ A ∆ e φ A − Q AeB λ A φ B (cid:17) == − (cid:16) φ A ∆ AA ′ ∆ A ′ B λ B + λ A ∆ AA ′ ∆ A ′ B φ B (cid:17) −− (cid:0) ∆ AA ′ Q DDA ′ B (cid:1)(cid:16) λ A φ B + φ A λ B (cid:17) . (2.19)If the last term vanishes, e.g. when ∆ ( AA ′ Q B ) A ′ DD = 0 holds, and if there is afun tion α : S → C su h that − ∆ AA ′ ∆ A ′ B λ B = 12 α λ A , − ∆ AA ′ ∆ A ′ B φ B = 12 α φ A , (2.20)then the ve tor (cid:28)eld ξ e := φ A δ e λ A + λ A δ e φ A (2.21)is δ e (and, in fa t, ∆ e )-divergen e free. Therefore, in parti ular when λ A and φ A are Weyl eigenspinors with the same eigenvalue (e.g. if φ A = λ A ), then inthe spe ial ase ∆ ( AA ′ Q B ) A ′ DD = 0 the (in general omplex) tangent ve tor(cid:28)eld ξ e is divergen e free on S (both with respe t to δ e and ∆ e ). Sin e in theGHP formalism ∆ ( AA ′ Q B ) A ′ DD = ( ð ′ ρ ′ ) o A o B − ( ð ρ ) ι A ι B , the vanishing of thisterm is equivalent to ð ρ = 0 , ð ′ ρ = 0 , ð ρ ′ = 0 and ð ′ ρ ′ = 0 ; i.e. the onvergen esare onstant on S . ξ e an be generalized to be Φ α δ e Ψ α + Ψ α δ e Φ α , whi h is δ e -divergen e free if Φ α and Ψ α are Dira eigenspinors with the same eigenvalue and ∆ ( AA ′ Q B ) A ′ DD = 0 . 9.4 The reality of the eigenvaluesThe ∆ e -divergen e of Z a − vanishes if it is built from eigenspinors with real eigen-value. Thus we should (cid:28)nd some riteria of the reality of the eigenvalues. Theusual proof of the reality of the eigenvalues is based on the existen e of a positivede(cid:28)nite Hermitian metri ompatible with the onne tion underlying the Dira operator. However, in the light of the non-existen e result even for the δ e -Dira operator (see the appendix), the ondition of this reality should be sear hedfor following a di(cid:27)erent strategy. First, one might be tempted to de(cid:28)ne theeigenvalue problem (2.10) with the additional requirement that µ A = cλ A forsome omplex onstant c . (In the language of Dira spinors a spinor Ψ α with µ A = λ A is alled a Majorana spinor.) However, by (2.10) this would imply that | c | = 1 and ¯ α = − α , and hen e that all the eigenvalues of − ∆ AA ′ ∆ A ′ B wouldbe non-positive. We will see that this annot be the ase: for round spheresthe eigenvalues of − ∆ AA ′ ∆ A ′ B , maybe apart from (cid:28)nitely many of them, areall positive.Another strategy is to introdu e a global Hermitian s alar produ t dire tlyon the spa e C ∞ ( S , S A ) of all smooth spinor (cid:28)elds on S , without trying to linkthis with any pointwise Hermitian s alar produ t G AA ′ . This is based on theintegral of the Nester(cid:21)Witten 2-form built from the Weyl spinors [23℄. For anypair ( λ A , ω A ) of spinor (cid:28)elds it an be rewritten in the form [24℄ H (cid:2) λ A , ¯ ω A ′ (cid:3) = 14 πG I S ¯ γ A ′ B ′ ¯ ω A ′ ∆ B ′ B λ B d S . (2.22)It is a straightforward al ulation to show that it is Hermitian in the sensethat H [ λ A , ¯ ω A ′ ] = H [ ω A , ¯ λ A ′ ] , where overline denotes omplex onjugation. Inparti ular, H [ λ A , ¯ λ A ′ ] is always real, but in general it may be negative or zeroeven for non-vanishing spinor (cid:28)elds. Then with the substitution ¯ ω A ′ := ∆ A ′ B π B we obtain H (cid:2) ∆ AB ′ ¯ π B ′ , ¯ λ A ′ (cid:3) = H (cid:2) λ A , ∆ A ′ B π B (cid:3) = 14 πG I S ¯ γ A ′ B ′ (cid:0) ∆ A ′ A π A (cid:1)(cid:0) ∆ B ′ B λ B (cid:1) d S = H (cid:2) π A , ∆ A ′ B λ B (cid:3) ; (2.23)i.e. the ∆ AA ′ -Dira operator is ompatible with the Hermitian s alar produ t.This implies that H (cid:2) ∆ AB ′ ∆ B ′ B σ B , ¯ λ A ′ (cid:3) = H (cid:2) σ A , ∆ A ′ B ∆ BB ′ ¯ λ B ′ (cid:3) (2.24)for any pair ( σ A , λ A ) of spinor (cid:28)elds; i.e. ∆ AA ′ ∆ A ′ B is formally self-adjointwith respe t to H .It might be worth noting that originally the Nester(cid:21)Witten integral wasintrodu ed as a Hermitian quadrati form on the spa e of Dira spinor (cid:28)elds[25, 26℄, and, with the representation Ψ α = ( λ A , ¯ µ A ′ ) , this an be written as H [ λ A , ¯ λ A ′ ] − H [ µ A , ¯ µ A ′ ] . A natural extension of (2.22) as a Hermitian bilinear10orm to Dira spinors (and of the original Nester(cid:21)Witten integral too) is theintegral of i √ α γ αeβ ε ef ∆ f Ψ β = (cid:16) ¯ γ A ′ B ′ ¯ ω A ′ ∆ B ′ B λ B − γ AB φ A ∆ BB ′ ¯ µ B ′ (cid:17) for any pair of Dira spinors Ψ α = ( λ A , ¯ µ A ′ ) and Φ α = ( φ A , ¯ ω A ′ ) . In fa t, (cid:0) Ψ α , Φ α (cid:1) H (cid:2) Ψ α , Φ + α (cid:3) : = 14 πG I S i √ + α γ αeβ ε ef ∆ f Ψ β d S == H (cid:2) λ A , ¯ φ A ′ (cid:3) − H (cid:2) ω A , ¯ µ A ′ (cid:3) is Hermitian, and redu es to (2.22) if at least one of Ψ α and Φ α has de(cid:28)nite η αβ - hirality (e.g. when ω A = 0 or µ A = 0 ). Interestingly enough, if the Dira spinors have de(cid:28)nite, but opposite η αβ - hirality, e.g. when φ A = 0 and µ A = 0 ,then they are orthogonal to ea h other: H [Ψ α , Φ + α ] = 0 .Returning to the hara terization of the reality of the eigenvalues of theDira operators, suppose that λ A satis(cid:28)es (2.12). Then (2.24) implies that α H [ λ A , ¯ λ A ′ ] = ¯ α H [ λ A , ¯ λ A ′ ] , i.e. α is real or purely imaginary provided H [ λ A , ¯ λ A ′ ] = 0 . The reality of α an be hara terized by the H -norm of the eigen-spinors. Indeed, for λ A and µ A satisfying (2.10) with non-zero α , (2.22) gives ¯ αH (cid:2) λ A , ¯ λ A ′ (cid:3) = − αH (cid:2) µ A , ¯ µ A ′ (cid:3) . This implies that H [ λ A , ¯ λ A ′ ] = ± H [ µ A , ¯ µ A ′ ] , and, for H [ λ A , ¯ λ A ′ ] = 0 , weobtain that α is real i(cid:27) H [ λ A , ¯ λ A ′ ] = − H [ µ A , ¯ µ A ′ ] , and α is purely imagi-nary i(cid:27) H [ λ A , ¯ λ A ′ ] = H [ µ A , ¯ µ A ′ ] . In subse tion 4.1 we give examples bothfor real and purely imaginary eigenvalues. α an be a more general om-plex number only if H [ λ A , ¯ λ A ′ ] = 0 , i.e. when H S γ AB λ A µ B d S = 0 . If σ A is an eigenspinor of − AB ′ ∆ B ′ B with eigenvalue β , then (2.24) gives that ( β − ¯ α ) H [ σ A , ¯ λ A ′ ] = 0 ; i.e. the eigenspinors of − AB ′ ∆ B ′ B with eigenval-ues α and β , satisfying ¯ α = ± β , are orthogonal to ea h other with respe t to H .3 The δ AA ′ (cid:21)Dira operatorSin e S is even dimensional, there is a notion of hirality in the spa e of surfa espinors too, whi h hirality remains inta t even if the gauge group SO (2) isenlarged to SO (2) × SO (1 , by in luding the boost gauge transformations inthe normal bundle of S . This hirality is represented by the spinor γ AB (thus we all it the ` γ - hirality'), and is preserved by δ e but not by ∆ e . (For the detailssee [18, 5℄.) Thus it seems useful to dis uss the onsequen es of the existen e ofthe γ - hirality in the ase of the δ AA ′ (cid:21)Dira operator.The δ AA ′ -Dira operator an be obtained from the ∆ AA ′ (cid:21)Dira operatorwith the formal substitution Q AeB = 0 ; and in this ase the eigenvalue problem(2.10) redu es to 11 δ A ′ A λ A = − α √ µ A ′ , i δ AA ′ ¯ µ A ′ = − α √ λ A . (3.1)It is easy to show (using e.g. the se ond order equation − δ AA ′ δ A ′ B λ B = α λ A )that this notion of eigenspinors oin ides with that de(cid:28)ned by (2.6) if a onstantHermitian metri G AA ′ exists on the spinor bundle. In this ase ¯ µ A ′ = G A ′ A λ A .Re alling that δ e γ AB = 0 , γ AB γ BC = δ AC and that γ AB ¯ γ A ′ B ′ a ts on ve torstangent to S as − δ ab , it is easy to verify that ¯ γ A ′ B ′ (cid:0) δ B ′ B λ B (cid:1) = δ A ′ A (cid:0) γ AB λ B (cid:1) , γ AB (cid:0) δ BB ′ δ B ′ C λ B (cid:1) = δ AA ′ δ A ′ B (cid:0) γ BC λ C (cid:1) . (3.2)Thus the δ AA ′ -Dira operator ommutes with the a tion of the γ -spinor as abase point preserving bundle map. In parti ular, if λ A is a Weyl eigenspinor of − δ AA ′ δ A ′ B with eigenvalue α , then γ AB λ B is also a Weyl eigenspinor withthe same eigenvalue. This implies that both λ A ± γ AB λ B are Weyl eigenspinorswith the same eigenvalue α , but they have de(cid:28)nite γ - hirality. In the GHPspinor dyad { o A , ι A } the right/left handed Weyl eigenspinors of − δ AA ′ δ A ′ B have the stru ture − λ ι A and λ o A , respe tively, where, by (2.15)-(2.16), thespinor omponents satisfy − ðð ′ λ = α λ , − ð ′ ð λ = α λ . (3.3)Thus the γ - hirality an be used to de ompose the spa e of the eigenspinorsfurther. Sin e, however, γ AB is annihilated only by δ e but not by ∆ e in general,this de omposition is possible only for the δ AA ′ -Dira operators. Using the listof the dimension of the kernel of the edth operators, by (3.3) we an determinethe number of the (e.g. right handed) eigenspinors of − δ AA ′ δ A ′ B with zeroeigenvalue: on topologi al 2-spheres there are no su h eigenspinors, on torithere are two, while on higher genus ( g > ) surfa es there are g − ones.Clearly, there is a natural one-to-one orresponden e between the eigen-spinors λ A of − δ AA ′ δ A ′ B with eigenvalue α and the eigenspinors ( λ A , ¯ µ A ′ ) of (3.1). Moreover, if λ A has de(cid:28)nite hirality, e.g. if γ AB λ B = ± λ A , then µ A has the same de(cid:28)nite hirality: γ AB µ B = ± µ A . Thus the natural one-to-one orresponden e above preserves the γ - hirality as well, and the spa e ofthe eigenspinors splits in a natural way to the dire t sum of the spa es of theright/left handed eigenspinors.By the analysis of subse tion 2.3, the Weyl eigenspinors de(cid:28)ne a olle tion of δ e -divergen e-free omplex ve tor (cid:28)elds on S . Indeed, taking into a ount that δ e ommutes with the proje tion Π ab , the omplex ve tor (cid:28)eld k a := Π ab λ B ¯ µ B ′ ,tangent to S , is δ e -divergen e free on S . Similarly, z a ± := Π ab ( λ B ¯ λ B ′ ± µ B ¯ µ B ′ ) is δ e -divergen e free on S for purely imaginary/real eigenvalue α . The ve tor(cid:28)elds k a and z a ± are vanishing for eigenspinors with de(cid:28)nite hirality, be ausethen the null ve tors λ A ¯ µ A ′ , λ A ¯ λ A ′ and µ A ¯ µ A ′ are all orthogonal to S .Similarly, the formal substitution Q AEE ′ B = 0 in (2.19) yields that ξ e = φ A δ e λ A + λ A δ e φ A is a δ e -divergen e-free omplex tangent ve tor (cid:28)eld on S if12 δ AA ′ δ A ′ B λ B = α λ A and − δ AA ′ δ A ′ B φ B = α φ A hold for some fun tion α : S → C , e.g. when λ A and φ A are Weyl eigenspinors with the same eigenvalue.If λ A and φ A have the same, de(cid:28)nite hirality, then the ve tor (cid:28)eld ξ a itself isvanishing.The analysis of subse tion 2.4 an be repeated to obtain riteria for thereality of the eigenvalue α . The only di(cid:27)eren e is that the Hermitian s alarprodu t on C ∞ ( S , S A ) should be de(cid:28)ned by h (cid:2) λ A , ¯ ω A ′ (cid:3) := 14 πG I S ¯ γ A ′ B ′ ¯ ω A ′ δ B ′ B λ B d S . (3.4)Then, for h [ λ A , ¯ λ A ′ ] = 0 , the non-zero α is real i(cid:27) h [ λ A , ¯ λ A ′ ] = − h [ µ A , ¯ µ A ′ ] ; andit is purely imaginary i(cid:27) h [ λ A , ¯ λ A ′ ] = h [ µ A , ¯ µ A ′ ] . However, for eigenspinors withde(cid:28)nite hirality the norm h [ λ A , ¯ λ A ′ ] is always zero, and hen e the orrespond-ing eigenvalue may in prin iple be a (not ne essarily real or purely imaginary) omplex number.4 On round spheres4.1 The spe trumLet S be a round sphere of radius r ; i.e. S is a transitivity surfa e of therotation group in a spheri ally symmetri spa etime, whose radius is de(cid:28)ned by πr := Area( S ) . Then the onne tion in the normal bundle is (cid:29)at, and by anappropriate boost gauge hoi e ρ, ρ ′ = const an be a hieved. To solve (2.11),it ould be a good strategy to solve (3.3) (cid:28)rst, and return to (2.11) later. Weexpand the spinor omponents λ and λ in terms of the spin weighted spheri alharmoni s: λ = ∞ X j = j X m = − j c j,m Y jm , λ = ∞ X j = j X m = − j c j,m − Y jm , (4.1)where j = , , , ... and m = − j, − j + 1 , ..., j ; and c j,m and c j,m are omplex onstants. Sin e (see. e.g. [20℄)ð s Y jm = − √ r q(cid:0) j + s + 1 (cid:1)(cid:0) j − s (cid:1) s +1 Y jm , (4.2)ð ′ s Y jm = 1 √ r q(cid:0) j − s + 1 (cid:1)(cid:0) j + s (cid:1) s − Y jm , (4.3)the eigenvalue equations (3.3) redu e to X j,m c j,m (cid:16) α − r (cid:0) j + 12 (cid:1) (cid:17) Y jm = 0 , X j,m c j,m (cid:16) α − r (cid:0) j + 12 (cid:1) (cid:17) − Y jm = 0 . (4.4)13e ause of the ompleteness of the spheri al harmoni s these imply that theeigenvalues α are dis rete: α = n r , n := j + 12 ∈ N , (4.5)and that, for a given allowed α and the orresponding j , λ = j X m = − j c m Y jm , λ = j X m = − j c m − Y jm (4.6)for some omplex onstants c m and c m . Thus the spa e of the right/left handedWeyl eigenspinors is spanned by c m and c m , respe tively, and hen e the di-mension of these spa es is j + 1 = 2 n . Then the eigenvalues an also beexpressed by the ( onstant) s alar urvature R of the intrinsi geometry of S as α = n R . Therefore, the (cid:28)rst eigenvalue already ontains all the informationabout (i.e. the only observable of) the geometry of S . The eigenvalues of the δ AA ′ -Dira operator are α = ± n/r , and the omponents of the orrespondingprimed Weyl spinor are ¯ µ ′ = ∓ i P m c m − Y jm and ¯ µ ′ = ∓ i P m c m Y jm . Wenote that for round spheres δ e - onstant Hermitian metri s G AA ′ do exist, andthe present parti ular results an be ompared with the general ones obtainedin pure Riemannian geometry: the spe trum (4.5) saturates the inequalities of[11, 12, 13, 15, 14, 16℄.By (2.15)-(2.16) the eigenvalue problem for the operators − AA ′ ∆ A ′ B and ∆ A ′ A an be solved in an analogous way. The stru ture of the unprimed Weylspinor part of the eigenspinors is similar to that given by (4.6). The onlydi(cid:27)eren e is that the primed Weyl spinor part hange slightly and the eigenvaluesare shifted by a term built from the onvergen es: α = 1 r (cid:0) n + 2 r ρρ ′ (cid:1) , n := j + 12 ∈ N . (4.7)In ontrast to (4.5) this an be zero or even negative in ertain spe ial geome-tries. For example, in spa etimes with onstant urvature the two onvergen es an be hosen to be ρ = − r q − Λ r and ρ ′ = r q − Λ r , Λ ∈ R , andhen e r ρρ ′ = Λ r . In Minkowski spa etime ( Λ = 0 ) α is zero, while inthe anti-de Sitter spa etime ( Λ < ) it is negative (i.e. α is purely imaginary) for n = 1 . (For a more general dis ussion of the kernel of the ∆ AA ′ -Dira operatoron round spheres, see [2℄.) Remarkably enough, by (4.7) the (cid:28)rst eigenvalue is onne ted with the Hawking quasi-lo al mass: α = 2 GE H ( S ) /r . Therefore,the (cid:28)rst two eigenvalues of − AA ′ ∆ A ′ B , or the (cid:28)rst eigenvalue of − δ AA ′ δ A ′ B and of − AA ′ ∆ A ′ B give the only two non-trivial gauge invariant observablesof the 2-surfa e in spa etime, namely the s alar urvature R and the length ρρ ′ of the mean urvature ve tor, or, in other ombination, the area of the surfa eand the Hawking energy.The omponents of the primed Weyl spinor part ¯ µ A ′ of the eigenspinors ofthe ∆ e -Dira operator an be al ulated easily from (2.10). Sin e, however, in14he present paper primarily we are interested in the δ e -divergen e free ve tor(cid:28)elds, we do not need them in what follows.4.2 The divergen e free ve tor (cid:28)eldsFor given j the δ e -divergen e free ve tor (cid:28)eld k a = Π ab λ B ¯ µ B ′ takes the form k a = ± i (cid:16)X k,m c k c m − Y jk − Y jm (cid:17) m a ± i (cid:16)X k,m c k c m Y jk Y jm (cid:17) ¯ m a , where m a = √ r (1 + ζ ¯ ζ )( ∂∂ ¯ ζ ) a . In parti ular, for j = the expli it form of thisve tor (cid:28)eld in the standard omplex stereographi oordinates ( ζ, ¯ ζ ) is ± k a = − i2 π
11 + ζ ¯ ζ (cid:16) c c − (cid:0) c c − + c − c (cid:1) ¯ ζ + c − c − ¯ ζ (cid:17) m a −− i2 π
11 + ζ ¯ ζ (cid:16) c c ζ + (cid:0) c c − + c − c (cid:1) ζ + c − c − (cid:17) ¯ m a . (4.8)Sin e there are no harmoni forms on spheres, by the Hodge de omposition the-orem the divergen e-free rk a an always be written as ε ab δ b F for some fun tion F : S → C . A short al ulation yields that this is indeed the ase with F = ∓ r √ π c − c − ¯ ζ − c c ζ + (cid:0) c c − + c − c (cid:1) ζ ¯ ζ ζ ¯ ζ , (4.9)where the irrelevant onstant of integration has been hosen to be zero. k a (andhen e the fun tion F also) is real pre isely when c − c − = a + i b, c c = − a + i b, c c − + c − c = 2 c (4.10)for some real onstants a , b and c . If we write c c − =: c + d + i e for some real d and e , and hen e c − c = c − d − i e , then − ( a + b ) = ( c c )( c − c − ) =( c c − )( c − c ) = c − d + e − i2 ed . Sin e its left hand side is nonpositive,the right hand side must also be real and nonpositive, implying that e = 0 , andhen e that d = a + b + c .Sin e S is two dimensional, in the ase of real F the level sets F = const arepre isely the integral urves of the ve tor (cid:28)eld k a , and F is given expli itly by F = ∓ r √ π a (cid:0) ζ + ¯ ζ (cid:1) + i b (cid:0) ¯ ζ − ζ (cid:1) + 2 cζ ¯ ζ ζ ¯ ζ = const ∓ r √ π (cid:16) ax + by + cz (cid:17) . (4.11)(Here x , y and z are the standard Cartesian oordinates de(cid:28)ned by x + i y :=2 rζ/ (1 + ζ ¯ ζ ) and z := r ( ζ ¯ ζ − / ((1 + ζ ¯ ζ ) .) However, the orresponding level15ets are just the integral urves of the Killing ve tors of the metri 2-sphere. Infa t, in these oordinates the standard rotation Killing 1-forms are K a = i √ r ζ ¯ ζ (cid:16)(cid:0) − ¯ ζ (cid:1) m a − (cid:0) − ζ (cid:1) ¯ m a (cid:17) ,K a = 1 √ r ζ ¯ ζ (cid:16)(cid:0) ζ (cid:1) m a + (cid:0) ζ (cid:1) ¯ m a (cid:17) , (4.12) K a = i r √
21 + ζ ¯ ζ (cid:16) ¯ ζm a − ζ ¯ m a (cid:17) . Comparing these with (4.8), we (cid:28)nd that rk a = ± √ π (cid:16)(cid:0) c − c − − c c (cid:1) K a − i (cid:0) c c + c − c − (cid:1) K a ++ (cid:0) c c − + c − c (cid:1) K a (cid:17) ; (4.13)i.e. on round spheres of radius r the δ e -divergen e free ve tor (cid:28)eld k a builtfrom the j = eigenspinors is /r -times a omplex ombination of the rotationKilling ve tors.By a similar analysis one an determine the ve tor (cid:28)elds k a for all j > . Inparti ular, for j = there are pre isely ten su h omplex independent ve tor(cid:28)elds. The fun tions F orresponding to the independent real ve tor (cid:28)elds rk a are ubi expressions of the Cartesian oordinates x , y and z , divided by r , e.g. ( x − xy ) /r , xyz/r , ... et . However, their signi(cid:28) an e and geometri meaningare still not quite lear.Sin e the eigenvalues α are real, the ve tor (cid:28)elds z a − are δ e divergen e free.Repeating the analysis above, a dire t al ulation shows that for j = theseve tor (cid:28)elds have the form r ( AK a + BK a + CK a ) for some real onstants A , B and C ; i.e. for j = the ve tor (cid:28)eld rz a − is a real linear ombination of the threerotation Killing ve tors. For j = the ve tor (cid:28)eld ξ a is also proportional to therotation Killing ve tor (cid:28)elds. Sin e, however, ξ a is built from the (cid:28)rst derivativeof the Weyl spinor parts of the eigenspinor, it s ales with a di(cid:27)erent power of theradius: ξ a = r ( AK a + BK a + CK a ) with appropriate ( omplex) onstants A , B and C . We ontinue the dis ussion of k a and z a − in a more general ontextin subse tions 5.3.1 and 5.3.2, respe tively.5 On large spheres near the null in(cid:28)nity5.1 Asymptoti ally (cid:29)at spa etimesLet the spa etime be asymptoti ally (cid:29)at at future null in(cid:28)nity, and let ( u, r, ζ, ¯ ζ ) be a Bondi-type oordinate system (see e.g. [27℄). Then the standard edthoperators, a ting on ( p, q ) type s alar f , take the form ð f = P ( ∂f /∂ ¯ ζ ) + ( ∂f /∂ζ ) − pβf + q ¯ β ′ f and ð ′ f = P ( ∂f /∂ζ ) + ¯ Q ( ∂f /∂ ¯ ζ ) + pβ ′ f − q ¯ βf , re-spe tively. (For the de(cid:28)nition of the GHP spin oe(cid:30) ients see [21, 20℄.) TheGHP spin frame is hosen su h that o A is parallelly propagated along the nullgeodesi generators of the null hypersurfa es u = const , and ι A is hosen su hthat m a := o A ¯ ι A ′ and ¯ m a := ι A ¯ o A ′ be tangents to the spa elike 2-surfa es u = const , r = const . In this ase ¯ β ′ = β − τ . In these oordinates andspin frame in an Einstein(cid:21)Maxwell spa etime the asymptoti form of the fun -tions P and Q and some of the spin oe(cid:30) ients that we need are well known[28℄ to be given by P = r √ (1 + ζ ¯ ζ ) + O ( r − ) , Q = − r √ (1 + ζ ¯ ζ ) σ + O ( r − ) , β = − r √ ζ − r √ ¯ ζσ + O ( r − ) and τ = r ð ′ σ − r (2 σ ð ¯ σ + ψ )+ O ( r − ) .Here ð and ð ′ denote the edth operators on the metri , unit sphere, σ is theasymptoti shear (i.e. σ = r σ + O ( r − ) ), the dot denotes di(cid:27)erentiation withrespe t to u , and ψ , ψ are the leading terms in the asymptoti expansionof the Weyl spinor omponents Ψ and Ψ , respe tively (see e.g. [27, 6℄). Inaddition, the Weyl spinor omponents satisfy ˙ ψ = ð ψ − σ ð ˙¯ σ + 4 Gϕ ¯ ϕ , (5.1) ˙ ψ = − ð ˙¯ σ − σ ¨¯ σ + 2 Gϕ ¯ ϕ , (5.2) ψ − ¯ ψ = ð ′ σ − ð ¯ σ + ¯ σ ˙ σ − ˙¯ σ σ . (5.3)Here ϕ n are the leading terms in the asymptoti expansion of the Maxwell spinor omponents: ϕ n = r n − ϕ n + O ( r n − ) . We will use these formulae in subse tion5.4.3.5.2 The eigenvalue problem on large spheresIn the Bondi type oordinate system the spa elike 2-surfa es u = const , r =const for large enough r are alled large spheres (and will be denoted by S r ), andour aim is to solve the eigenvalue problem (3.3) on these surfa es asymptoti ally.To do so, let us write the omponents λ A of the spinor (cid:28)eld λ A in the GHPspin frame ε AA := { o A , ι A } , A = 0 , , as λ A =: λ (0) A + r λ (1) A + ... and, similarly,we expand the eigenvalue as r α =: α + r α + ... . (Thus we adopt the onvention that an index between parentheses, e.g. 0 or 1 here, is referring tothe order of approximation.) Then substituting all these into equation (3.3) weobtain ð ð ′ λ (0)0 + α λ (0)0 = 0 (5.4) ð ð ′ λ (1)0 + α λ (1)0 = − α λ (0)0 + 2 ð ′ (cid:0) σ ð ′ λ (0)0 (cid:1) + 2 ð (cid:0) ¯ σ λ (0)0 (cid:1) (5.5)and ð ′ ð λ (0)1 + α λ (0)1 = 0 (5.6) ð ′ ð λ (1)1 + α λ (1)1 = − α λ (0)1 + 2 ð ′ (cid:0) σ ð ′ λ (0)1 (cid:1) + 2¯ σ ð λ (0)1 . (5.7)17he zeroth order equations (5.4) and (5.6) are just the eigenvalue equations(3.3) on the unit sphere. Thus in the zeroth order the eigenvalues are α = n , n := j + ∈ N , and the omponents λ (0)0 and λ (0)1 of the eigenspinors are givenexpli itly by (4.6), where in general the expansion oe(cid:30) ients are still arbitraryfun tions of the retarded time oordinate u . Sin e the stru ture of the left handside of equation (5.5) is similar to the homogeneous (5.4), and that of (5.7) tothe homogeneous (5.6), the general solution of the homogeneous equations anbe added to the (cid:28)rst order orre tions λ (1)0 and λ (1)1 , yielding an ambiguity inthe order O ( r − ) .Next let us al ulate the (cid:28)rst-order orre tion to the zeroth-order expressionof the (cid:28)rst eigenvalue, α = 1 , and the orresponding eigenspinors. Sin e λ (0)0 = (0) c Y
12 12 + (0) c − Y − and λ (0)1 = (0) c − Y
12 12 + (0) c − − Y − ,where the oe(cid:30) ients (0) c mA are fun tions of u , one has ð λ (0)0 = 0 , ð ′ λ (0)1 = 0 , ð ′ λ (0)0 = 0 , ð λ (0)1 = 0 , and ð ð ′ λ (0)0 = − λ (0)0 , ð ′ ð λ (0)1 = − λ (0)1 . Usingthese the equations (5.5) and (5.7), respe tively, redu e to ð ð ′ λ (1)0 + λ (1)0 = − α λ (0)0 + 2 (cid:0) ð ′ σ (cid:1)(cid:0) ð ′ λ (0)0 (cid:1) + 2 (cid:0) ð ¯ σ (cid:1) λ (0)0 , (5.8) ð ′ ð λ (1)1 + λ (1)1 = − α λ (0)1 . (5.9)To solve (5.9) let us write λ (1)1 =: P j,m (1) c jm − Y jm with arbitrary fun tions (1) c jm = (1) c jm ( u ) . We obtain that α (1) = 0 (i.e. in parti ular the (cid:28)rst eigen-value is real in the (cid:28)rst two orders), and all the expansion oe(cid:30) ients (1) c jm must be zero for j ≥ . Hen e λ (1)1 is a ombination only of the two − spinweighted spheri al harmoni s − Y ± , i.e. the orre tion to λ (0)1 has the stru -ture similar to that of λ (0)1 itself. Thus λ (1)1 , as the (cid:28)rst-order orre tion to thezeroth-order eigenspinor, represents only a pure (gauge) ambiguity.To solve (5.8) let us observe that the operator ð a ting on s alars with pos-itive spin weight on topologi al 2-spheres is surje tive (see e.g. [20, 22℄). Thusthe asymptoti shear σ an always be derived from an appropriate omplexs alar S of spin weight zero (indeed, of type (1 , ): σ = ð S , where the am-biguities in S are the elements of ker ð , and, on any given surfa e u = const ,they form a four omplex dimensional spa e. Then taking into a ount α (1) = 0 and the ommutator ( ð ð ′ − ð ′ ð ) f = − ( p − q ) f a ting on the ( p, q ) types alar f , (5.8) an be written into the form ð ′ (cid:16) ð λ (1)0 − ð (cid:0) S − ¯ S (cid:1)(cid:0) ð ′ λ (0)0 (cid:1) − (cid:0) ð ′ ð ¯ S (cid:1) λ (0)0 (cid:17) = 0 . (5.10)However, dim ker ð ′ = 0 for positive spin weight s alars on topologi al 2-spheres,and hen e the expression between the big parentheses itself is zero. Nevertheless,using the ommutator of ð and ð ′ above, this expression an also be rewrittenas a pure ð-derivative, and hen e we have18 ð (cid:16) λ (1)0 − ð (cid:0) S − ¯ S (cid:1)(cid:0) ð ′ λ (0)0 (cid:1) − (cid:0) S + ¯ S (cid:1) λ (0)0 − (cid:0) ð ð ′ ¯ S (cid:1) λ (0)0 (cid:17) = 0 . (5.11)Finally, sin e dim ker ð = 2 for ð a ting on s alars of spin weight on topologi al2-spheres, from (5.11) we an dedu e that λ (1)0 = ð (cid:0) S − ¯ S (cid:1)(cid:0) ð ′ λ (0)0 (cid:1) + (cid:0) ð ð ′ ¯ S + 12 (cid:0) S + ¯ S (cid:1)(cid:1) λ (0)0 + λ, (5.12)where λ is an arbitrary spin weight solution of ð λ = 0 . Thus, for j = ,equations (5.4)(cid:21)(5.7) an be solved expli itly in terms of the omplex potential S for the asymptoti shear σ .The ambiguities in (5.12), oming from the ambiguity of the potential S (i.e.formally from the non-triviality of the kernel of ð a ting on zero spin weights alars) and the ambiguity of the solution of the inhomogeneous equation (5.5)(i.e. from the non-triviality of the kernel of ð a ting on spin weight s alars), an be summarized as λ (1)0 λ (1)0 + X m = − (1) c m Y m , (5.13)where (1) c ± are arbitrary omplex fun tions of u . Here we used the expansionof the produ ts of spin weighted spheri al harmoni s of the form Y M Y m and Y M − Y m in terms of Y m and Y m ′ , whi h an easily be derivedby dire t al ulation from the expli it expression of the harmoni s given e.g.in [29℄. Though in these expansions the spheri al harmoni s Y m do appear,all these are an eled from (5.12). Thus the ambiguities in S des ribed by the Y m spheri al harmoni s are all an eled from the eigenspinors. The remainings alar ambiguity in S yields an ambiguity in λ (1)0 similar to that oming from thesolutions of the homogeneous equation. Hen e all these an be parameterizedonly by two (rather than the originally expe ted six) omplex fun tions of u .Although there is no anoni al isomorphism between the spa es of the eigen-spinors on two di(cid:27)erent surfa es even if these spa es an be mapped to ea hother isomorphi ally, in the asymptoti ally (cid:29)at ontext we an introdu e a nat-ural equivalen e between the spa e of the eigenspinors on the large spheres, say S ′ and S ′′ , at di(cid:27)erent retarded times u = u ′ and u = u ′′ . Namely, we requirethat the zeroth-order solutions be independent of the retarded time oordinate u ,i.e. we hoose the expansion oe(cid:30) ients (0) c mA onstant. (It is easy to see thatthis requirement an be extended to the ase when the uts of I + that the largespheres de(cid:28)ne, say ˆ S ′ and ˆ S ′′ , are related to ea h other only by a proper BMS su-pertranslation. The basis of this equivalen e is the fa t that (1) the spinor basesare uniquely determined on the large spheres S ′ and S ′′ su h that these basesare related via the spinor onstituent of the null geodesi generators of I + ;(2) the oordinates ( ζ ′ , ¯ ζ ′ ) on S ′ determine ( ζ ′′ , ¯ ζ ′′ ) on S ′′ uniquely via the null19eodesi generators of I + ; (3) in these bases and oordinates the zeroth-ordersolutions ( λ ′ (0)0 , λ ′ (0)1 ) on S ′ and ( λ ′′ (0)0 , λ ′′ (0)1 ) on S ′′ have the same stru ture ifwe expand them in terms of the spin weighted spheri al harmoni s. Then weidentify ( λ ′ (0)0 , λ ′ (0)1 ) with ( λ ′′ (0)0 , λ ′′ (0)1 ) pre isely when their spinor omponentsare ombined from the spin weighted spheri al harmoni s by the same omplex oe(cid:30) ients. This ondition is analogous to a hoi e of a onformal gauge on I + . Then an eigenspinor λ ′ A on S ′ will be alled equivalent to the eigenspinor λ ′′ A on S ′′ if their zeroth-order parts are the same in the sense above.)Though this ondition does not rule out the u -dependen e of the (cid:28)rst order orre tion terms λ (1) A , this makes the whole solution onsiderably more (cid:16)rigid(cid:17),and essentially their u -dependen e is already ontrolled: all of their ambiguitieswith arbitrary u -dependen e have the form X m = − (1) c m ( u ) Y m , X m = − (1) c m ( u ) − Y m , (5.14)i.e. they belong to the kernel of ð and ð ′ , respe tively, while the u -dependen eof the remaining u -dependent parts (of λ (1)0 , see (5.12)) omes only from the u -dependen e of the asymptoti shear. Thus the role of the equivalen e is toprovide a ommon, universal (i.e. ut-independent) parameter spa e, namelythe spa e of the zeroth order solutions, by means of whi h the solutions (upto the ambiguities (5.14)) an be parameterized. As we will see in subse tions5.4.1 and 5.4.2, the general expression of our 2-surfa e observable is not sensitiveto this ambiguity, and this makes it possible to be able to ompare angularmomenta de(cid:28)ned at di(cid:27)erent retarded times in subse tion 5.4.3.5.3 The δ e (cid:21)divergen e free ve tor (cid:28)elds for j = From λ (0) A and λ (1) A we an ompute the omponents of ¯ µ A ′ using (3.1) and theasymptoti form of the edth operators. For the eigenvalue α (0) = ± we have ¯ µ ′ = ∓ i √ (cid:16) ð ′ λ (0)0 + 1 r ð ′ λ (1)0 − r (cid:0) ð ¯ σ (cid:1) λ (0)0 (cid:17) , (5.15) ¯ µ ′ = ± i √ (cid:16) ð λ (0)1 + 1 r ð λ (1)1 (cid:17) , (5.16)and in the next two subse tions we dis uss the δ e (cid:21)divergen e-free ve tor (cid:28)elds k a and z a − built from λ A and ¯ µ A ′ .5.3.1 The ve tor (cid:28)eld k a For the ve tor (cid:28)eld k a we obtain 20 k a = i r √ (cid:16) ð ′ (cid:0) λ (0)0 λ (0)1 (cid:1) ˆ m a − ð (cid:0) λ (0)0 λ (0)1 (cid:1) ¯ˆ m a (cid:17) ++ i r √ (cid:16) h λ (1)1 (cid:0) ð ′ λ (0)0 (cid:1) + λ (0)1 (cid:0) ð ′ λ (1)0 (cid:1) − λ (0)0 λ (0)1 0 ð ¯ σ i ˆ m a −− h λ (1)0 (cid:0) ð λ (0)1 (cid:1) + λ (0)0 (cid:0) ð λ (1)1 (cid:1)i ¯ˆ m a (cid:17) + O (cid:0) r − (cid:1) , (5.17)where ˆ m a := √ (1 + ζ ¯ ζ )( ∂/∂ ¯ ζ ) a , the omplex null ve tor on the unit sphere,normalized with respe t to the unit sphere metri . Sin e rk a is divergen e free,it an always be written as the dual of the gradient of some fun tion F : rk a = ε ab δ b F = − i (cid:16) m a ð ′ F − ¯ m a ð F (cid:17) == i r (cid:16)(cid:0) ð F (cid:1) − r σ (cid:0) ð ′ F (cid:1)(cid:17) ¯ˆ m a − i r (cid:16)(cid:0) ð ′ F (cid:1) − r ¯ σ (cid:0) ð F (cid:1)(cid:17) ˆ m a . (5.18)Writing the fun tion F as F =: r F ( − + rF ( − + O (1) and omparing (5.17)with (5.18) we obtain a system of partial di(cid:27)erential equations for F ( − and F ( − . Their integrability ondition is satis(cid:28)ed by (5.12), and the solution is F ( − = ∓ √ λ (0)0 λ (0)1 , (5.19) F ( − = ∓ √ n(cid:0) ð S (cid:1) ð ′ (cid:0) λ (0)0 λ (0)1 (cid:1) + (cid:0) ð ′ ¯ S (cid:1) ð (cid:0) λ (0)0 λ (0)1 (cid:1) −− (cid:0) S + ¯ S (cid:1) ð ′ ð (cid:0) λ (0)0 λ (0)1 (cid:1) + 12 (cid:0) S − ¯ S (cid:1)(cid:16) λ (0)0 λ (0)1 + ð ′ ð (cid:0) λ (0)0 λ (0)1 (cid:1)(cid:17) ++ λ (0)0 λ (1)1 + λλ (0)1 o , (5.20)where the irrelevant onstants of integration have been hosen to be zero. Apartfrom the last two (ambiguous) terms in (5.20) (whi h have the stru ture of F ( − ) all the terms are proportional to λ (0)0 λ (0)1 , whi h is essentially F ( − .Although r F ( − has been given expli itly by (4.9), here we give another, and,from the points of view of later appli ations, a more useful expression.Let us de(cid:28)ne the real (essentially the Y m spheri al harmoni ) fun tions t = ζ + ¯ ζ ζ ¯ ζ , t = i ¯ ζ − ζ ζ ¯ ζ , t = ζ ¯ ζ −
11 + ζ ¯ ζ , (5.21)and introdu e the oe(cid:30) ients 21 := 14 √ π (cid:16) (0) c − c − − (0) c c (cid:17) = 12 √ π a, (5.22) R := − i4 √ π (cid:16) (0) c − c − + (0) c c (cid:17) = 12 √ π b, (5.23) R := 14 √ π (cid:16) (0) c c − + (0) c − c (cid:17) = 12 √ π c, (5.24) R := 14 √ π (cid:16) (0) c c − − (0) c − c (cid:17) = 12 √ π d ; (5.25)whi h are onstant by our requirement imposed at the end of the previoussubse tion. Here the se ond equalities hold if the reality ondition (4.10) isimposed, and hen e d is determined up to sign by the parameters in R i , i =1 , , , as d = a + b + c , i.e. formally ( R , R i ) is a real null ve tor withrespe t to the onstant Lorentzian metri . Then a simple al ulation gives λ (0)0 λ (0)1 = √ (cid:16) R + R i t i (cid:17) , (5.26)yielding a parameterization of F ( − and F ( − in terms of R i and R : F ( − = ∓ (cid:16) R + R i t i (cid:17) , (5.27) F ( − = ∓ (cid:16)(cid:0) ð S (cid:1)(cid:0) ð ′ t i (cid:1) + (cid:0) ð ′ ¯ S (cid:1)(cid:0) ð t i (cid:1) + 12 (cid:0) S + ¯ S (cid:1) t i (cid:17) R i ∓∓ (cid:0) S − ¯ S (cid:1) R ∓ (cid:0) G + G i t i (cid:1) , (5.28)where the last term of (5.28) omes from the last two ambiguous terms of (5.20)for some (in general omplex) fun tions G and G i of u . Introdu ing the newnotation K i a := ε ijk K jk a for the Killing (cid:28)elds of the metri sphere of radius r and de(cid:28)ning the antisymmetri matrix M ij by R i =: ε ijk M jk , the leading-orderterm in rk a an in fa t be written as K ij a M ij = − R i K i a , as it ould be expe tedby (4.13). (Here raising and lowering of the boldfa e indi es are de(cid:28)ned by thenegative de(cid:28)nite onstant metri η ij := − δ ij .) The ve tor (cid:28)eld rk a itself is ∓ rk a = 2i (cid:16) ¯ˆ m a (cid:0) ð t i (cid:1) − ˆ m a (cid:0) ð ′ t i (cid:1)(cid:17) R i ++ i r (cid:16) ¯ˆ m a h (cid:0) ð ð ′ ¯ S (cid:1)(cid:0) ð t i (cid:1) + ð (cid:0) ¯ S − S (cid:1) t i + (cid:0) S + ¯ S (cid:1)(cid:0) ð t i (cid:1)i −− ˆ m a h (cid:0) ð ′ ð S (cid:1)(cid:0) ð ′ t i (cid:1) + ð ′ (cid:0) S − ¯ S (cid:1) t i + (cid:0) S + ¯ S (cid:1)(cid:0) ð ′ t i (cid:1)i(cid:17) R i ++ i r (cid:16) ¯ˆ m a ð (cid:0) S − ¯ S (cid:1) + ˆ m a ð ′ (cid:0) ¯ S − S (cid:1)(cid:17) R ++ 2i r (cid:16) ¯ˆ m a (cid:0) ð t i (cid:1) − ˆ m a (cid:0) ð ′ t i (cid:1)(cid:17) G i + O ] (cid:0) r − (cid:1) . (5.29)22ne an see that the oe(cid:30) ients of R i are real, but the oe(cid:30) ient of R is imag-inary. Consequently, in a general radiative spa etime (i.e. when the potential S for the asymptoti shear annot be hosen to be real) the reality of the whole k a annot be ensured, even if R i and G i are hosen to be real. Indeed, thevanishing of R is equivalent to d = 0 , whi h would imply a = b = c = 0 , too.Next re all that if ξ := − (1 − ζ ) , ξ := − i(1 + ζ ) and ξ := − ζ , then thegeneral BMS ve tor (cid:28)eld has the form ˆ K a = (cid:16) H + (cid:0) c i + ¯ c i (cid:1) t i u (cid:17)(cid:0) ∂∂u (cid:1) a + c i ξ i (cid:0) ∂∂ζ (cid:1) a + ¯ c i ¯ ξ i (cid:0) ∂∂ ¯ ζ (cid:1) a == (cid:16) H + (cid:0) c i + ¯ c i (cid:1) t i u (cid:17)(cid:0) ∂∂u (cid:1) a − c i (cid:0) ð ′ t i (cid:1) ˆ m a − c i (cid:0) ð t i (cid:1) ¯ˆ m a , (5.30)where H = H ( ζ, ¯ ζ ) is an arbitrary real fun tion and c i are omplex onstants.This is a rotation BMS ve tor (cid:28)eld tangent to the u = const ut if H = 0 andthe onstants c i are purely imaginary c i = i R i . Therefore, taking the real part of rk a , we have three independent real divergen e free ve tor (cid:28)elds that are rotationBMS (cid:28)elds at the future null in(cid:28)nity, and, onversely, every rotation BMS ve tor(cid:28)eld determines a ve tor (cid:28)eld rk a . Thus in the asymptoti ally (cid:29)at ontext therole of the eigenvalue equation for the lowest eigenvalue is the unique extension ofthe BMS rotation ve tor (cid:28)elds o(cid:27) the future null in(cid:28)nity into a neighbourhoodof the future null in(cid:28)nity. In addition, taking the imaginary part of rk a wehave one divergen e free ve tor (cid:28)eld that is asymptoti ally vanishing as /r .However, the parameterization of the latter is (cid:28)xed by those of the real part of rk a . All the ambiguities in rk a (in luding the addition of `gauge solutions') anbe written as rk a rk a + r G ′ i K i a for some free (in general omplex) fun tions G ′ i of u .5.3.2 The ve tor (cid:28)eld z a − Sin e the ve tor (cid:28)eld z a − is real, the fun tion F = r F ( − + rF ( − + O (1) , forwhi h rz a − = ε ab δ b F , an also be hosen to be real. Following the strategy of theprevious subse tion we an determine the expli it form of the omponents of z a − in terms of the omponents of the spinor (cid:28)elds λ A and ¯ µ A ′ , and omparing themwith the de(cid:28)ning equation for F , we obtain a system of di(cid:27)erential equations for F ( − and F ( − . Apart from the gauge terms, the solution will be an expressionof λ (0)1 (cid:0) ð ¯ λ (0)0 ′ (cid:1) − λ (0)0 (cid:0) ð ′ ¯ λ (0)1 ′ (cid:1) = R + i R i t i , (5.31)where now the onstants R and R i are real and are given by23 := − √ π (cid:16) (0) c − ¯ c − + (0) ¯ c c + (0) ¯ c − c − + (0) c ¯ c (cid:17) , (5.32) R := i4 √ π (cid:16) (0) ¯ c c − − (0) c − ¯ c − (0) c ¯ c − + (0) ¯ c − c (cid:17) , (5.33) R := 14 √ π (cid:16) (0) ¯ c c − − (0) c − ¯ c + (0) c ¯ c − − (0) ¯ c − c (cid:17) , (5.34) R := i4 √ π (cid:16) (0) c − ¯ c − + (0) ¯ c c − (0) ¯ c − c − − (0) c ¯ c (cid:17) . (5.35)Note that R is independent of the R i , in ontrast to R and R i of (5.22)-(5.25).In terms of (5.31) the fun tions F ( − and F ( − take the simple, expli it form F ( − = − t i R i , (5.36) F ( − = − (cid:16)(cid:0) ð S (cid:1)(cid:0) ð ′ t i (cid:1) + (cid:0) ð ′ ¯ S (cid:1)(cid:0) ð t i (cid:1) + 12 (cid:0) S + ¯ S (cid:1) t i (cid:17) R i ++ i (cid:0) S − ¯ S (cid:1) R − t i G i , (5.37)where the last term represents the ambiguity oming from the gauge solutionsin the eigenspinor omponents λ (1) A ; and, for the sake of simpli ity, the irrelevant onstants of integration have been hosen to be zero. Comparing (5.36), (5.37)with (5.27), (5.28) we see that, apart from onstants and the ± sign, the oef-(cid:28) ients of R i oin ide, and the imaginary part of (5.28) (i.e. the oe(cid:30) ient of R ) is just the oe(cid:30) ient of the independent parameter R in (5.37). Therefore,the orresponding ve tor (cid:28)elds oin ide: the ve tor (cid:28)elds rz a − parameterized by R i are just the real part of the ve tor (cid:28)elds rk a , and the asymptoti ally vanish-ing ve tor (cid:28)eld, parameterized by R , is just the imaginary part of rk a of theprevious subse tion.5.4 2-surfa e observables at I + O [ N a ] In [8℄ we showed that (1) the basi Hamiltonian of va uum general relativity ona ompa t 3-manifold Σ with smooth 2-boundary S is fun tionally di(cid:27)erentiablewith respe t to the ADM anoni al variables if the area 2-form is (cid:28)xed on S ,the lapse fun tion is vanishing on S , and the shift ve tor N a is tangent to S anddivergen e free with respe t to the onne tion δ e ; (2) the evolution equationspreserve these boundary onditions; (3) the basi Hamiltonians form a losedPoisson algebra H , in whi h the onstraints form an ideal C ; (4) the value ofthe basi Hamiltonian on the onstraint surfa e, given expli itly by the integral O (cid:2) N a (cid:3) := − πG I S N e A e d S , (5.38)24s a well de(cid:28)ned, 2+2 ovariant, gauge invariant observable; and (5) this O pro-vides a Lie algebra (anti-)homomorphism of the Lie algebra of the δ e -divergen e-free ve tor (cid:28)elds on S into the quotient Lie algebra H / C of observables. How-ever, it should be noted that, independently of the quasi-lo al anoni al analysisabove, (5.38) already had appeared as a suggestion for the angular momentumof bla k hole horizons [30, 31, 32℄.We also onsidered the limit of this observable when the 2-surfa e S tendsto the spatial or the future null in(cid:28)nity. We showed that from the requirementof the (cid:28)niteness of the limit at spatial in(cid:28)nity it follows that N a is ne essarilya ombination of the asymptoti rotation Killing ve tors, and hen e the orre-sponding observable is just the familiar expression of spatial angular momentumthere. Unfortunately, however, at null in(cid:28)nity the general expression still on-tains a huge ambiguity. In fa t, if the fun tion ν de(cid:28)ned by N a = ε ab δ b ν iswritten as ν = r ν ( − + rν ( − + O (1) , then the r → ∞ limit of O [ N a ] is (cid:28)nitepre isely when ν ( − is a linear ombination of the (cid:28)rst four (i.e. j = 0 and 1)ordinary spheri al harmoni s. In this ase N a tends to a rotation BMS ve tor(cid:28)eld, and for the observable O [ N a ] , asso iated with a large sphere S r of radius r , we obtain O (cid:2) N a (cid:3) = i8 πG I (cid:16) (cid:0) ψ + σ ð ¯ σ (cid:1)(cid:0) ð ′ ν ( − (cid:1) − (cid:0) ¯ ψ + ¯ σ ð ′ σ (cid:1)(cid:0) ð ν ( − (cid:1) ++ σ (cid:0) ð ′ ν ( − (cid:1) − ¯ σ (cid:0) ð ν ( − (cid:1)(cid:17) d S + O (cid:0) r − (cid:1) , (5.39)where d S is the area element on the unit sphere. (In [8℄ ν ( − and ν ( − were denoted by ν (2) and ν (1) , respe tively.) Though this formula redu es tothe standard expression of angular momentum for stationary systems (whenthe potential S for the asymptoti shear an be hosen to be real), in generalspa etimes this is still ambiguous unless the fun tion ν ( − has been spe i(cid:28)ed.Or, in other words, the limit of O [ N a ] at the null in(cid:28)nity depends also on theway how the ve tor (cid:28)eld N a tends to the rotation BMS ve tor (cid:28)eld leaving the ut in question (cid:28)xed. Thus to have a well de(cid:28)ned angular momentum expressionin a general, radiative spa etime within the framework de(cid:28)ned by the observable O [ N a ] , a pres ription for ν ( − is needed.5.4.2 Spe tral angular momentum and a measure of the magneti part of the asymptoti shear at I + The aim of the present subse tion is to use the asymptoti rotation Killing(cid:28)elds rk a and rz a − as the ve tor (cid:28)eld N a and to al ulate the r → ∞ limitof the orresponding observable, hoping to obtain a reasonable de(cid:28)nition ofspatial angular momentum even in the presen e of outgoing gravitational radi-ation. One way of doing this might be based on the observation that N a A a = N a m a ( ¯ β − β ′ )+ N a ¯ m a ( β − ¯ β ′ ) = N ′ ¯ τ + N ′ τ . Substituting the omponents ofthe ve tor (cid:28)eld and the asymptoti form of the spin oe(cid:30) ient τ here, a ratherlengthy al ulation gives the desired expression.25owever, a mu h more e onomi method (and we adopt it here) is to use(5.39) by writing ν = F if N a is rk a or rz a − . Sin e only the se ond ð and ð ′ derivatives of ν ( − appear in (5.39), all the ambiguities are an eled from lim r →∞ O [ N a ] , i.e. the observable lim r →∞ O [ N a ] is well de(cid:28)ned and does notdepend on the gauge ambiguities of the solution λ (1) A . Thus, for the sake ofsimpli ity, the ambiguous terms in (5.28) and (5.37) an be hosen to be zero: G = 0 and G i = 0 .To determine the expli it form of the observable lim r →∞ O [ N a ] , re all thatthe fun tions ν ( − and ν ( − are linear fun tions of the parameters R i and R . Therefore, the r → ∞ limit of the observables O [ rk a ] and O [ rz a − ] have thestru ture ±
12 lim r →∞ O (cid:2) rk a (cid:3) =: R i J i + i R M , (5.40)
12 lim r →∞ O (cid:2) rz a − (cid:3) =: R i J i + R M , (5.41)respe tively. From the a tual form of F ( − it follows that J i and M are real.Re alling that the ve tor (cid:28)elds rk a and rz a − tend (or an be hosen to tend)asymptoti ally to the ombination − R i K i a of the rotation Killing (cid:28)elds of themetri sphere of radius r with real R i , whi h have in fa t unique extension toreal BMS ve tor (cid:28)elds tangent to the u = const ut of I + , the observable J i may be interpreted as the spatial angular momentum asso iated with the u = const ut of the future null in(cid:28)nity. Remarkably enough, the ve tor (cid:28)elds rk a and rz a − de(cid:28)ne the same observables J i and M , i.e. the J i 's and the M 's inequations (5.40) and (5.41) oin ide.The expli it form of J i is J i = i8 πG I n (cid:16) ψ + 2 σ ð ¯ σ (cid:17)(cid:0) ð ′ t i (cid:1) − (cid:16) ¯ ψ + 2¯ σ ð ′ σ (cid:17)(cid:0) ð t i (cid:1) ++ σ (cid:16) ð (cid:0) ð ′ S (cid:1)(cid:0) ð ′ t i (cid:1) + 12 (cid:0) ð ′ S (cid:1) t i (cid:17) −− ¯ σ (cid:16) ð ′ (cid:0) ð ¯ S (cid:1)(cid:0) ð t i (cid:1) + 12 (cid:0) ð ¯ S (cid:1) t i (cid:17)o d S . (5.42)It ould be interesting to note that the (cid:28)rst line gives just the spatial part ofBramson's angular momentum expression [33℄: in fa t, taking into a ount that ð ( σ ¯ σ )( ð ′ t i ) = ð ( σ ¯ σ ð ′ t i ) + σ ¯ σ t i , the (cid:28)rst line of the integrand an bewritten as the imaginary part of ( ψ + 2 σ ð ¯ σ + ð ( σ ¯ σ ))( ð ′ t i ) . Therefore,the whole J i an be interpreted as Bramson's angular momentum plus some` orre tion terms' built from the potential S in a gauge invariant way.Apart from the Weyl spinor omponent ψ all the terms of the integrand in(5.42) an be written into the form of a bilinear expression of the real and theimaginary parts of the potential S . In fa t, a trivial al ulation gives that theterms in the integrand of (5.42) ontaining t i algebrai ally an be written as26 (cid:16) ð (cid:0) S + ¯ S (cid:1) ð ′ (cid:0) S − ¯ S (cid:1) + ð ′ (cid:0) S + ¯ S (cid:1) ð (cid:0) S − ¯ S (cid:1)(cid:17) t i ; while the remaining terms ontaining the asymptoti shear as n (cid:16) ð (cid:0) S + ¯ S (cid:1) ð ð ′ (cid:0) S + ¯ S (cid:1) − ð (cid:0) S − ¯ S (cid:1) ð ð ′ (cid:0) S − ¯ S (cid:1)(cid:17)(cid:0) ð ′ t i (cid:1) −− (cid:16) ð ′ (cid:0) S + ¯ S (cid:1) ð ′ ð (cid:0) S + ¯ S (cid:1) − ð ′ (cid:0) S − ¯ S (cid:1) ð ′ ð (cid:0) S − ¯ S (cid:1)(cid:17)(cid:0) ð t i (cid:1)o ++ 14 n (cid:16) ð (cid:0) S − ¯ S (cid:1) ð ð ′ (cid:0) S + ¯ S (cid:1) − ð (cid:0) S + ¯ S (cid:1) ð ð ′ (cid:0) S − ¯ S (cid:1)(cid:17)(cid:0) ð ′ t i (cid:1) ++ (cid:16) ð ′ (cid:0) S − ¯ S (cid:1) ð ′ ð (cid:0) S + ¯ S (cid:1) − ð ′ (cid:0) S + ¯ S (cid:1) ð ′ ð (cid:0) S − ¯ S (cid:1)(cid:17)(cid:0) ð t i (cid:1)o . (5.43)However, the (cid:28)rst two lines of (5.43), i.e. the terms quadrati in ( S + ¯ S ) andin ( S − ¯ S ) , an be written into a total divergen e. Indeed, by taking totalderivatives and using the ommutator of the edth operators, for any fun tion f we obtain (cid:0) ð f (cid:1) ð (cid:0) ð ′ f (cid:1)(cid:0) ð ′ t i (cid:1) = ð (cid:16)(cid:0) ð f (cid:1) ð (cid:0) ð ′ f (cid:1)(cid:0) ð ′ t i (cid:1) + 12 (cid:0) ð ð ′ f (cid:1) (cid:0) ð ′ t i (cid:1)(cid:17) ++ ð ′ (cid:16)(cid:0) ð f (cid:1)(cid:0) ð ð ′ f (cid:1) t i − (cid:0) ð f (cid:1) ð ′ (cid:0) ð f (cid:1)(cid:0) ð ′ t i (cid:1)(cid:17) ++ (cid:16)(cid:0) ð f (cid:1)(cid:0) ð ′ f (cid:1) − (cid:0) ð ð ′ f (cid:1) (cid:17) t i , implying that ( ð f ) ð ( ð ′ f )( ð ′ t i ) − ( ð ′ f ) ð ′ ( ð f )( ð t i ) is a total diver-gen e. Applying this to f = S + ¯ S and to f = S − ¯ S in (5.43) we (cid:28)nally obtainthat the (cid:28)rst two lines in (5.43) do, indeed, form a total divergen e. This im-plies, in parti ular, that in stationary spa etimes, when S is real (see [34℄), J i redu es to the standard expression [35℄ i8 πG I (cid:16) ψ (cid:0) ð ′ t i (cid:1) − ¯ ψ (cid:0) ð t i (cid:1)(cid:17) d S . We also note that in axi-symmetri spa etime with Killing ve tor K a the ob-servable O [ K a ] on axi-symmetri surfa es is just the Komar expression [8℄. We all the J i given by (5.42) the spe tral angular momentum at I + .The expli it form of the other observable M is M = 116 πG I (cid:16) σ ¯ σ − (cid:0) ð ′ ¯ S (cid:1)(cid:0) ð ¯ S (cid:1) − (cid:0) ð ′ S (cid:1)(cid:0) ð S (cid:1)(cid:17) d S . (5.44)Clearly, this is real and is vanishing for purely ele tri σ , i.e. when S isreal. However, this is, in fa t, non-negative and zero pre isely for purely ele tri asymptoti shear; i.e. it de(cid:28)nes a measure of the presen e of the magneti part27f the asymptoti shear. To see this, let us rewrite its integrand (by integrationby parts and by using ð ð ′ S = ð ′ ð S ) as (cid:0) ð S (cid:1) ð ′ (cid:0) ¯ S − S (cid:1) + (cid:0) ð ′ ¯ S (cid:1) ð (cid:0) S − ¯ S (cid:1) ≃≃ (cid:0) ð ′ ð S (cid:1)(cid:0) ¯ S − S (cid:1) + (cid:0) ð ′ ¯ S (cid:1) ð (cid:0) S − ¯ S (cid:1) == (cid:0) ð ð ′ S (cid:1)(cid:0) ¯ S − S (cid:1) + (cid:0) ð ′ ¯ S (cid:1) ð (cid:0) S − ¯ S (cid:1) ≃≃ (cid:16) ð ′ (cid:0) ¯ S − S (cid:1)(cid:17)(cid:16) ð (cid:0) S − ¯ S (cid:1)(cid:17) , where ≃ means `equal up to total divergen es'. Sin e in stationary spa etimes(and hen e in the absen e of outgoing gravitational radiation) the asymptoti shear is always purely ele tri [34℄, the non-vanishing of M indi ates non-trivialdynami s of the gravitational (cid:28)eld near the future null in(cid:28)nity.From (5.42) and (5.44) it is lear that both J i and M are well de(cid:28)ned inthe sense that they depend only on the ut of I + . In parti ular, they do notdepend on the hoi e of the origin of u , and hen e J i is free of supertranslationambiguities. The notion of spe tral angular momentum is based on the solutionof a ertain ellipti equation on the ut rather than on the BMS ve tor (cid:28)elds.The latter is used only to interpret J i as spatial angular momentum. On theother hand, sin e the ve tor (cid:28)eld N e in the observable O [ N e ] must be tangentto the 2-surfa e, in the theoreti al framework based on O [ N e ] we annot askfor the e(cid:27)e t of (any form of) translations or boosts. Thus the spe tral angularmomentum J i should probably be interpreted only as the spin part of the (asyet not known) total relativisti angular momentum.5.4.3 Comparison of angular momenta on di(cid:27)erent utsIn the previous subse tion, we saw that J i is unambiguously asso iated withany ut of I + . In the present subse tion, we ask how the angular momentaasso iated with two di(cid:27)erent uts of I + , say ˆ S and ˜ˆ S , an be ompared. Firstwe dis uss the theoreti al basis of this omparison, and then we derive theformula for the (cid:29)ux of spe tral angular momentum arried away by the outgoinggravitational radiation in Einstein(cid:21)Maxwell spa etimes. (The hat over a symbolis referring to the unphysi al spa etime, indi ating that that may be ill-de(cid:28)nedin the physi al spa etime.)Mathemati ally J i is an element of the dual of the spa e R of the rotationBMS ve tor (cid:28)elds that are tangent to ˆ S . Hen e the spe tral angular momentaon the uts ˆ S and ˜ˆ S an be ompared only if we an (cid:28)nd a natural isomorphism I : R → ˜ R , yielding the identi(cid:28) ation of the orresponding dual spa es aswell. Formally, this isomorphism is analogous to that we already dis ussed in anutshell at the end of subse tion 5.2 in a slightly di(cid:27)erent ontext. Thus, let us(cid:28)x the Bondi onformal gauge on I + , and hen e both ˆ S and ˜ˆ S inherit the unitsphere metri , and let ˆ n a be the future pointing tangent of the null geodesi generators of I + su h that ˆ n a = − ˆ ∇ a Ω | I + . The spinor onstituent ˆ ι A of ˆ n A ˆ n a ˆ ∇ a ˆ ι B = 0 , and (cid:28)x itsphase. Then the spinor ˆ ι A an be ompleted to a spin frame { ˆ o A , ˆ ι A } at thepoints of the uts ˆ S and ˜ˆ S su h that the omplex null ve tors ˆ m a := ˆ o A ¯ˆ ι A ′ and ¯ˆ m a := ˆ ι A ¯ˆ o A ′ are tangent to the surfa es. This ˆ o A is uniquely determined by the uts, and hen e we have a uniquely determined omplex null basis { ˆ m a , ¯ˆ m a } onboth ˆ S and ˜ˆ S . (If the uts interse t ea h other, then, of ourse, the spinors ˆ o A at the points of interse tion do not oin ide unless the uts are tangent to ea hother there.) In addition, if we (cid:28)x a omplex stereographi oordinate system ( ζ, ¯ ζ ) on one of the uts, then by the ondition ˆ n a ˆ ∇ a ζ = 0 this determines a omplex stereographi oordinate system on the other ut (and, in fa t, on thewhole I + ) in a unique way. But then, re alling the stru ture (5.30) of therotation BMS ve tor (cid:28)elds, we have an isomorphism I : R → ˜ R : the ve tor(cid:28)eld ˆ K a ∈ R is identi(cid:28)ed with ˆ K a ∈ ˜ R via I pre isely when their omponents c i oin ide. It is this isomorphism by means of whi h we identify the dual spa es R ∗ and ˜ R ∗ with ea h other, and one an subtra t J i from ˜ J i dire tly. Thus weare ready to al ulate how J i hanges from ut to ut.Thus let u be the Bondi time oordinate whose origin is hosen to be ˆ S ,introdu e the 1-parameter family of uts ˆ S u obtained from ˆ S by BMS timetranslations along ˆ n a , denote the spe tral angular momentum on ˆ S u by J i ( u ) ,and al ulate its derivative with respe t to u at u = 0 . If, for the sake ofbrevity, we write the integrand of J i as ( ψ + F ) ð ′ t i − ( ¯ ψ + ¯ F ) ð t i (by using ð ð ′ t i = ð ′ ð t i = − t i ), then by (5.1) and (5.3) we (cid:28)nd ˙ J i = i8 πG I (cid:16) G (cid:0) ϕ ¯ ϕ (cid:0) ð ′ t i (cid:1) − ϕ ¯ ϕ (cid:0) ð t i (cid:1)(cid:1) ++ t i (cid:0) ¯ σ ˙ σ + 2 ð ′ (cid:0) σ ð ˙¯ σ (cid:1) − ð ′ ˙ F (cid:1) −− t i (cid:0) σ ˙¯ σ + 2 ð (cid:0) ¯ σ ð ′ ˙ σ (cid:1) − ð ˙¯ F (cid:1)(cid:17) d S . (5.45)The (cid:28)rst line is just the angular momentum (cid:29)ux arried away by the ele tro-magneti radiation, while the remaining terms an be interpreted as that arriedaway by the gravitational waves. In fa t, the (cid:28)rst two terms an also be writtenas Φ i := 18 π lim r →∞ I S r n a T ab (cid:16) ˆ m b (cid:0) ð ′ t i (cid:1) − ¯ˆ m b (cid:0) ð t i (cid:1)(cid:17) d S r , where T ab = 2 ϕ AB ¯ ϕ A ′ B ′ , the energy-momentum tensor of the Maxwell (cid:28)eld.Re alling the form (5.30) of the rotation BMS ve tor (cid:28)eld, Φ i du is just the an-gular momentum urrent arried away by the ele tromagneti radiation betweenthe u and u + du retarded times.Moreover, re all that by (5.2) the time derivative of the Bondi(cid:21)Sa hs energy-momentum, P a := − πG H ( ψ + σ ˙¯ σ ) t a d S , is the integral of − πG ( ˙ σ ˙¯ σ +2 Gϕ ¯ ϕ ) t a , where t a := (1 , t i ) (see e.g. [34, 27, 6℄). Thus the vanishing of theoutgoing energy (cid:29)ux is equivalent to ˙ σ = 0 and ϕ = 0 . Clearly, if ˙ σ = 0 ,then the orresponding shear potential is also time independent: ˙ S = 0 . Then,29owever, equation (5.45) shows, in parti ular, that in the absen e of outgoingenergy (cid:29)ux the outgoing spe tral angular momentum (cid:29)ux is also vanishing. Ob-viously, every angular momentum expression shares this property whi h an bewritten as the integral of the imaginary part of ( ψ + F ) ð ′ t i with some gaugeinvariant expression F of S and ¯ S .Although we ould onsider a more general family of uts obtained from ˆ S by a general supertranslation along H ˆ n a with a general H = H ( ζ, ¯ ζ ) , butthe resulting formula does not seem to yield mu h deeper understanding ofgravitational radiation. Te hni ally, the use of su h a more general foliation of I + yields only that the ˆ o A spinor of the spin frame adapted to the foliationundergoes the spe ial null rotation ˆ o A ˆ o A +( ð H )ˆ ι A , and the u -derivatives ˙ ψ and ˙ σ have to be substituted by H ˙ ψ +3( ð H ) ψ and H ˙ σ − ð H , respe tively.The ultimate answer whether or not the spe tral angular momentum andthe observable M have physi al signi(cid:28) an e will be given by the pra ti e. Inparti ular, it ould be interesting to see whether or not the inequalities like in[36, 37℄ an be proven for J i too, or whether in the uni(cid:28)ed model of spatialand null in(cid:28)nity [38℄ the u → −∞ limit of J i redu es to the ADM angularmomentum.6 Appendix6.1 Hermitian (cid:28)bre metri s on the spinor bundleClearly, any Hermitian (cid:28)bre metri G AA ′ on S A ( S ) an also be onsidered as areal se tion G a of the (dual) Lorentzian ve tor bundle V a ( S ) , de(cid:28)ned to be thepull ba k to S of the spa etime otangent bundle T ∗ M . This G AA ′ is positivede(cid:28)nite (and hen e nonsingular) i(cid:27) it is future pointing and timelike in V a ( S ) .Su h a metri an be spe i(cid:28)ed by four real fun tions on S .Next introdu e the inverse G AA ′ of G AA ′ by G AA ′ G BA ′ = δ AB . Then from ( ε AB ε A ′ B ′ G BB ′ ) G CA ′ = δ AC G e G f g ef we see that the inverse G AA ′ is just the ontravariant form of G AA ′ , i.e. G AA ′ = ε AB ε A ′ B ′ G BB ′ (or, equivalently,the symple ti and the Hermitian metri s are ompatible in the sense that ε AB G AA ′ G BB ′ = ε A ′ B ′ ) i(cid:27) G a G b g ab = 2 . This normalization redu es the inde-pendent omponents of G AA ′ to three. However, su h a (cid:28)bre metri an be spe- ialized further by requiring its ompatibility with the hirality: G AA ′ γ AB ¯ γ A ′ B ′ = G BB ′ . Sin e γ AB ¯ γ A ′ B ′ , as a base-point preserving bundle map V a ( S ) → V a ( S ) a ts as identity pre isely on se tions orthogonal to S (and as minus theidentity pre isely on se tions tangent to S ), this ompatibility is equivalent tothe orthogonality of G a to S . The independent omponents of su h a metri isonly one. In a (cid:28)xed, normalized GHP spin frame { o A , ι A } , adapted to S , it hasthe general form G AA ′ = go A ¯ o A ′ + (1 /g ) ι A ¯ ι A ′ , where g is an arbitrary, stri tlypositive real fun tion on S . Note that for orientable S in a time and spa eorientable spa etime there is no obstru tion to the global existen e of su h a G AA ′ , be ause in this ase the normal bundle of S in the spa etime is globallytrivializable. 30n general 2-surfa es there does not seem to be any natural hoi e for su ha G AA ′ , or, equivalently, for su h a g . If, however, the 2-surfa e is mean- onvex,i.e. when the dual mean urvature ve tor of S is timelike, then there is su h a hoi e. In fa t, the mean urvature ve tor Q b := Q aab = − ρ ′ o A ¯ o A ′ + ρι A ¯ ι A ′ ) and its dual, ˜ Q b := ⊥ ε ba Q a = 2( ρ ′ o A ¯ o A ′ − ρι A ¯ ι A ′ ) , are globally de(cid:28)ned andorthogonal to S , furthermore | Q | := ˜ Q a ˜ Q a = − Q a Q a = − ρρ ′ . Then for mean onvex surfa es either ρ > and ρ ′ < or ρ < and ρ ′ > , and it is naturalto de(cid:28)ne G AA ′ := ±√ Q AA ′ / | ˜ Q | = ± ( ρ ′ o A ¯ o A ′ − ρι A ¯ ι A ′ ) / p | ρρ ′ | with the signyielding future pointing G a .6.2 Non-existen e of onstant Hermitian (cid:28)bre metri sWe show that the existen e of a positive de(cid:28)nite Hermitian metri on the spinorbundle ompatible with the ovariant derivative operator δ e is equivalent to thetriviality of the holonomy of δ e on the normal bundle.Let G AA ′ be a Hermitian s alar produ t on the spinor bundle S A ( S ) andassume that δ e G AA ′ = 0 . Considering this metri to be a Lorentzian ve tor (cid:28)eldon S , the integrability ondition of δ e G AA ′ = 0 is δ c δ d − δ d δ c ) G b = G a f abcd .Contra ting this with the area 2-form ε cd and taking into a ount the expli itexpression for the urvature f abcd given in subse tion 2.1, we obtain (cid:16) Rε ab − ε cd (cid:0) δ c A d (cid:1) ⊥ ε ab (cid:17) G b = 0 . (6.1)This implies that δ e admits a onstant se tion of the ve tor bundle V a ( S ) onlyif δ e is (cid:29)at. Restri ting G a to be normal to S we (cid:28)nd in parti ular that δ e admits a onstant se tion of the normal bundle only if A e is (cid:29)at. (By (6.1) δ e admits a onstant se tion of the tangent bundle only if ( S , q ab ) is (cid:29)at, whi h,by the Gauss(cid:21)Bonnet theorem, an happen only if S is a torus.) Therefore, thene essary ondition of the existen e of a δ e - onstant positive de(cid:28)nite Hermitianmetri , being ompatible with γ AB , is the (cid:29)atness of the onne tion δ e on thenormal bundle. Then, however, the globality of G a implies that the onne tionis holonomi ally trivial too. To see this, onsider a losed urve γ : [0 , → S with the base point p := γ (0) = γ (1) , and suppose, on the ontrary, that theholonomy H γ : N p S → N p S , as an element of the stru ture group SO (1 , ,is di(cid:27)erent from identity. Sin e G a is globally de(cid:28)ned on S and onstant withrespe t to δ e , the holonomy H γ a ts on G ap (the value of G a at p ) as the identity.Sin e the only element of SO (1 , leaving the nonzero ve tor G a (cid:28)xed is theidentity, the holonomy H γ is the identity, and hen e the whole holonomy groupat every point p , must be trivial.Conversely, the triviality of the holonomy of the onne tion δ e on the normalbundle and the global trivializability of N S imply the existen e of a globallyde(cid:28)ned orthonormal δ e - onstant frame (cid:28)eld { t a , v a } with future pointing andtimelike t a . Then e.g. G AA ′ := √ t AA ′ is a desired positive de(cid:28)nite Hermitians alar produ t on the spinor bundle. 31he author is grateful to Jörg Frauendiener and Ted Newman for the usefuldis ussions on the angular momentum at null in(cid:28)nity. This work was partiallysupported by the Hungarian S ienti(cid:28) Resear h Fund (OTKA) grants T042531and K67790.Referen es[1℄ J. 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