Evolution of angular momentum and center of mass at null infinity
Po-Ning Chen, Jordan Keller, Mu-Tao Wang, Ye-Kai Wang, Shing-Tung Yau
aa r X i v : . [ g r- q c ] F e b EVOLUTION OF ANGULAR MOMENTUM AND CENTEROF MASS AT NULL INFINITY
PO-NING CHEN, JORDAN KELLER,MU-TAO WANG, YE-KAI WANG, AND SHING-TUNG YAU
Abstract.
We study how conserved quantities such as angular mo-mentum and center of mass evolve with respect to the retarded timeat null infinity, which is described in terms of a Bondi-Sachs coordinatesystem. These evolution formulae complement the classical Bondi massloss formula for gravitational radiation. They are further expressed interms of the potentials of the shear and news tensors. The consequencesthat follow from these formulae are (1) Supertranslation invariance ofthe fluxes of the CWY conserved quantities. (2) A conservation law ofangular momentum `a la Christodoulou. (3) A duality paradigm for nullinfinity. In particular, the supertranslation invariance distinguishes theCWY angular momentum and center of mass from the classical defini-tions. Introduction
In this article, we study the evolution of angular momentum and centerof mass at null infinity of asymptotically flat vacuum spacetimes. Theseevolution formulae complement the classical Bondi mass loss formula forgravitational radiations. We are particularly interested in the total flux ofangular momentum and center of mass.For a good notion of conserved quantities, one expects that the total fluxis independent of the choice of coordinate systems. However, as indicatedby Penrose [19], the notion of “angular momentum carried away by gravita-tional radiation” can be shifted by supertranslations, an infinite dimensionalsymmetry at null infinity. Such ambiguity has been a crucial obstacle to aclear understanding of conserved quantities at null infinity. In this article,we consider both the classical and the Chen-Wang-Yau (CWY) [4] defini-tions for angular momentum and center of mass at null infinity. A key result
P.-N. Chen is supported by NSF grant DMS-1308164 and Simons Foundation collabora-tion grant is the supertranslation invariance of the flux of the CWY angular momen-tum and center of mass. This invariance distinguishes the CWY definitionsfrom the classical definitions.Consider the future null infinity I + of an asymptotically flat spacetime,which is described in terms of a Bondi-Sachs coordinate system. I + isidentified with I × S , where I ⊂ ( −∞ , + ∞ ) is an interval parametrized bythe retarded time u and S is the standard unit 2-sphere equipped with thestandard round metric σ AB . Let m denote the mass aspect, N A the angularmomentum aspect, C AB the shear tensor, and N AB the news tensor of I + .One can view m as a smooth function, N A a smooth one-form, and C AB and N AB smooth symmetric traceless 2-tensors (with respect to σ AB ) on S that depend on u . In particular, ∂ u C AB = N AB . See a brief description of I + in the Bondi-Sachs coordinates and the definitions of these quantitiesin Section 2.All integrals in this paper on the sphere are taken over the standardtwo-sphere S with the standard round metric σ AB . We take the standardformulae for energy and linear momentum: E = Z S mP k = Z S m ˜ X k , k = 1 , , X k , k = 1 , , R restrictedto the unit sphere S .Furthermore, we consider the classical angular momentum˜ J k = Z S ǫ AB ∇ B ˜ X k [ N A − C DA ∇ B C DB ] , (1.2)and the classical center of mass˜ C k = Z S ∇ A ˜ X k [ N A − u ∇ A m − C DA ∇ B C DB − ∇ A ( C DE C DE )] , (1.3)where ∇ A denotes the covariant derivative with respect to σ AB , and ǫ AB denotes the volume form of σ AB and k = 1 , ,
3. The indexes are raised,lowered, and contracted with respect to σ AB . Our definition is that ofDray-Streubel [12]. See Section III.B of Flanagan-Nichols [13] for details. Remark 1.1.
In the above definitions of conserved quantities, we omit theconstant π .Furthermore, we consider the CWY angular momentum J k and centerof mass C k as the limits of the CWY quasi-local angular momentum andcenter of mass [4, 5] on I + evaluated in [15]. J k = Z S ǫ AB ∇ B ˜ X k (cid:18) N A − C AB ∇ D C DB − c ∇ A m (cid:19) VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 3 C k = Z S ∇ A ˜ X k " N A − u ∇ A m − C AB ∇ D C DB − ∇ A (cid:0) C DE C DE (cid:1) − c ∇ A m + 2 ǫ AB ( ∇ B c ) m + Z S X k cm −
14 ˜ X k ∇ A F AB ∇ D F DB where c and c are the potentials of C AB , as given in (2.9) and F AB = ( ǫ AD ∇ B ∇ D c + ǫ BD ∇ A ∇ D c ) . For definiteness, the potentials are assumedto be supported in the ℓ ≥ J k and C k E are thelimit of the Chen-Wang-Yau quasi-local angular momentum and center ofmass (omitting constant 1 / π ) under the zero linear momentum assumption Z S m ( u, x ) ˜ X i = 0 . (1.4)The CWY angular momentum and center of mass modify the classical defi-nitions as follows: J k = ˜ J k − Z S ǫ AB ∇ B ˜ X k c ∇ A mC k = ˜ C k + Z S ∇ A ˜ X k (cid:0) − c ∇ A m + 2 ǫ AB ( ∇ B c ) m (cid:1) + Z S X k cm −
14 ˜ X k ∇ A F AB ∇ D F DB (1.5)The correction terms come from solving the optimal isometric embeddingequation in the theory of Wang-Yau quasilocal mass [25, 26] and are non-local. They provide the reference terms that are critical in the Hamiltonianapproach of defining conserved quantities. See [16] for a definition of angularmomentum in the context of perturbations of Kerr, in which the referencingis achieved by the uniformization theorem.The ten conserved quantities ( E, P k , ˜ J k , ˜ C k ), or ( E, P k , J k , C k ), are func-tions on I that depend on the retarded time u . We compute the derivativesof these conserved quantities with respect to u . In particular, for the classicalangular momentum and center of mass, we obtain Theorem 1.2.
The classical angular momentum ˜ J k and center of mass ˜ C k , k = 1 , , evolve according to the following: ∂ u ˜ J k = 14 Z S h ǫ AE ∇ E ˜ X k ( C AB ∇ D N BD − N AB ∇ D C BD ) + ˜ X k ǫ AB ( C DA N DB ) i , (1.6) ∂ u ˜ C k = 14 Z S h ∇ A ˜ X k (cid:16) u ∇ A | N | + C AB ∇ D N BD − N AB ∇ D C BD (cid:17)i . (1.7) P.-N. CHEN, J. KELLER, M.-T. WANG, Y.-K. WANG, AND S.-T. YAU
The evolution formulae (1.6) and (1.7) can be further expressed in termsof the potentials of C AB and N AB : Theorem 1.3.
Suppose c and c are the potentials of C AB and n and n arethe potentials of N AB , as given in (2.9) and (2.10) , then ∂ u ˜ J k = 18 Z S ˜ X k ([ c, ∆(∆ + 2) n ] + [ c, ∆(∆ + 2) n ] ) ∂ u ˜ C k = 18 Z S ˜ X k (cid:16) u [((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n ]+ [(∆ + 2) c, (∆ + 2) n ] + [(∆ + 2) c, (∆ + 2) n ] (cid:17) , (1.8) where [ · , · ] is the Poisson bracket on S defined in (4.1) and [ · , · ] is anotherbracket on S defined in (4.2) . The Bondi-Metzner-Sachs (BMS) group acts on I + . It includes super-translations which we will review in further details in Section 5. The ambigu-ity of supertranslations has presented an essential difficulty to understandingthe structure of I + since the 1960s. Among ( m, N A , C AB , N AB ), only N AB is a supertranslation invariant quantity. It is natural to ask whether totalflux of angular momentum is invariant under a supertranslation. For theclassical angular momentum, we prove that Corollary 1.4 (Theorem 5.1) . Suppose I + extends from u = −∞ to u =+ ∞ and the news tensor decays as N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ , then the total flux of the classical angular momentum ˜ J k is supertranslationinvariant if and only if lim u → + ∞ m ( u, x ) − lim u →−∞ m ( u, x ) (1.9) is supported in the l ≤ modes. In particular, if lim u → + ∞ m ( u, x ) − lim u →−∞ m ( u, x ) contains l ≥ Theorem 1.5 (Theorem 5.4) . Suppose the news tensor decays as N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ . Then the total flux of J k is supertranslation invariant. Remark 1.6.
In the above statement, supertranslation invariant meansthat it is equivariant under ordinary ( l = 1) translation and is invariant VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 5 under higher mode ( l ≥
2) of the supertranslation. See the statement ofTheorem 5.4 for further details.We also show that the invariance under supertranslation distinguishesthe CWY center of mass from the classical center of mass. Indeed, the totalflux of the classical center of mass is invariant under supertranslation if andonly if lim u → + ∞ m ( u, x ) − lim u →−∞ m ( u, x ) is a constant function on S .On the other hand, the total flux of the CWY center of mass is alwayssupertranslation invariant. See the statement of Theorem 5.5.Next, we show that if a spacetime admits a Bondi-Sachs coordinate systemwith vanishing news tensor, then ( E, P k , J k , C k ) are constant (independentof the retarded time u ) and supertranslation invariant. See the statementof Theorem 6.2 for further details.While our focus is on the study of angular momentum and center of massin a Bondi-Sachs coordinate system, we show that the evolution formulaefor the classical angular momentum can be carried over to the frameworkof the stability of Minkowski spacetime [9] if we take Rizzi’s definition ofangular momentum [20, 21]. This provides a conservation law of angularmomentum that complements the conservation law for linear momentum ofChristodoulou [7, Equation (13)].Another natural consequence of (1.8) is a duality paradigm among setsof null infinity data ( m, N A , C AB , N AB ), through replacing the potentials( c, c, n, n ) by ( − c, c, − n, n ). Corollary 1.7.
Given a set of null infinity data ( m, N A , C AB , N AB ) definedon [ u , u ] × S , there exists a dual set of null infinity data ( m ∗ , N ∗ A , C ∗ AB , N ∗ AB ) that has the same (classical) energy, linear momentum, angular momentum,and center-of-mass. These are dual sets of null infinity data that are indistinguishable in termsof the classical conserved quantities.The paper is organized as follows. In Section 2, we introduce the defini-tions and integration by parts formulae used throughout the paper. The fluxof classical conserved quantities is computed in Section 3 and is rewrittenin terms of the potentials in Section 4. The aforementioned consequencesof flux formulae are presented in Section 5 to Section 7. In the last section,we consider the case of quadrupole moment radiation. With the future the-oretical and numerical investigation in mind, we express the flux formulaein terms of the spherical harmonics expansion of potentials explicitly.2.
Background information
In this section, we describe the Bondi-Sachs coordinate system and recallseveral useful formulae for functions and tensors on S . P.-N. CHEN, J. KELLER, M.-T. WANG, Y.-K. WANG, AND S.-T. YAU
Bondi-Sachs coordinates.
In terms of a Bondi-Sachs coordinate sys-tem ( u, r, x , x ), near I + of a vacuum spacetime, the metric takes the form g αβ dx α dx β = − U V du − U dudr + r h AB ( dx A + W A du )( dx B + W B du ) . (2.1)The index conventions here are α, β = 0 , , , A, B = 2 ,
3, and u = x , r = x . See [2, 17] for more details of the construction of the coordinatesystem.The metric coefficients U, V, h AB , W A of (2.1) depend on u, r, θ, φ , butdet h AB is independent of u and r . These gauge conditions thus reducethe number of metric coefficients of a Bondi-Sachs coordinate system to six(there are only two independent components in h AB ). On the other hand,the boundary conditions U → V → W A → h AB → σ AB are imposedas r → ∞ (such boundary conditions may not be satisfied in a radiativespacetime). Here σ AB denotes a standard round metric on S . The specialgauge choice implies a hierarchy among the vacuum Einstein equations, see[17, 14].Assuming the outgoing radiation condition [2, 22, 17], the boundary con-dition and the vacuum Einstein equation imply that as r → ∞ , all metriccoefficients can be expanded in inverse integral powers of r . In particular(see Chrusciel-Jezierski-Kijowski [10, (5.98)-(5.100)] for example), U = 1 − r | C | + O ( r − ) ,V = 1 − mr + 1 r (cid:18) ∇ A N A + 14 ∇ A C AB ∇ D C BD + 116 | C | (cid:19) + O ( r − ) ,W A = 12 r ∇ B C AB + 1 r (cid:18) N A − ∇ A | C | − C AB ∇ D C BD (cid:19) + O ( r − ) ,h AB = σ AB + C AB r + 14 r | C | σ AB + O ( r − )where m = m ( u, x A ) is the mass aspect, N A = N A ( u, x A ) is the angularaspect and C AB = C AB ( u, x A ) is the shear tensor of this Bondi-Sachs co-ordinate system. Note that our convention of angular momentum aspectdiffers from that of Chrusciel-Jezierski-Kijowski [10], N A = − N A ( CJK ) .Here we take norm, raise and lower indices of tensors with respect to themetric σ AB . We also define the news tensor N AB = ∂ u C AB .2.2. Integral formulae on 2-sphere.
Let σ AB be the standard roundmetric on S with respect to which the indexes of tensors are raised or The outgoing radiation condition assumes the traceless part of the r − term in theexpansion of h AB is zero. The presence of this traceless term will lead to a logarithmicterm in the expansions of W A and V . Spacetimes with metrics which admit an expansionin terms of r − j log i r are called “polyhomogeneous” and are studied in [11]. They do notobey the outgoing radiation condition or the peeling theorem [23], but they do appear asperturbations of the Minkowski spacetime by the work of Christodoulou-Klainerman [9]. VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 7 lowered. Let ∇ A be covariant derivative with respect to σ AB . Let ǫ AB bethe volume form. The following identity ǫ AB ǫ CD = σ AC σ BD − σ AD σ BC (2.2)and its contraction ǫ AB ǫ AC = σ BC (2.3)will be used frequently.The curvature formula on S gives ∇ A ∇ B ∇ C u − ∇ B ∇ A ∇ C u = σ AC ∇ B u − σ BC ∇ A u for a smooth function u on S . In particular, we have ∇ D ∇ D ∇ A u = ∇ A (∆ + 1) uǫ AB ∇ A ∇ B ∇ C u = ǫ BC ∇ B u. (2.4)Let ˜ X k , k = 1 , , S of the standard coordinatefunctions in R . It is well-known that they are eigenfunctions for σ AB :∆ ˜ X k = − X k . ˜ X k also satisfies the Hessian equation ∇ A ∇ B ˜ X k = − ˜ X k σ AB . (2.5)In general, an eigenfunction f with∆ f = − ℓ ( ℓ + 1) f (2.6)is said to be of mode ℓ . We need the following integration by parts lemma: Lemma 2.1.
Suppose u and v are smooth functions on S of mode m and n respectively. Then Z S ˜ X k ǫ AB ∇ A u ∇ B v = 0 unless m = n .Proof. Integrating by parts, we obtain Z S ˜ X k ǫ AB ∇ A u ∇ B v = Z S ( Y A ∇ A v ) u, where Y A = ǫ AB ∇ B ˜ X k is a rotation Killing field. Since ∆ commutes with Y A ∇ A , Y A ∇ A v is of the same mode as v . (cid:3) The following integrating by parts formulae will be useful in the latersections.
Lemma 2.2.
For any smooth functions u, v on S , we have Z S ˜ X k ǫ AB ∇ A (∆ u ) ∇ B v = Z S ˜ X k ǫ AB ∇ A u ∇ B (∆ v ) (2.7) Z S ˜ X k ǫ AB ∇ A ∇ D u ∇ B ∇ D v = − Z S ˜ X k ǫ AB ∇ A u ∇ B (∆ + 2) v. (2.8) P.-N. CHEN, J. KELLER, M.-T. WANG, Y.-K. WANG, AND S.-T. YAU
Proof.
We prove the second formula and the first formula follows similarly.Integrating by parts the left hand side, we obtain − Z S ∇ D ˜ X k ǫ AB ∇ A u ∇ B ∇ D v − Z S ˜ X k ǫ AB ∇ A u ∇ D ∇ B ∇ D v Integrating the first term by parts again, we obtain Z S ∇ B ∇ D ˜ X k ǫ AB ∇ A u ∇ D v − Z S ˜ X k ǫ AB ∇ A u ∇ D ∇ B ∇ D v By (2.4), this is equal to − Z S ˜ X k ǫ AB ∇ A u ∇ B v − Z S ˜ X k ǫ AB ∇ A u ∇ B (∆ + 1) v. (cid:3) Lemma 2.3.
For any smooth function u on S , we have Z S [2 ∇ A ∇ B u ∇ A ∇ B u − (∆ u ) ] = Z S u ∆(∆ + 2) u Z S ˜ X i [2 ∇ A ∇ B u ∇ A ∇ B u − (∆ u ) ] = Z S ˜ X i [(∆ + 2) u ] . Proof.
We use the following formulae in the derivation∆ |∇ u | = 2 |∇ u | + 2 ∇ u · ∇ (∆ + 1) u ∆( u ) = 2 |∇ u | + 2 u ∆ u ∆( u ∆ u ) = (∆ u ) + 2 ∇ u · ∇ (∆ u ) + u ∆ u. We prove the second formula and the first one follows similarly. Integrat-ing by parts twice gives Z S ˜ X i ∇ A ∇ B u ∇ A ∇ B u = Z S u ∇ A ∇ B ( ˜ X i ∇ A ∇ B u )We compute ∇ A ∇ B ( ˜ X i ∇ A ∇ B u )=( ∇ A ∇ B ˜ X i ) ∇ A ∇ B u + 2 ∇ B ˜ X i ∇ A ∇ A ∇ B u + ˜ X i ∇ A ∇ B ∇ A ∇ B u = − ˜ X i ∆ u + 2 ∇ B ˜ X i ∇ B (∆ + 1) u + ˜ X i ∆(∆ + 1) u = ˜ X i ∆ u + 2 ∇ B ˜ X i ∇ B (∆ + 1) u , where we use ∇ A ∇ A ∇ B u = ∇ B (∆ + 1) u .On the other hand, we have the identity:2 ∇ B u ∇ B v = ∆( uv ) − u ∆ v − v ∆ u and thus2 ∇ B ˜ X i ∇ B (∆ + 1) u = ∆( ˜ X i (∆ + 1) u ) − ˜ X i ∆(∆ + 1) u + 2 ˜ X i (∆ + 1) u. VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 9
Putting all together gives: Z S ˜ X i ∇ A ∇ B u ∇ A ∇ B u = Z S ˜ X i ∆ u + Z u [∆( ˜ X i (∆ + 1) u ) − ˜ X i ∆(∆ + 1) u + 2 ˜ X i (∆ + 1) u ]= Z S ˜ X i [(∆ u ) + 2 u ∆ u + 2 u ] . Therefore, Z S ˜ X i [2 ∇ A ∇ B u ∇ A ∇ B u − (∆ u ) ] = Z S ˜ X i [(∆ u ) +4 u ∆ u +4 u ] = Z S ˜ X i [(∆+2) u ] . (cid:3) Closed and Co-closed Decomposition.
In this subsection, we con-sider symmetric traceless 2-tensors C AB and N AB on S with the decompo-sition (see [15, Appendix B] for a derivation) C AB = ∇ A ∇ B c − σ AB ∆ c + 12 ( ǫ EA ∇ E ∇ B c + ǫ EB ∇ E ∇ A c ) (2.9) N AB = ∇ A ∇ B n − σ AB ∆ n + 12 ( ǫ EA ∇ E ∇ B n + ǫ EB ∇ E ∇ A n ) (2.10)for smooth functions c, c, n, n on S that are referred as potentials of C AB and N AB . The potentials are unique up to their 0 and 1 mode. In the casewe consider when C AB and N AB depend on u , all c, c, n, n depend on u aswell. Proposition 2.4.
Closed and co-closed parts of a symmetric traceless 2-tensors on S are dual to each other in the following sense. (1) Denote the space of symmetric traceless 2-tensors on S by d Sym.Then the map ε : d Sym → d Sym , ε ( C AB ) = ǫ DA C DB satisfies ε ( ∇ A ∇ B c − σ AB ∆ c ) = 12 ( ǫ EA ∇ E ∇ B c + ǫ EB ∇ E ∇ A c ) , (2.11) ε (cid:18)
12 ( ǫ EA ∇ E ∇ B c + ǫ EB ∇ E ∇ A c ) (cid:19) = −∇ A ∇ B c + 12 σ AB ∆ c. (2.12)(2) The following identity holds for symmetric traceless 2-tensors ǫ BD ∇ D C BA = ǫ DA ∇ B C BD . (2.13) In other words, we have a commutative diagram of isomorphisms d Sym ε −−−−→ d Sym y div y div Λ ∗ −−−−→ Λ , where Λ denotes the space of 1-forms and ( ∗ ω ) A = ǫ BA ω B is theHodge star on 1-forms. Proof.
We use (2.2) and (2.3) in the derivation. Since ǫ AB ε ( C AB ) = 0 and σ AB ε ( C AB ) = 0, ε ( C AB ) is symmetric and traceless. In particular, ǫ DA C DB = 12 ( ǫ DA C DB + ǫ DB C DA ) (2.14)and (2.11) and (2.12) follow by direct computation.To verify (2.13), note that both sides are equal to ∇ D C DE after contractedwith ǫ AE . (cid:3) In the following two lemmas, we express several integrals involving theshear tensor and the news tensor in terms of their potentials. These formulaewill help us to derive Theorem 1.3 from Theorem 1.2.
Lemma 2.5.
Suppose Y A is either ∇ A ˜ X k or ǫ AB ∇ B ˜ X k , and C AB and N AB are given by (2.9) and (2.10) , then Z S Y A C AB ∇ D N BD = − Z S Y A [(∆ + 2) n ∇ A (∆ + 2) c + (∆ + 2) n ∇ A (∆ + 2) c ]+ 14 Z S Y A ǫ DA [ ∇ D ((∆ + 2) c )(∆ + 2) n − ∇ D ((∆ + 2) c )(∆ + 2) n ](2.15) Proof.
First of all, note that ∇ B Y A C AB = 0 , ǫ BD ∇ D Y A C AB = 0 . From ∇ D N BD = 12 ∇ B (∆ + 2) n + 12 ǫ BD ∇ D (∆ + 2) n, we integrate by parts to get Z S Y A C AB ∇ D N BD = − Z S Y A ( ∇ B C AB (∆+2) n + ǫ BD ∇ D C AB (∆+2) n ) . By (2.13) ǫ DB ∇ D C BA = ǫ AD ∇ B C BD and ∇ B C BD = ∇ D (∆ + 2) c + ǫ BD ∇ B (∆ + 2) c , we obtain the desiredformula. (cid:3) The above generalizes the integral identities derived in [15, (65), (66)]: Z S Y A F BA ∇ D F DB = 0 , Z S Y A F BA ∇ D F DB = 0for Y A = ǫ AB ∇ B ˜ X k . VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 11
Skew-symmetrizing (2.15), we obtain: Z S Y A ( C AB ∇ D N BD − N AB ∇ D C BD )= 14 Z S Y A [(∆ + 2) c ∇ A (∆ + 2) n − (∆ + 2) n ∇ A (∆ + 2) c ]+ 14 Z S Y A [(∆ + 2) c ∇ A (∆ + 2) n − (∆ + 2) n ∇ A (∆ + 2) c ]+ 14 Z S Y A ǫ DA ∇ D [(∆ + 2) c (∆ + 2) n − (∆ + 2) c (∆ + 2) n ] . (2.16)Next we prove Lemma 2.6. Z S N AB N AB = 12 Z S n ∆(∆ + 2) n + n ∆(∆ + 2) n Z S ˜ X k N AB N AB = 12 Z S ˜ X k h ((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n i . Proof.
Using the formula ǫ CA ǫ BD = δ BA σ CD − δ DA σ CB and ǫ AB ǫ AE = σ BE ,we compute that N AB N AB = ∇ A ∇ B n ∇ A ∇ B n −
12 (∆ n ) + ∇ A ∇ B n ∇ A ∇ B n −
12 (∆ n ) + 2 ǫ AC ∇ A ∇ B n ∇ C ∇ B n (2.17)Integrating by parts yields Z S ǫ AC ∇ A ∇ B n ∇ C ∇ B n = − Z S ǫ AC ∇ B ∇ A ∇ B n ∇ C n = − Z S ǫ AC ∇ A (∆ + 1) n ∇ C n = 0The first formula now follows from the first formula in Lemma 2.3. Thesecond formula follows from the second and third formula in Lemma 2.3. (cid:3) The second formula of Lemma 2.6 can be polarized and we obtain Z S ˜ X k C AB N AB = 12 Z S ˜ X k h (∆ + 2) c (∆ + 2) n + (∆ + 2) c (∆ + 2) n − ǫ AB ( ∇ A c ∇ B (∆ + 2) n + ∇ A n ∇ B (∆ + 2) c ) i (2.18)3. Evolution of Conserved quantities
In this section, we compute the evolution of the classical angular momen-tum and center of mass. These formulae will be used to calculate the totalflux of the conserved quantities.
Let’s first review the evolution of the metric under the Einstein equation.It is well-known (see [10, (5.102)] for example) that the evolution of the massaspect function is given by ∂ u m = − N AB N AB + 14 ∇ A ∇ B N AB . (3.1)The modified mass aspect function b m is defined to be [24] b m = m − ∇ A ∇ B C AB = m −
18 ∆(∆ + 2) c (3.2)and satisfies ∂ u b m = − N AB N AB . (3.3)Therefore, ∂ u E = − Z S N AB N AB ∂ u P k = − Z S ˜ X k N AB N AB , k = 1 , , . We also recall the evolution of N A (see [10, (5.103)] for example): ∂ u N A = ∇ A m − ∇ D ( ∇ D ∇ E C EA − ∇ A ∇ E C ED )+ 14 ∇ A ( C BE N BE ) − ∇ B ( C BD N DA ) + 12 C AB ∇ D N DB . The formula can be rewritten in the following form:
Proposition 3.1.
The angular momentum aspect N A evolves according to ∂ u N A = ∇ A m + 14 ǫ AB ∇ B ( ǫ P Q ∇ P ∇ E C EQ ) + 18 ∇ A ( C BE N BE )+ 18 ǫ AB ∇ B ( ǫ P Q C EP N EQ ) + 12 C AB ∇ D N DB . (3.4) Proof.
We rewrite the terms − ∇ D ( ∇ D ∇ E C EA −∇ A ∇ E C ED ) and − ∇ B ( C BD N DA ).First we check the following identity directly: ǫ AB ∇ B ( ǫ P Q ∇ P ∇ E C EQ ) = −∇ D ( ∇ D ∇ E C EA − ∇ A ∇ E C ED ) . As for the term C DB N DA , we use the following general formulae for sym-metric traceless 2-tensors on the 2-sphere: C DB N DA + N DB C DA = ( C DE N DE ) σ AB C DB N DA − N DB C DA = − ( ǫ P Q C EP N EQ ) ǫ AB Therefore,2 C DB N DA = ( C DE N DE ) σ AB − ( ǫ P Q C EP N EQ ) ǫ AB . (cid:3) VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 13
Equation (3.4) is indeed equivalent to equation (4) on page 48 of [8]. Weapply (3.4) to derive the evolution of the classical angular momentum andcenter of mass.
Theorem 3.2 (Theorem 1.2) . The classical angular momentum and centerof mass evolve according to the following: ∂ u ˜ J k = 14 Z S h ǫ AE ∇ E ˜ X k ( C AB ∇ D N BD − N AB ∇ D C BD ) + ˜ X k ǫ AB ( C DA N DB ) i , (3.5) ∂ u ˜ C k = 14 Z S h ∇ A ˜ X k (cid:16) u ∇ A | N | + C AB ∇ D N BD − N AB ∇ D C BD (cid:17)i , (3.6) where k = 1 , , . Proof.
By (1.2), ∂ u ˜ J k = Z S ǫ AB ∇ B ˜ X k [ ∂ u N A − ∂ u ( C DA ∇ B C DB )] . First, we deal with the term ǫ AB ∇ B ( ǫ P Q ∇ P ∇ E C EQ ) on the right handside of (3.4) and claim that Z S Y A ǫ AB ∇ B ( ǫ P Q ∇ P ∇ E C EQ ) = 0 (3.7)for Y A = ∇ A ˜ X k or ǫ AB ∇ B ˜ X k . Integrating by parts, the integral becomes Z S ǫ AB ∇ A Y B ( ǫ P Q ∇ P ∇ E C EQ ) . Since ǫ P Q ∇ P ∇ E C EQ = − ∆(∆ + 2) c and ǫ AB ∇ A Y B is zero or 2 ˜ X k , theintegral vanishes.Hence, we obtain ∂ u ˜ J k = Z S ǫ AB ∇ B ˜ X k (cid:20) ǫ AE ∇ E ( ǫ P Q C EP N EQ ) + 12 C AB ∇ D N DB − ∂ u ( C DA ∇ B C DB ) (cid:21) since the integral of ∇ A m + ∇ A ( C BE N BE ) against ǫ AB ∇ B ˜ X k vanishes.Integrating by parts the first term and use ǫ AB ǫ AE = δ BE , we obtain thedesired formula.We now turn to the formula for ˜ C k . By (1.3) and (3.7), ∂ u ˜ C k = Z S ∇ A ˜ X k (cid:20) ∂ u N A − ∇ A m + u ∇ A | N | − ∂ u ( C DA ∇ B C DB ) − ∇ A ∂ u ( C DE C DE ) (cid:21) = Z S ∇ A ˜ X k h u ∇ A | N | + 18 ∇ A ( C BE N BE ) + 12 C AB ∇ D N DB − ∂ u ( C DA ∇ B C DB ) − ∇ A ∂ u ( C DE C DE ) i since the integral of ǫ AB ∇ B ( ǫ P Q C EP N EQ ) against ∇ A ˜ X k vanishes. Wearrive at the desired formula since ∂ u ( C DE C DE ) = 2( C BE N BE ). (cid:3) Evolution formulae in terms of potentials
In this section, we rewrite the evolution formulae in terms of the potentialsof the shear and the news tensor.4.1.
Energy and linear momentum.
First we recall the formulae for theenergy and linear momentum.
Proposition 4.1.
Suppose C AB and N AB are given as in (2.9) and (2.10) ,we have ∂ u E = − Z S [ n ∆(∆ + 2) n + n ∆(∆ + 2) n ] ∂ u P k = − Z S ˜ X k [((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n ] . Proof.
These follow from Lemma 2.6. (cid:3)
Proof of Theorem 1.3.
We first prove the following Proposition:
Proposition 4.2.
The evolution formulae of the conserved quantities canbe written as ∂ u ˜ J k = 18 Z S ˜ X k ǫ AB [ ∇ A c ∇ B ∆(∆ + 2) n + ∇ A c ∇ B ∆(∆ + 2) n ] ∂ u ˜ C k = 18 Z S u ˜ X k [((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n ]+ 116 Z S [ ˜ X k (∆(∆ + 2) c (∆ + 2) n − ∆(∆ + 2) n (∆ + 2) c )]+ 116 Z S [ ˜ X k (∆(∆ + 2) c (∆ + 2) n − ∆(∆ + 2) n (∆ + 2) c ] . Proof.
We write4 ∂ u ˜ J k = Z S − ˜ X k ǫ AB C DB N DA + Z S Y Ak ( C AB ∇ D N BD − N AB ∇ D C BD ) = (1)+(2)and compute (1) and (2) separately. VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 15
Note that (1) = − R S ˜ X k ε ( C AB ) N AB and recall that ε ( C AB ) has po-tentials − c and c . Applying (2.18), we get(1) = − Z S ˜ X k [ − (∆ + 2) c (∆ + 2) n + (∆ + 2) c (∆ + 2) n + Z S ǫ AB ( ∇ A c ∇ B (∆ + 2) n − ∇ A n ∇ B (∆ + 2) c )]= − Z S ˜ X k [ − (∆ + 2) c (∆ + 2) n + (∆ + 2) c (∆ + 2) n ] − Z S ˜ X k ǫ AB [ ∇ A c ∇ B (∆ + 2) n − ∇ A n ∇ B (∆ + 2) c ]= − Z S ˜ X k [ − (∆ + 2) c (∆ + 2) n + (∆ + 2) c (∆ + 2) n ] − Z S ˜ X k ǫ AB [ ∇ A c ∇ B (∆ + 2) n + ∇ A c ∇ B (∆ + 2) n ]where we used (2.7) in the last equality.Applying (2.16) to Y A = Y Ak , we have(2) = 12 Z S ˜ X k ǫ AB [ ∇ A (∆ + 2) c ∇ B (∆ + 2) n + ∇ A (∆ + 2) c ∇ B (∆ + 2) n ]+ 12 Z S ˜ X k [(∆ + 2) c (∆ + 2) n − (∆ + 2) c (∆ + 2) n ]Therefore,(1) + (2) = − Z S ˜ X k ǫ AB [ ∇ A c ∇ B (∆ + 2) n + ∇ A c ∇ B (∆ + 2) n ]+ 12 Z S ˜ X k ǫ AB [ ∇ A (∆ + 2) c ∇ B (∆ + 2) n + ∇ A (∆ + 2) c ∇ B (∆ + 2) n ]= 12 Z S ˜ X k (cid:2) ǫ AB ∇ A ∆ c ∇ B (∆ + 2) n + ∇ A ∆ c ∇ B (∆ + 2) n (cid:3) , and the desired formula follows by (2.7).As for the evolution of the center of mass, we apply (2.15) and note that Z S ∇ A ˜ X k ǫ DA ∇ D [(∆ + 2) c (∆ + 2) n + (∆ + 2) c (∆ + 2) n ] = 0 . Therefore, ∂ u ˜ C k = 18 Z S u ˜ X k [((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n ] − Z S [ ∇ A ˜ X k ( ∇ A (∆ + 2) c (∆ + 2) n − ∇ A (∆ + 2) n (∆ + 2) c )] − Z S [ ∇ A ˜ X k ( ∇ A (∆ + 2) c (∆ + 2) n − ∇ A (∆ + 2) n (∆ + 2) c ]= 18 Z S u ˜ X k [((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n ]+ 116 Z S ˜ X k [∆(∆ + 2) c (∆ + 2) n − ∆(∆ + 2) n (∆ + 2) c ]+ 116 Z S ˜ X k [∆(∆ + 2) c (∆ + 2) n − ∆(∆ + 2) n (∆ + 2) c ] (cid:3) To obtain the formulae given in Theorem 1.3, we rewrite the above for-mulae in terms of bracket operators on S . Definition 4.3.
For two smooth functions u and v on S , denote[ u, v ] = ǫ AB ∇ A u ∇ B v (4.1)and [ u, v ] = 12 ((∆ u ) v − (∆ v ) u ) . (4.2)In view of Definition 4.3, we can write ∂ u ˜ J k = 18 Z S ˜ X k ([ c, ∆(∆ + 2) n ] + [ c, ∆(∆ + 2) n ] )and similarly for the center of mass. This proves Theorem 1.3.5. Supertranslation invariance of the total flux
Total flux of classical conserved quantities.
We study the effect ofsupertranslation on the total flux of conserved quantities along null infinityor, equivalently, the difference of conserved quantities at timelike infinityand spatial infinity. As in the previous section, suppose I = ( −∞ , ∞ ) and I + is complete extending from spatial infinity ( u = −∞ ) to timelike infinity( u = + ∞ ). A supertranslation is a change of coordinates (¯ u, ¯ x A ) → ( u, x A )such that u = ¯ u + f ( x ) , x A = ¯ x A on I + . Let m , C AB , and N AB denote themass aspect, the shear, and the news, respectively, in the ( u, x A ) coordinatesystem. Since the spherical coordinate is unchanged, we use x to denoteeither x A or ¯ x A throughout this section. It is well-known (see [10, (C.117)and (C.119)] for example) that the shear ¯ C AB (¯ u, x ), and the news ¯ N AB (¯ u, x ) VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 17 in the (¯ u, x ) coordinate system are given by¯ C AB (¯ u, x ) = C AB (¯ u + f ( x ) , x ) − ∇ A ∇ B f + ∆ f σ AB (5.1)¯ N AB (¯ u, x ) = N AB (¯ u + f ( x ) , x ) (5.2)We assume that there exists a constant ε > N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ . (5.3)Note that the limits of the shear tensor existlim u →±∞ C AB ( u, x ) = C AB ( ± )as a result of (5.3).Similarly, (5.3) implies that the limits of the angular momentum existlim u →±∞ ˜ J k ( u ) = ˜ J k ( ± ) . Denote the corresponding quantities after supertranslation by ˜ J kf ( ± ).By (3.5), the total angular momentum flux is˜ J k (+) − ˜ J k ( − )= 14 Z + ∞−∞ Z S h Y A (cid:0) C AB ∇ D N BD − N AB ∇ D C BD (cid:1) + ˜ X k ǫ AB C DA N DB i ( u, x ) dS du = 14 Z + ∞−∞ Z S (cid:2) −∇ D Y A C AB N BD + Y A (cid:0) −∇ D C AB N BD − N AB ∇ D C BD (cid:1)(cid:3) ( u, x ) dS du + 14 Z + ∞−∞ Z S ˜ X k ǫ AB ( C DA N DB )( u, x ) dS du (5.4)in the ( u, x ) coordinates and˜ J kf (+) − ˜ J kf ( − )= 14 Z + ∞−∞ Z S (cid:2) −∇ D Y A ¯ C AB ¯ N BD + Y A (cid:0) −∇ D ¯ C AB ¯ N BD − ¯ N AB ∇ D ¯ C BD (cid:1)(cid:3) (¯ u, x ) dS d ¯ u + 14 Z + ∞−∞ Z S ˜ X k ǫ AB ( ¯ C DA ¯ N DB )(¯ u, x ) dS d ¯ u (5.5)in the (¯ u, x ) coordinates.Applying the chain rule on (5.1) yields ∇ D ¯ C AB (¯ u, x ) = N AB (¯ u + f, x ) ∇ D f + ( ∇ D C AB )(¯ u + f, x ) − ∇ D F AB , ∇ D ¯ C BD (¯ u, x ) = N BD (¯ u + f, x ) ∇ D f + ( ∇ D C BD )(¯ u + f, x ) − ∇ B (∆ + 2) f. To simplify notation, we introduce the u independent symmetric traceless2-tensor F AB = 2 ∇ A ∇ B f − ∆ f σ AB and thus ∇ D F BD = ∇ B (∆ + 2) f . Equation (5.5) can be rewritten as˜ J kf (+) − ˜ J kf ( − )= 14 Z ∞−∞ Z S (cid:2) −∇ D Y A ( C AB − F AB ) N BD + Y A ω A (cid:3) (¯ u + f, x ) dS d ¯ u + 14 Z + ∞−∞ Z S h ˜ X k ǫ AB (cid:0) C DA − F DA (cid:1) N DB i (¯ u + f, x ) dS d ¯ u (5.6)where ω A ( u, x ) = (cid:0) − N AB ( u, x ) ∇ D f − ∇ D C AB + ∇ D F AB (cid:1) N BD ( u, x ) − N AB ( u, x ) (cid:0) N BD ( u, x ) ∇ D f + ∇ D C BD ( u, x ) − ∇ D F BD ( x ) (cid:1) . Note that the integrand is evaluated at (¯ u + f, x ) in equation (5.6), to whichthe change of variable will be applied.By the decaying assumption of the news (5.3), we can apply change ofvariable u = ¯ u + f to (5.6) and rewrite it as˜ J kf (+) − ˜ J kf ( − )= 14 Z + ∞−∞ Z S h −∇ D Y A ( C AB − F AB ) N BD + Y A ω A + ˜ X k ǫ AB (cid:0) C DA − F DA (cid:1) N DB i ( u, x ) dS du (5.7)Combining (5.4) and (5.7), we obtain (cid:16) ˜ J kf (+) − ˜ J kf ( − ) (cid:17) − (cid:16) ˜ J k (+) − ˜ J k ( − ) (cid:17) = 14 Z ∞−∞ Z S − Y A | N | ∇ A f dS du + 14 Z ∞−∞ Z S h − Y A F AB ∇ D N BD + Y A N AB ∇ D F BD − ˜ X k ǫ AB F DA N DB i dS du where we used the identity 2 N AB N BD = | N | δ DA .Observe that the second integral is of the same form as ∂ u ˜ J given in (3.5)and one can thus simplify it as in the proof of Proposition 4.2 to get (cid:16) J kf (+) − J kf ( − ) (cid:17) − (cid:16) J k (+) − J k ( − ) (cid:17) = 14 Z ∞−∞ Z S f Y A ∇ A | N | dS du + 14 Z ∞−∞ Z S ˜ X k ǫ AB ∇ A n ∇ B ∆(∆ + 2) f dS du Integrating by parts, we arrive at (cid:16) ˜ J kf (+) − ˜ J kf ( − ) (cid:17) − (cid:16) ˜ J k (+) − ˜ J k ( − ) (cid:17) = 14 Z + ∞−∞ Z S f Y A ∇ A (cid:0) | N | − ∆(∆ + 2) n (cid:1) dS du = Z S − f Y A ∇ A ( m (+) − m ( − )) dS (5.8) VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 19 where m ( ± ) = lim u →±∞ m ( u, x ) . Here we used the mass loss formula (3.1) in the form ∂ u m = ∆(∆ + 2) n − | N | . Note that m (+) − m ( − ) is of the same mode as Y A ∇ A ( m (+) − m ( − ))because Y A is a Killing field.In summary, we obtain a necessary and sufficient condition for the totalflux of the classical angular momentum to be supertranslation invariant. Theorem 5.1.
Suppose the news tensor decays as N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ . The total flux of the classical angular momentum ˜ J k is supertranslationinvariant if and only if m (+) − m ( − ) (as a function on S ) is supported in the l ≤ modes.Moreover, the above condition holds when the rescaled curvature compo-nents P (see Definition A.5) at I + satisfy lim u →∞ P − lim u →−∞ P (5.9) is supported in the l ≤ modes. Remark 5.2.
Theorem 5.1 is motivated by the investigation in [7], which isbuilt on the framework of stability of Minkowski spacetime. Indeed, equation(11) and (12) of [7] Z + − Z − = ∇ Φ div (cid:0) Σ + − Σ − (cid:1) = Z + − Z − imply lim u →∞ c ( u, x ) = lim u →−∞ c ( u, x ). Using moreover (10) of [7]∆Φ = − F − ¯ F ) , we get ∇ A (cid:0) F − ∆(∆ + 2) c | + ∞−∞ (cid:1) = 0 . Moreover, the total flux of the classical center of mass is supertranslationinvariant under the same condition
Theorem 5.3.
Suppose the news tensor decays as N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ , The total flux of the classical center of mass ˜ C k is supertranslation invariantif and only if m (+) − m ( − ) is a constant function on S . Proof.
Denoting ˜ C k ( ± ) = lim u →±∞ ˜ C k ( u ), by (3.6) we have˜ C k (+) − ˜ C k ( − )= 14 Z + ∞−∞ Z S u | N | ( u, x ) ˜ X k + ∇ A ˜ X k (cid:2) C AB ∇ D N BD − N AB ∇ D C BD (cid:3) ( u, x ) dS du. (5.10)On the other hand,˜ C kf (+) − ˜ C kf ( − )= 14 Z + ∞−∞ Z S ¯ u | ¯ N | (¯ u, x ) ˜ X k + ∇ A ˜ X k (cid:2) ¯ C AB ∇ D ¯ N BD − ¯ N AB ∇ D ¯ C BD (cid:3) (¯ u, x ) dS d ¯ u. Proceed in the same way as in the case of angular momentum, we obtain (cid:16) ˜ C kf (+) − ˜ C kf ( − ) (cid:17) − (cid:16) ˜ C k (+) − ˜ C k ( − ) (cid:17) = 14 Z + ∞−∞ Z S − ˜ X k | N | f − ∇ A ˜ X k | N | ∇ A f dS du + 14 Z + ∞−∞ Z S ∇ A ˜ X k ( − ∇ A ∇ B f + ∆ f σ AB ) ∇ D N BD + ∇ A ˜ X k N AB ∇ B (∆ + 2) f dS du We simplify the second integral as Z S − X k ∇ A f ∇ D N AD + 2 ∇ A ˜ X k ∇ A f ∇ B ∇ D N BD − ∇ A ˜ X k ∇ B N AB · f = Z S (cid:16) ∇ A ˜ X k ∇ A f ∇ B ∇ D N BD + ˜ X k ∇ A ∇ B N AB · f (cid:17) and the mass loss formula ∂ u m = ∇ A ∇ B N AB − | N | implies that (cid:16) ˜ C kf (+) − ˜ C kf ( − ) (cid:17) − (cid:16) ˜ C k (+) − ˜ C k ( − ) (cid:17) = Z S X k f (cid:0) m (+) − m ( − ) (cid:1) + 2 ∇ A ˜ X k (cid:0) m (+) − m ( − ) (cid:1) ∇ A f = Z S (cid:16) X k (cid:0) m (+) − m ( − ) (cid:1) − ∇ A ˜ X k ∇ A (cid:0) m (+) − m ( − ) (cid:1)(cid:17) f. Hence, ˜ C k (+) − ˜ C k ( − ) is invariant under arbitrary supertranslation if andonly if 6 ˜ X k (cid:0) m (+) − m ( − ) (cid:1) − ∇ A ˜ X k ∇ A (cid:0) m (+) − m ( − ) (cid:1) is supported in the l ≤ X k and summing over k = 1 , ,
3, weget m (+) − m ( − ) is supported in the l ≤ m (+) − m ( − ) contains a l = 2 mode, then6 ˜ X k (cid:0) m (+) − m ( − ) (cid:1) − ∇ A ˜ X k ∇ A (cid:0) m (+) − m ( − ) (cid:1) contains a l = 3 mode.Simiarly, if m (+) − m ( − ) contains a l = 1 mode, then 6 ˜ X k (cid:0) m (+) − m ( − ) (cid:1) − ∇ A ˜ X k ∇ A (cid:0) m (+) − m ( − ) (cid:1) contains a l = 2 mode. Thus, m (+) − m ( − ) is VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 21 constant if and only if ˜ C k (+) − ˜ C k ( − ) is invariant under arbitrary super-translation (cid:3) Total flux of the CWY conserved quantities.
In this subsection,we show that the total flux of the CWY angular momentum and center ofmass is supertranslation invariant. We decompose f into its modes: f = α + α i ˜ X i + f l ≥ and let J k ( ± ) be the limits of the CWY angular momentum in the u co-ordinate and J kf ( ± ) be the limits of the CWY angular momentum in the ¯ u coordinate. We have Theorem 5.4.
Suppose the news tensor decays as N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ . Then the total flux of J k is supertranslation invariant. Namely, (cid:16) J kf (+) − J kf ( − ) (cid:17) − (cid:16) J k (+) − J k ( − ) (cid:17) = α i ε ikj ( P j (+) − P j ( − )) . Proof.
Note that J k = ˜ J k − Z S Y A c ∇ A m. (5.11)The assumption (5.3) on the decay of news tensor implies that the limitof mass aspect function is invariant of supertranslationlim ¯ u →±∞ ¯ m (¯ u, x ) = lim u →±∞ m ( u, x ) or ¯ m ( ± ) = m ( ± ) . (5.12)Moreover, we havelim ¯ u →±∞ ¯ C AB (¯ u, x ) = lim u →±∞ C AB ( u, x ) − ∇ A ∇ B f + ∆ f σ AB . (5.13)If we denote the closed potential of lim ¯ u → + ∞ ¯ C AB and lim ¯ u → + ∞ C AB by¯ c (+) and c (+) respectively, we have¯ c (+) = c (+) − f ℓ ≥ (5.14)as functions on S . Evaluating the definition of the CWY angular momen-tum (5.11) at + ∞ gives J k (+) = ˜ J k (+) − Z S Y A c (+) ∇ A m (+) , and J kf (+) = ˜ J kf (+) − Z S Y A ¯ c (+) ∇ A ¯ m (+) . Taking the difference and applying (5.12) and (5.14), we derive J kf (+) − J k (+) = ˜ J kf (+) − ˜ J k (+) + 2 Z S f ℓ ≥ Y A ∇ A m (+) . We derive a similar equation at −∞ and thus (cid:16) J kf (+) − J kf ( − ) (cid:17) − (cid:16) J k (+) − J k ( − ) (cid:17) = (cid:16) ˜ J kf (+) − ˜ J kf ( − ) (cid:17) − (cid:16) ˜ J k (+) − ˜ J k ( − ) (cid:17) + 2 Z S f ℓ ≥ Y A ∇ A ( m (+) − m ( − ))= − Z S f ℓ ≤ Y A ∇ A ( m (+) − m ( − ))by (5.8). It follows that (cid:16) J kf (+) − J kf ( − ) (cid:17) − (cid:16) J k (+) − J k ( − ) (cid:17) = α i ε ikj ( P j (+) − P j ( − )) . (cid:3) Let C k ( ± ) be the limits of the CWY center of mass in the u coordinateand C kf ( ± ) be the limits of the CWY center of mass in the ¯ u coordinate. Theorem 5.5.
Suppose the news tensor decays as N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ , then the total flux of C k is supertranslation invariant. Namely, (cid:16) C kf (+) − C kf ( − ) (cid:17) − (cid:16) C k (+) − C k ( − ) (cid:17) = α (cid:16) P k (+) − P k ( − ) (cid:17) + α k ( E (+) − E ( − )) . Proof.
We write C k = ˜ C k − Z S c ∇ A ˜ X k ∇ A m + 3 Z S c ˜ X k m + Ξ( c, m ) , (5.15)where Ξ is an integral over S that involves only c and m . Since the lastthree integrals have limits at u = ±∞ , the mass loss formula now implies C k (+) − C k ( − )= 14 Z + ∞−∞ Z S u | N | ( u, x ) ˜ X k + ∇ A ˜ X k (cid:2) C AB ∇ D N BD − N AB ∇ D C BD (cid:3) ( u, x ) dS du − Z S c (+) ∇ A ˜ X k ∇ A m (+) + Z S c ( − ) ∇ A ˜ X k ∇ A m ( − )+ 3 Z S c (+) ˜ X k m (+) − Z S c ( − ) ˜ X k m ( − )+ Ξ( c (+) , m (+)) − Ξ( c ( − ) , m ( − )) . VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 23
By (5.13), c ( ± ) is invariant under supertranslation. We apply (5.12) and(5.14) to get (cid:16) C kf (+) − C kf ( − ) (cid:17) − (cid:16) C k (+) − C k ( − ) (cid:17) = (cid:16) ˜ C kf (+) − ˜ C kf ( − ) (cid:17) − (cid:16) ˜ C k (+) − ˜ C k ( − ) (cid:17) + 2 Z S f ℓ ≥ ∇ A ˜ X k ∇ A (cid:0) m (+) − m ( − ) (cid:1) − Z S f ℓ ≥ ˜ X k (cid:0) m (+) − m ( − ) (cid:1) = − Z S f ℓ ≤ ∇ A ˜ X k ∇ A (cid:0) m (+) − m ( − ) (cid:1) + 6 Z S f ℓ ≤ ˜ X k (cid:0) m (+) − m ( − ) (cid:1) . We obtain (cid:16) C kf (+) − C kf ( − ) (cid:17) − (cid:16) C k (+) − C k ( − ) (cid:17) = 2 Z S ( α ˜ X k + α k )( m (+) − m ( − ))= α (cid:16) P k (+) − P k ( − ) (cid:17) + α k ( E (+) − E ( − )) . (cid:3) Spacetime with zero news
In this section, we consider a non-radiative spacetime in the sense thatthe news vanishes. This includes all model spacetimes such as Minkowskiand Kerr. First, we show that the CWY angular momentum and center ofmass are constant.
Lemma 6.1.
Suppose the news N AB ( u, x ) ≡ in a Bondi-Sachs coordinatesystem ( u, x ) , then the CWY angular momentum J k ( u ) and CWY center ofmass C k ( u ) are constant, i.e. independent of the retarded time u .Proof. The assumption implies ∂ u m ( u, x ) = 0 , ∂ u C AB ( u, x ) = 0 and thus m ( u, x ) ≡ ˚ m ( x ) , C AB ( u, x ) ≡ ˚ C AB ( x )and both potentials c and c are independent of u as well.We recall the definition of CWY angular momentum J k ( u ) = Z S Y A (cid:18) N A − C AB ∇ D C DB − c ∇ A m (cid:19) . (6.1)Since c and m are both independent of u , our previous calculation shows ∂ u Z S Y A (cid:18) N A − C AB ∇ D C DB (cid:19) = 14 Z S h Y A ( C AB ∇ D N BD − N AB ∇ D C BD ) + ˜ X k ǫ AB ( C DA N DB ) i , the conclusion follows. On the other hand, the CWY center of mass C k is given by Z S ∇ A ˜ X k (cid:18) N A − C AB ∇ D C DB − ∇ A (cid:0) C DE C DE (cid:1)(cid:19) − ∇ A ˜ X k ( c + u ) ∇ A m + Z S (cid:18) X k cm + 2 ∇ A ˜ X k ǫ AB ( ∇ B c ) m −
116 ˜ X k ∇ A (∆ + 2) c ∇ A (∆ + 2) c (cid:19) (6.2)Since all m , c , and c are independent of u , ∂ u C k = ∂ u Z S ∇ A ˜ X k (cid:18) N A − C AB ∇ D C DB − ∇ A (cid:0) C DE C DE (cid:1)(cid:19) − Z S ∇ A ˜ X k ∇ A m (6.3)Our previous calculation shows that the first term on the right hand side is Z S ∇ A ˜ X k (cid:18) ∇ A m + 14 C AB ∇ D N BD − N AB ∇ D C BD (cid:19) , and the conclusion follows. (cid:3) Finally, we show that in a spacetime with vanishing news tensor, theangular momentum and center of mass themselves, not just their total flux,are invariant under supertranslation.We pin down the exact formula for the angular momentum aspect on aspacetime with vanishing news. In this case, we have ∂ u N A ( u, x ) = ∇ A m ( u, x ) − ∇ B P BA ( u, x )where P BA ( u, x ) = ( ∇ B ∇ E C EA − ∇ A ∇ E C EB )( u, x )Therefore ∂ u N A ( u, x ) = ∇ A ˚ m ( x ) − ∇ B ˚ P BA ( x )is independent of u . Integrating gives N A ( u, x ) = N A ( u , x ) + ( u − u )( ∇ A ˚ m − ∇ B ˚ P BA ) (6.4)for any u and fixed u .Suppose (¯ u, x ) is another Bondi-Sachs coordinate system that is relatedto ( u, x ) by a supertranslation u = ¯ u + f for f ∈ C ∞ ( S ).Recall the mass aspect ¯ m (¯ u, x ), the shear ¯ C AB (¯ u, x ), and the news ¯ N AB (¯ u, x )in the (¯ u, x ) coordinate system are related to the mass aspect m ( u, x ), theshear C AB ( u, x ), and the news N AB ( u, x ) in the ( u, x ) coordinate systemthrough: VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 25 ¯ m (¯ u, x ) = m (¯ u + f, x ) + 12 ( ∇ B N BD )(¯ u + f, x ) ∇ D f + 14 ( ∂ u N BD )(¯ u + f, x ) ∇ B f ∇ D f + 14 N BD (¯ u + f, x ) ∇ B ∇ D f ¯ C AB (¯ u, x ) = C AB (¯ u + f ( x ) , x ) − ∇ A ∇ B f + ∆ f σ AB ¯ N AB (¯ u, x ) = N AB (¯ u + f ( x ) , x ) (6.5)In particular, ¯ N AB (¯ u, x ) ≡ J k (¯ u ) and ¯ C k (¯ u ) are independent of ¯ u .In addition, we have ¯ m (¯ u, x ) = ˚¯ m ( x ) = ˚ m ( x )¯ C AB (¯ u, x ) = ˚¯ C AB ( x ) = ˚ C AB ( x ) − F AB ˚¯ c = ˚ c − f ℓ ≥ ˚¯ c = ˚ c (6.6)where F AB = 2 ∇ A ∇ B f − ∆ f σ AB .Finally the angular momentum aspect transforms by¯ N A (¯ u, x ) = N A (¯ u + f, x ) + 3 m (¯ u + f, x ) ∇ A f − P BA (¯ u + f, x ) ∇ B f = N A (¯ u + f, x ) + 3˚ m ∇ A f −
34 ˚ P BA ∇ B f. See [10, (C.123)]. Note that the convention of angular momentum aspectthere is − N A .Combining with (6.4) and setting u = ¯ u + f , we obtain¯ N A (¯ u, x ) = N A ( u , x )+(¯ u − u + f )( ∇ A ˚ m − ∇ B ˚ P BA )+3˚ m ∇ A f −
34 ˚ P BA ∇ B f (6.7)for any ¯ u and fixed u .Now fixing ¯ u = ¯ u , we consider the angular momentums¯ J = ¯ J (¯ u ) = Z S Y A (cid:18) ¯ N A −
14 ¯ C AB ∇ D ¯ C DB − ˚¯ c ∇ A ˚¯ m (cid:19) (¯ u , x ) J = J ( u ) = Z S Y A (cid:18) N A −
14 ˚ C AB ∇ D ˚ C DB − ˚ c ∇ A ˚ m (cid:19) ( u , x ) ¯ C = ¯ C (¯ u ) = Z S ∇ A ˜ X k (cid:18) ¯ N A −
14 ¯ C AB ∇ D ¯ C DB − ∇ A ( ¯ C DE ¯ C DE ) (cid:19) (¯ u , x )+ Z S (cid:16) X k ˚¯ c ˚¯ m − ∇ A ˜ X k (˚¯ c + ¯ u ) ∇ A ˚¯ m (cid:17) + Z S (cid:18) ∇ A ˜ X k ǫ AB ( ∇ B ˚¯ c )˚¯ m −
116 ˜ X k ∇ A (∆ + 2)˚¯ c ∇ A (∆ + 2)˚¯ c (cid:19) C = C ( u ) = Z S ∇ A ˜ X k (cid:18) N A −
14 ˚ C AB ∇ D ˚ C DB − ∇ A ( ˚ C DE ˚ C DE ) (cid:19) ( u , x )+ Z S (cid:16) X k ˚ c ˚ m − ∇ A ˜ X k (˚ c + u ) ∇ A ˚ m (cid:17) + Z S (cid:18) ∇ A ˜ X k ǫ AB ( ∇ B ˚ c ) m −
116 ˜ X k ∇ A (∆ + 2)˚ c ∇ A (∆ + 2)˚ c (cid:19) We prove the following theorem:
Theorem 6.2.
On a spacetime with vanishing news, the CWY angular mo-mentum and center of mass satisfy ¯ J − J = − Z S Y A f ℓ ≤ ∇ A ˚ m (6.8)¯ C − C = Z S (cid:16) f ℓ ≤ ˜ X k ˚ m − f ℓ ≤ ∇ A ˜ X k ∇ A ˚ m (cid:17) (6.9) Proof.
Taking the difference of ¯ J and J and applying (6.6), we obtain¯ J − J = Z S Y A (cid:2) ¯ N A (¯ u , x ) − N A ( u , x ) (cid:3) + 14 Z S Y A h ˚ C AB ∇ D F BD + F AB ∇ D ˚ C BD − F AB ∇ D F BD i + 2 Z S Y A f ℓ ≥ ∇ A ˚ m We observe that R S Y A ( F AB ∇ D F BD ) = 0 and compute Z S Y A (cid:2) ¯ N A (¯ u , x ) − N A ( u , x ) (cid:3) = Z S Y A (cid:20) (¯ u − u + f )( ∇ A ˚ m − ∇ B ˚ P BA ) + 3˚ m ∇ A f −
34 ˚ P BA ∇ B f (cid:21) = Z S Y A (cid:20) f ∇ A ˚ m − f ∇ B ˚ P BA + 3˚ m ∇ A f −
34 ˚ P BA ∇ B f (cid:21) = Z S Y A (cid:20) − f ∇ A ˚ m − f ∇ B ˚ P BA −
34 ˚ P BA ∇ B f (cid:21) , where we use R S Y A ∇ B ˚ P BA = 0 and R S Y A ∇ A ˚ m = 0. Therefore, VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 27 ¯ J − J = 14 Z S Y A h ˚ C AB ∇ D F BD + F AB ∇ D ˚ C BD − f ∇ B ˚ P BA − P BA ∇ B f i − Z S Y A f ℓ ≤ ∇ A ˚ m. By Theorem B.1, the first integral vanishes and the result follows.Taking the difference of ¯ C and C and applying (6.6), we obtain¯ C − C = Z S ∇ A ˜ X k (cid:2) ¯ N A (¯ u , x ) − N A ( u , x ) (cid:3) + 14 Z S ∇ A ˜ X k h ˚ C AB ∇ D F BD + F AB ∇ D ˚ C BD − F AB ∇ D F DB + 12 ∇ A ( C BD F BD ) − ∇ A ( F BD F BD ) i + Z S (cid:16) − f ℓ ≥ ˜ X k ˚ m + (2 f ℓ ≥ − ¯ u + u ) ∇ A ˜ X k ∇ A ˚ m (cid:17) We observe that Z S ∇ A ˜ X k (cid:20) − F AB ∇ D F DB − ∇ A ( F BD F BD ) (cid:21) = 0and compute Z S ∇ A ˜ X k (cid:2) ¯ N A (¯ u , x ) − N A ( u , x ) (cid:3) = Z S ∇ A ˜ X k (cid:20) (¯ u − u + f )( ∇ A ˚ m − ∇ B ˚ P BA ) + 3˚ m ∇ A f −
34 ˚ P BA ∇ B f (cid:21) = Z S ∇ A ˜ X k (cid:20) (¯ u − u + f ) ∇ A ˚ m − f ∇ B ˚ P BA + 3˚ m ∇ A f −
34 ˚ P BA ∇ B f (cid:21) = Z S ∇ A ˜ X k (cid:20) (¯ u − u − f ) ∇ A ˚ m − f ∇ B ˚ P BA −
34 ˚ P BA ∇ B f (cid:21) + Z S (6 f ˜ X k ˚ m ) , where we use R S ∇ A ˜ X k ∇ B ˚ P BA = 0.Putting everything together, we arrive at¯ C − C = 14 Z S ∇ A ˜ X k (cid:20) ˚ C AB ∇ D F BD + F AB ∇ D ˚ C BD + 12 ∇ A ( ˚ C BD F BD ) − f ∇ B ˚ P BA − P BA ∇ B f (cid:21) + Z S (cid:16) f ℓ ≤ ˜ X k ˚ m − f ℓ ≤ ∇ A ˜ X k ∇ A ˚ m (cid:17) By Theorem B.2, the first integral vanishes and the result follows. (cid:3) Conservation law of angular momentum and a dualityparadigm for null infinity
Conservation law of angular momentum.
In this subsection, wederive a conservation law of angular momentum at I + `a la Christodoulou[7].Suppose I = ( −∞ , + ∞ ) and I + is complete extending from spatial in-finity ( u = −∞ ) to timelike infinity ( u = + ∞ ). Integrating the formula inProposition 3.1 from −∞ to + ∞ and projecting onto the ℓ = 1 modes, weobtain ǫ AE ∇ E N A (+ ∞ ) ℓ =1 − ǫ AE ∇ E N A ( −∞ ) ℓ =1 = G ℓ =1 , (7.1)where G = Z + ∞−∞ ∇ A ∇ A ( ǫ P Q C EP N EQ ) + 12 ǫ AE ∇ E ( C AB ∇ D N DB )Equation (7.1) should be considered as a conservation law for angularmomentum that complements the conservation law for linear momentum ofChristodoulou [7, Equation (13)], which in our notation is b m (+ ∞ ) ℓ =0 , − b m ( −∞ ) ℓ =0 , = − F ℓ =0 , , where F = 18 Z ∞−∞ N AB N AB (7.2)and follows from (3.3).The above discussion can be carried over under the framework of stabilityof Minkowski spacetime, provided that we take Rizzi’s definition of angularmomentum [20, 21]. Recall from [7, 8] that two symmetric traceless 2-tensorsΣ and Ξ are defined bylim C + u ,r →∞ r b χ = Σ , lim C + u ,r →∞ r b χ = Ξwith ∂ Σ ∂u = −
12 Ξ . (7.3)See Definition A.5 for the curvature components and their limits at nullinfinity.Rizzi’s definition of angular momentum [20, (3)] is given by (omitting theconstant π ) L (Ω ( i ) ) = Z S Ω A ( i ) (cid:0) I A − Σ AB ∇ C Σ CB (cid:1) , i = 1 , , assumes that the curvature component β satisfies lim r →∞ r β A = − I A . Here Ω A ( i ) corresponds to ǫ AB ∇ B ˜ X i . In the appendix, we show that I A VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 29 and Σ AB correspond to N A and − C AB in Bondi-Sachs coordinate system.Hence Rizzi’s definition coincides with (1.2).Using Bianchi identities, Rizzi derived the evolution formula [20, (4)] ∂L∂u = Z S Ω A (cid:20) Ξ AB ∇ C Σ CB + 12 (cid:0) Σ CB ∇ B Ξ CA − Σ AB ∇ C Ξ CB (cid:1)(cid:21) = 12 Z S Ω A (cid:0) Ξ AB ∇ C Σ CB − Σ AB ∇ C Ξ CB (cid:1) + ∇ A Ω B Σ CB Ξ CA , (7.5)where the second line is obtained by integrating by parts the term Ω A Σ CB ∇ B Ξ CA . Remark 7.1.
The definition we take has the opposite sign to [20, (3),(4)].The discrepancy comes from the fact that Kerr spacetime has angular mo-mentum − ma under our definition.According to the main theorem of [9], B = O ( | u | − ) (7.6)as | u | → ∞ , where B = lim r →∞ r β . (see also [7], the paragraph afterequation (8) where B is denoted by B there)Estimate (7.6) and equation (2) of [7]divΞ = B (7.7)imply that Ξ = O ( | u | − ) (7.8)as | u | → ∞ and Σ → Σ ± (7.9)as u → ±∞ .By (7.8) and (7.9), R ∞−∞ ∂L∂u du is finite and furnishes the difference of theangular momenta at timelike infinity ( u → ∞ ) and spatial infinity ( u →−∞ ).We can write this in the spirit of [7]. In general, the peeling fails and β decays as β = o ( r − ). Christodoulou [8] observed that Bianchi equationnevertheless implies that R = lim C + u ,r →∞ r Dβ exists. Moreover, one has R = ∇ P + ∗∇ Q + 2Σ · B, (7.10)where ( P, Q ) = lim r →∞ ( r ρ, r σ ) and ∇ , ∗ , · are taken with respect to stan-dard metric σ on S . In order to exhibit a physically reasonable initial data set that has a com-plete Cauchy development without peeling, Christodoulou made the crucialassumption lim u →−∞ uR = R − = 0 , (7.11)which we adopt here.From (7.11) he derived [8, (5)] β = B ∗ r − log r + Br − + o ( r − )uniformly in u with 1-forms B ∗ and B on S satisfying [8, (6)] ∂B ∗ ∂u = 0 (7.12) ∂B∂u = 12 R. (7.13)Moreover, using Bianchi equation, he derived thatlim C + u ,r →∞ r α = A ∗ = 0exists. A ∗ is a symmetric traceless 2-tensor that is independent of u andsatisfies div A ∗ = − B ∗ . (7.14) Definition 7.2.
For a function f on S , we denote the projection of f on thesum of zeroth and first eigenspaces of ∆ by f [1] . Namely, f [1] = f ℓ =0 + f ℓ =1 .For a 1-form ω A = ∇ A f + ǫ AB ∇ B g , we denote ω A [1] = ∇ A f ℓ =1 + ǫ AB ∇ B g ℓ =1 .Since the spherical tangent vectors ∂ A have length O ( r ), we have thecorrespondence B A = − I A . (7.15)By (7.14), B ∗ [1] = 0 and we integrate(7.10) to get( I A ( u , x ) − I A ( u , x )) [1] = − Z u u ∇ A P ℓ =1 + ǫ AB ∇ B Q ℓ =1 + (2Σ AB B B ) [1] du. By (7.6) and (7.9), the last term is integrable on ( −∞ , ∞ ). For the firsttwo terms, we observe that equations (10, 11) of [8] ∂P∂u = −
12 div B + 12 Σ · ∂ Ξ ∂u (7.16) ∂Q∂u = −
12 curl B + 12 Σ ∧ ∂ Ξ ∂u (7.17)infer that P ℓ =1 ( u, x ) = a i ˜ X i + O ( | u | − ) and Q ℓ =1 ( u, x ) = b i ˜ X i + O ( | u | − )for some constants a i , b i independent of u . Thanks to the main theorem of[9], P − P ℓ =0 , Q − Q ℓ =0 = O ( | u | − ), we have a i = 0 , b i = 0. Thus P ℓ =1 , Q ℓ =1 are also integrable on ( −∞ , ∞ ). VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 31
We conclude lim u →∞ I A [1] ( u, x ) − lim u →−∞ I A [1] ( u, x ) exists and is givenby − Z ∞−∞ (cid:0) ∇ A P ℓ =1 + ǫ AB ∇ B Q ℓ =1 + (2Σ AB B B ) [1] (cid:1) du ′ . By (7.3) and Rizzi’s definition (7.4), we can interpret the following formulaas a conservation law of angular momentumlim u →∞ (cid:0) I A − Σ AB ∇ C Σ CB (cid:1) [1] − lim u →−∞ (cid:0) I A − Σ AB ∇ C Σ CB (cid:1) [1] = 12 Z ∞−∞ −∇ A P ℓ =1 − ǫ AB ∇ B Q ℓ =1 + (cid:0) Ξ AB ∇ C Σ CB − Σ AB ∇ C Ξ CB (cid:1) [1] du. (7.18)From Proposition A.2 and Proposition A.6, it follows that the co-closed partof the above conservation law is equivalent to the total flux of the classicalangular momentum in a Bondi-Sachs coordinate system.7.2. A duality paradigm for null infinity.
In this subsection, we de-scribe a duality paradigm for null infinity which creates a pair of dual space-times with the same classical conserved quantities.
Corollary 7.3 (Corollary 1.7) . Given a set of null infinity data ( m, N A , C AB , N AB ) defined on [ u , u ] × S , there exists a dual set of null infinity data ( m ∗ , N ∗ A , C ∗ AB , N ∗ AB ) that has the same (classical) energy, linear momentum, angular momentum,and center of mass.Proof. Define C ∗ AB = ε ( C AB ) on [ u , u ] × S . Then N ∗ AB = ∂ u C ∗ AB = ε ( N AB ). Define m ∗ ( u, x ) by the differential equation ( m ∗ ( u , x ) = m ( u , x ) ∂ u m ∗ = ∇ A ∇ B N ∗ AB − N ∗ AB N ∗ AB and then define N ∗ A by the differential equation N ∗ A ( u , x ) = N A ( u , x ) ∂ u N ∗ A = ∇ A m ∗ − ∇ D ( ∇ D ∇ E C ∗ EA − ∇ A ∇ E C ∗ ED )+ ∇ A ( C ∗ BE N ∗ BE ) − ∇ B ( C ∗ BD N ∗ DA ) + C ∗ AB ∇ D N ∗ DB . For this subsection alone, we denote the classical conserved quantities of theinfinity data ( m, N A , C AB , N AB ) by E, P k , J k , C k and denote the classicalconserved quantities of the data ( m ∗ , N ∗ A , C ∗ AB , N ∗ AB ) by E ∗ , P ∗ k , J ∗ k , C ∗ k ,we have E ∗ ( u ) = E ( u ) , P ∗ k ( u ) = P k ( u )and since C ∗ A D ∇ B C ∗ BD = C DA ∇ B C BD , C ∗ DE C ∗ DE = C DE C DE , J ∗ k ( u ) = J k ( u ) , C ∗ k ( u ) = C k ( u ) . It remains to show that the evolutions of the conserved quantities areidentical. Recall that the potentials of C ∗ AB and N ∗ AB are given by ( − c, c ) and ( − n, n ). We observe that replacing ( c, c, n, n ) by ( − c, c, − n, n ) does notchange the following expressions ∂ u E = − Z S [ n ∆(∆ + 2) n + n ∆(∆ + 2) n ] ,∂ u P k = − Z S ˜ X k [((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n ] ,∂ u J k = 18 Z S ˜ X k ǫ AB [ ∇ A c ∇ B ∆(∆ + 2) n + ∇ A c ∇ B ∆(∆ + 2) n ] ∂ u C k = 18 Z S ˜ X k [((∆ + 2) n ) + ((∆ + 2) n ) − ǫ AB ∇ A n ∇ B (∆ + 2) n ]+ 116 Z S [ ˜ X k (∆(∆ + 2) c (∆ + 2) n − ∆(∆ + 2) n (∆ + 2) c )]+ 116 Z S [ ˜ X k (∆(∆ + 2) c (∆ + 2) n − ∆(∆ + 2) n (∆ + 2) c ]This finishes the proof. (cid:3) The case of quadrupole moments
In this section, we consider the case of generalized quadrupole moments.Namely, all c, c, n, n are ( −
6) eigenfunctions (or ℓ = 2 spherical harmonics).Therefore, c = P c ij ( u ) ˜ X i ˜ X j , c = P c ij ( u ) ˜ X i ˜ X j n = P n ij ( u ) ˜ X i ˜ X j , n = P n ij ( u ) ˜ X i ˜ X j with ∂ u c ij = n ij and ∂ u c ij = n ij .8.1. Classical conserved quantities.
Next we compute the evolution ofclassical angular momentum and center of mass for quadrupole moments.
Lemma 8.1.
Suppose f ij and g ij are both symmetric, traceless × ma-trices. Then Z S ( f ij ˜ X i ˜ X j ) = 8 π X ij f ij (8.1) Z S ˜ X p ǫ AB ∇ A ( f ij ˜ X i ˜ X j ) ∇ B ( g kl ˜ X k ˜ X l ) = 16 π X j f ij g jk ǫ ikp . (8.2) Proof.
Both formulae follow from Lemma 5.3 of [6] Z S ˜ X i ˜ X j ˜ X k ˜ X l = 4 π
15 ( δ ij δ kl + δ ik δ jl + δ il δ jk ) . (cid:3) Combining the above lemma with Theorem 1.3 and Proposition 4.1, weconclude that
VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 33
Proposition 8.2.
Suppose c = P c ij ( u ) ˜ X i ˜ X j , c = P c ij ( u ) ˜ X i ˜ X j n = P n ij ( u ) ˜ X i ˜ X j , n = P n ij ( u ) ˜ X i ˜ X j , then ∂ u E = − π X ij n ij + X ij n ij ) ∂ u P k = − π X n ij n jp ǫ ipk ∂ u ˜ J k = 16 π X ( c ij n jp + c ij n jp ) ǫ ipk ∂ u ˜ C k = 32 uπ X n ij n jp ǫ ipk . (8.3)8.2. CWY angular momentum and center of mass.
Next we com-pute the evolution of the CWY angular momentum and center of mass forquadrupole moments. We need the following lemma.
Lemma 8.3.
Suppose the potentials of the news tensor are of mode ℓ = 2 .Namely, N AB = ∇ A ∇ B n −
12 ∆ nσ AB + 12 ( ǫ AC ∇ B ∇ C n + ǫ BC ∇ A ∇ C n ) where n = P ij n ij ˜ X i ˜ X j and n = P ij n ij ˜ X i ˜ X j satisfy P i n ii = P i n ii = 0 .Introduce two ℓ = 2 spherical harmonics Q = X i,k,l ( n ik n il ˜ X k ˜ X l ) − X i,j n ij ,Q = X i,k,l ( n ik n il ˜ X k ˜ X l ) − X i,j n ij . Then (1) the ℓ = 2 component of N AB N AB is − Q − Q, (2) the odd mode component of N AB N AB is ǫ ikm n ij n kl ˜ X m ( δ jl − ˜ X j ˜ X l ) . Here ǫ ijk is the Levi-Civita symbol in three dimensions.Proof. Note that the even-mode components ( ℓ = 0 , ,
4) and odd-modecomponents ( ℓ = 1 ,
3) of N AB N AB are given by ∇ A ∇ B n ∇ A ∇ B n −
12 (∆ n ) + ∇ A ∇ B n ∇ A ∇ B n −
12 (∆ n )
24 P.-N. CHEN, J. KELLER, M.-T. WANG, Y.-K. WANG, AND S.-T. YAU and ( ∇ A ∇ B n −
12 ∆ nσ AB )( ǫ AC ∇ B ∇ C n + ǫ BC ∇ A ∇ C n )= 2 n ij ∇ A ˜ X i ∇ B ˜ X j ( ǫ AC ∇ B ∇ C n + ǫ BC ∇ A ∇ C n )= 2 n ij ∇ A ˜ X i ∇ B ˜ X j · n kl ( ǫ AC ∇ B ˜ X k ∇ C ˜ X l + ǫ BC ∇ A ˜ X k ∇ C ˜ X l )= 8 ǫ ikm n ij n kl ˜ X m ( δ jl − ˜ X j ˜ X l )respectively. In the last equality we use the identity ǫ AB ∇ A ˜ X i ∇ B ˜ X j = ǫ ijk ˜ X k and ∇ B ˜ X i ∇ B ˜ X j = δ ij − ˜ X i ˜ X j .For (1), we compute ∇ A ∇ B n ∇ A ∇ B n = 20 n − Q + 43 X ij n ij . Since the space of ℓ = 4 spherical harmonics is spanned by˜ X i ˜ X j ˜ X k ˜ X l + 135 (cid:16) δ ij δ kl + δ ik δ jl + δ il δ jk (cid:17) − (cid:16) ˜ X i ˜ X j δ kl + ˜ X i ˜ X k δ jl + ˜ X i ˜ X l δ jk + ˜ X j ˜ X k δ il + ˜ X j ˜ X l δ ik + ˜ X k ˜ X l δ ij (cid:17) , (8.4)the ℓ = 2 component of n is Q . Putting these together, we obtain (1). (cid:3) In the case of quadrupole moments, the CWY angular momentum andcenter of mass take the form: J k = Z S ǫ AB ∇ B ˜ X k [ N A − C DA ∇ B C DB − c ∇ A m ] (8.5) C k = Z S ∇ A ˜ X k [ N A − u ∇ A m − C DA ∇ B C DB − ∇ A ( C DE C DE ) − c ǫ AB ∇ B m ] , (8.6)Therefore, J k = ˜ J k − Z S ˜ X k ǫ AB ∇ A c ∇ B b mC k = ˜ C k + 2 Z S ˜ X k ǫ AB ∇ A c ∇ B b m + 14 Z S ˜ X k ǫ AB ∇ A c ∇ B ∆(∆ + 2) c, where we use (3.2) and (2.7).The evolution formulae for J k and C k are thus ∂ u J k = ∂ u ˜ J k − Z S ˜ X k ǫ AB ∇ A n ∇ B b m − Z S ˜ X k ǫ AB ∇ A c ∇ B ∂ u b m∂ u C k = ∂ u ˜ C k + 2 Z S ˜ X k ǫ AB ∇ A n ∇ B b m + 2 Z S ˜ X k ǫ AB ∇ A c ∇ B ∂ u b m + 14 Z S ˜ X k ǫ AB ∇ A n ∇ B ∆(∆ + 2) c + 14 Z S ˜ X k ǫ AB ∇ A c ∇ B ∆(∆ + 2) n VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 35
By Lemma 2.1, only the ℓ = 2 mode components of b m and ∂ u b m willsurvive in the above integrals.Denote the ℓ = 2 mode of b m by b m ℓ =2 = b m kl ˜ X k ˜ X l . By Lemma 8.3, weget ∂ u b m kl = 67 X i ( n ik n il + n ik n il ) − δ kl X i,j ( n ij + n ij ) . By (8.2), we obtain the evolution equation of J k and C k : Proposition 8.4.
Suppose c = P c ij ( u ) ˜ X i ˜ X j , c = P c ij ( u ) ˜ X i ˜ X j , n = P n ij ( u ) ˜ X i ˜ X j , n = P n ij ( u ) ˜ X i ˜ X j , then ∂ u E = − π X ij n ij + X ij n ij ) ∂ u P k = − π X i,j,p n ij n jp ǫ ipk ∂ u J k = 16 π X i,j,p (3 c ij n jp + 3 c ij n jp − n ij b m jp − c ij ∂ u b m jp ) ǫ ipk ∂ u C k = 16 π X i,j,p (2 n ij b m jp + 2 c ij ∂ u b m jp + 6 n ij c jp + 6 c ij n jp ) ǫ ipk + 32 uπ X n ij n jp ǫ ipk , (8.7) where b m kl is given by ∂ u b m kl = 67 [ X i ( n ik n il + n ik n il ) − δ kl X i,j ( n ij + n ij )] Appendix A. Christodoulou-Klainerman connectioncoefficients and curvature components inBondi-Sachs formalism
We write the limit of connection coefficients and curvature componentsdefined in [9, 7, 8] in terms of the Bondi-Sachs metric coefficients.We choose the null vector fields L = ∂∂r and L = U (cid:0) ∂ u − W D ∂ D − V ∂ r (cid:1) ,which satisfy h L, L i = − Definition A.1.
The second fundamental forms and torsion are defined by χ AB = h D A L, ∂ B i = 12 tr χg AB + b χ AB χ AB = h D A L, ∂ B i = 12 tr χg AB + b χ AB ζ A = 12 h D A L, L i Their limit as r → ∞ are defined byΣ = lim r →∞ b χ Ξ = lim r →∞ r − b χZ = lim r →∞ rζ. They are related to the metric coefficients in the corresponding Bondi-Sachs coordinate system as follows:
Proposition A.2. Σ AB = − C AB Ξ AB = N AB Z A = − ∇ B C AB Proof.
Starting with g AB = r σ AB + rC AB + O (1), the determinant conditiongives tr χ = r and we compute χ AB = rσ AB + 12 C AB + O ( r − )to get Σ AB = − C AB . Direct computation gives χ AB = r ( − σ AB + ∂ u C AB ) + O (1)and hence tr χ = − r + O ( r − ) and b χ AB = r∂ u C AB + O (1). The limit oftorsion follows from ζ A = − r W ( − A + O ( r − ). (cid:3) Definition A.3.
The mass aspect and conjugate mass aspect function ofChristodoulou-Klainerman are defined by µ = K + 14 tr χ tr χ − div ζ µ = K + 14 tr χ tr χ + div ζ. Here K denotes the Gauss curvature of the two-sphere r =const. Theirlimits are defined by N = lim r →∞ r µN = lim r →∞ r µ. We express them in terms of the Bondi-Sachs metric coefficients as follows:
Proposition A.4. N = 2 m + 12 ∇ A ∇ B C AB , N = 2 m − ∇ A ∇ B C AB Proof.
We compute K = r + r ∇ A ∇ B C AB + O ( r − ) and tr χ tr χ = − r + r (2 m − ∇ A ∇ B C AB ) and the assertion follows. (cid:3) VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 37
We turn to curvature components. The convention of Riemann curvaturetensor is R ( X, Y ) Z = ( D X D Y − D Y D X − D [ X,Y ] ) ZR ( X, Y, W, Z ) = h R ( X, Y ) Z, W i . Definition A.5.
Define the curvature components α AB = R ( ∂ A , L, ∂ B , L ) β A = 12 R ( ∂ A , L, L, L ) ρ = 14 R ( L, L, L, L ) σ/ǫ AB = 12 R ( ∂ A , ∂ B , L, L ) β A = 12 R ( ∂ A , L, L, L )Here /ǫ AB dx A ∧ dx B is the area form of the two-sphere with respect to g AB .Their limits are defined by A AB = lim r →∞ r − α AB B A = lim r →∞ rβ A P = lim r →∞ r ρQ = lim r →∞ r σB A = lim r →∞ r β A Note that (
A, B ) were denoted by (
A, B ) in [7].We express them in terms of the Bondi-Sachs metric coefficients as follows:
Proposition A.6. A AB = − ∂ u N AB B A = ∇ B N AB P = − m − C AB N AB Q = ǫ AB (cid:18) − C DA N DB − ∇ A ∇ D C DB (cid:19) B A = − N A Proof.
The formula for A is obtained from (6) of [7], 2 ∂ Ξ ∂u = − A , which isthe rescaled limit of the propagation equation b D b χ = − α .The formula for B is obtained from (2) of [7], ∇ B Ξ AB = B A , which is therescaled limit of the Codazzi equation / div b χ − b χ · ζ = (cid:0) / ∇ tr χ − tr χζ (cid:1) + β . The formula for P and Q are obtained from (3) of [7], ǫ AB ∇ A Z B = Q −
12 Σ ∧ Ξ , ∇ A Z A = N + P −
12 Σ · Ξ , which is the rescaled limit of the Hodge system /curlζ = σ − b χ ∧ b χ, / div ζ = µ + ρ − b χ · b χ. Finally, we consider the Codazzi equation / div b χ + b χ · ζ = 12 (cid:0) / ∇ tr χ + tr χζ (cid:1) − β. Its leading order at O ( r − ) leads to (1) of [7] and its subleading order at O ( r − ) leads to (cid:18) − ∂ A | C | + 12 C BD ∇ D C AB + 14 ∇ A C ED C DE + 12 ∇ D C DE C AE (cid:19) + 14 C AB ∇ D C BD = ζ ( − A − B A . We simplify the second term in the parentheses by the identity ∇ ( D C B ) A = ∇ A C BD + ∇ E C AE σ BD − ∇ E C E ( D C B ) A and the left-hand side becomes ∂ A | C | + C AB ∇ D C BD . Direct computation yields ζ ( − A = − N A + ∂ A | C | + C AB ∇ D C BD and the formula for B follows. (cid:3) Appendix B. Integration by Part Formula
In this section, we prove two integration formula that will be used tocompute angular momentum and center of mass in spacetime with vanishingnews.
Theorem B.1.
Let F AB = 2 ∇ A ∇ B f − ∆ f σ AB and P BA = ∇ B ∇ D C DA −∇ A ∇ D C DB . Then Z S Y A (cid:18) C AB ∇ D F DB + 14 F AB ∇ D C DB − P BA ∇ B f − ∇ B P BA f (cid:19) = 0 . Proof.
We integrate by parts the last two terms to get Z S − Y A ( ∇ B ∇ D C AD − ∇ A ∇ D C BD ) ∇ B f + 12 ∇ B Y A ∇ B ∇ D C AD · f. = Z S ∇ B Y A ∇ D C AD ∇ B f + 12 Y A ∇ D C AD ∆ f − Y A ∇ D C BD ∇ A ∇ B f + Z S Y A ∇ D C AD f − ∇ B Y A ∇ D C AD ∇ B f = Z S − Y A ∇ D C BD ( ∇ A ∇ B f −
12 ∆ f σ AB ) + 14 Y A ∇ D C AD (∆ + 2) f (cid:3) VOLUTION OF CONSERVED QUANTITIES AT NULL INFINITY 39
Theorem B.2.
Let F AB = 2 ∇ A ∇ B f − ∆ f σ AB and P BA = ∇ B ∇ D C DA −∇ A ∇ D C DB . Then Z S ∇ A ˜ X k (cid:18) C AB ∇ D F BD + F AB ∇ D C BD + 12 ∇ A ( C BD F BD ) − f ∇ B P BA − P BA ∇ B f (cid:19) = 0 . Proof.
We integrate by parts the last two terms to get Z S ∇ A ˜ X k ( − P BA ) ∇ B f = Z S − ∇ A ˜ X k ( ∇ B ∇ D C DA − ∇ A ∇ D C DB ) ∇ B f = Z S − X k ∇ D C DA ∇ A f + 2 ∇ A ˜ X k ∇ D C DA ∆ f + 4 ˜ X k ∇ D C DB ∇ B f − ∇ A ˜ X k ∇ D C DB ∇ A ∇ B f = Z S X k ∇ D C DA ∇ A f − ∇ A ˜ X k ∇ D C DB F AB + ∇ A ˜ X k ∇ D C DA ∆ f = Z S − ∇ D ˜ X k C DA ∇ A f − X k C DA ∇ D ∇ A f − ∇ A ˜ X k ∇ D C DB F AB + ∇ A ˜ X k ∇ D C DA ∆ f = Z S ∇ A ˜ X k ∇ D C DA ∇ A f + 12 ∆ ˜ X k C DA F DA − ∇ A ˜ X k ∇ D C DB F AB + ∇ A ˜ X k ∇ D C DA ∆ f = Z S −∇ A ˜ X k C DA ∇ D (∆ + 2) f − ∇ A ˜ X k ∇ A ( C DB F DB ) − ∇ A ˜ X k ∇ D C DB F AB . (cid:3) References [1] A. Ashtekar and R. O. Hansen,
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