Supertranslation invariance of angular momentum
aa r X i v : . [ g r- q c ] F e b Supertranslation invariance of angular momentum
Po-Ning Chen
University of California Riverside,Department of Mathematics,900 University Avenue,Riverside, California 92521, USA
Mu-Tao Wang
Columbia University,Department of Mathematics,2990 Broadway, New York,New York 10027, USA
Ye-Kai Wang
National Cheng Kung University,Department of Mathematics,1 Dasyue Road,Tainan City 70101, Taiwan
Shing-Tung Yau
Harvard University,Department of Mathematics,One Oxford Street, Cambridge,Massachusetts 02138, USA (Dated: February 8, 2021) bstract LIGO’s successful detection of gravitational waves has revitalized the theoretical understandingof the angular momentum carried away by gravitational radiation. An infinite dimensional super-translation ambiguity has presented an essential difficulty for decades of study. Recent advanceswere made to address and quantify the supertranslation ambiguity in the context of compact binarycoalescence. Here we present the first definition of angular momentum in general relativity that iscompletely free from supertranslation ambiguity. The new definition was derived from the limit ofthe quasilocal angular momentum defined previously by the authors. A new definition of center ofmass at null infinity is also proposed and shown to be supertranslation invariant. Together withthe classical Bondi-Sachs energy-momentum, they form a complete set of conserved quantities atnull infinity that transform according to basic physical laws.
PACS numbers: . INTRODUCTION The definitions of conserved quantities such as mass and angular momentum have beenamong the most difficult problems since the genesis of general relativity. According to Ein-stein’s equivalence principle, there is no density for gravitation and no canonical coordinatesystem for spacetime. The issue is further complicated by the nonlinear nature of Einstein’seponymous equation. One of the most important problems is the definition of angular mo-mentum for a distant observer, or angular momentum at null infinity, a notion that hasbeen studied for decades [3, 4, 13]. An essential difficulty is presented by the ambiguityof supertranslations, an infinite dimensional subgroup of the Bondi-Metzner-Sachs (BMS)group [5]. According to Penrose [12], the very concept of angular momentum gets shifted bythese supertranslations and “it is hard to see in these circumstances how one can rigorouslydiscuss such questions as the angular momentum carried away by gravitational radiation”(page 654 of [12]). The astronomical event GW150914 [1] observed by LIGO correspondsto the coalescence of a binary black hole, and recent advances [2] were made to address andquantify the supertranslation ambiguity in general compact binary coalescences. Neverthe-less, to deal with general isolated systems it is desirable to have a rigorous definition thatis free from any supertranslation ambiguity. In this article, we propose the first definitionof angular momentum at null infinity that is supertranslation invariant. The definition isderived as the limit of quasilocal angular momentum that was proposed in [8] and evalu-ated at null infinity in [7]. Comparing with existing definitions, the new definition containsan important correction term (that has never appeared in any previous definitions), whichcomes from solving the optimal isometric embedding equation in the theory of Wang-Yauquasilocal mass [15]. This provides the reference term that is critical in the Hamiltonianapproach of defining conserved quantities. Our theory also produces a definition of centerof mass at null infinity which is shown to be supertranslation invariant as well.
II. CONSERVED QUANTITIES IN BONDI-SACHS COORDINATES
We consider a Bondi-Sachs coordinate system ( u, r, x , x ) in which the physical spacetimemetric takes the form − U V du − U dudr + r h AB ( dx A + W A du )( dx B + W B du ) , A, B = 2 , . (1)3he future null infinity I + corresponds to the idealized null hypersurface r = ∞ and canbe viewed I + = I × S with coordinates ( u, x ) where u ∈ I and x = ( x , x ) ∈ S , the unitsphere. The outgoing radiation condition [14] implies the following expansions in inversepowers of r : V = 1 − mr + 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 + O ( r − ) , where σ AB ( x ) is a standard round metric on S and ∇ A denotes the covariant derivativewith respect to σ AB . The indices are contracted, raised, and lowered with respect to themetric σ AB . Defining on I + ( r = ∞ ) are the mass aspect m = m ( u, x ), the angular aspect N A = N A ( u, x ), and the shear C AB = C AB ( u, x ) of this Bondi-Sachs coordinate system. Wealso define the news N AB = ∂ u C AB .The standard formulae for the Bondi-Sachs energy-momentum [5] at a u cut along I + are E ( u ) = Z S m ( u, · ) , P k ( u ) = Z S m ( u, · ) ˜ X k , k = 1 , , X k , k = 1 , , R restricted to S .In order to define the angular momentum, we consider the decomposition of C AB into C AB = ∇ A ∇ B c − σ AB ∆ c + 12 ( ǫ EA ∇ E ∇ B c + ǫ EB ∇ E ∇ A c ) (3)where ǫ AB denotes the volume form of σ AB . c = c ( u, x ) and c = c ( u, x ) are the closed andco-closed potentials of C AB ( u, x ). They are chosen to be of ℓ ≥ I + consists of the BMS fields [2, 14]. We say a BMS field Y is a rotation BMS field if in a Bondi-Sachs coordinate system ( u, x ), Y = ˆ Y A ∂∂x A (4)where ˆ Y A ( x ) is a rotation Killing field on S . We define the angular momentum with respectto a rotation BMS field Y in the following: Definition of angular momentum:
For a rotation BMS field Y that is tangent to u cuts on I + , the angular momentum of a u cut is defined to be : J ( u, Y ) = Z S Y A (cid:18) N A − C AB ∇ D C DB − c ∇ A m (cid:19) ( u, · ) (5)4uppose (¯ u, ¯ x ) is another Bondi-Sachs coordinate system, we define similarly: J (¯ u, ¯ Y ) = Z S ¯ Y A (cid:18) ¯ N A −
14 ¯ C DA ¯ ∇ B ¯ C DB − ¯ c ¯ ∇ A ¯ m (cid:19) (¯ u, · ) , (6)where ¯ Y is a rotation BMS field that is tangent to ¯ u cuts.A supertranslation is a change of Bondi-Sachs coordinates (¯ u, ¯ x ) → ( u, x ) such that u = ¯ u + f ( x ) , x = ¯ x (7)on I + for a function f that is defined on S . Two rotation BMS fields ¯ Y and Y are said tobe related by the supertranslation f if, in the ( u, x ) coordinate system¯ Y = ˆ Y A ∂∂x A + ˆ Y ( f ) ∂∂u and Y = ˆ Y A ∂∂x A (8)for a rotation Killing field ˆ Y on S . In this case, ¯ Y is tangent to the ¯ u cuts while Y istangent to the u cuts.We show (Theorem 1) that the total fluxes of J ( u, Y ) and J (¯ u, ¯ Y ) are the same when Y and ¯ Y are related by a supertranslation f of harmonic mode ℓ ≥
2, thus removing thesupertranslation ambiguity.The expression (5) originated from the definition of quasilocal angular momentum,which was proposed in [8] to complement the definitions of quasilocal mass-energy-momentum in [15]. Their limits at null infinity were evaluated in [7]. The expression R S Y A (cid:0) N A − C AB ∇ D C DB (cid:1) ( u, · ) := ˜ J ( u, Y ) in (5) already appeared in other definitions ofangular momentum [3, 10], while the last term in (5) that involves c is new and plays anindispensable role in supertranslation invariance. This term arises naturally from solvingthe optimal isometric embedding in the theory of Wang-Yau quasilocal mass [15] and pro-vides the reference term that is critical in the Hamiltonian approach of defining conservedquantities. Rizzi’s angular momentum definition [13] is essentially ˜ J ( u, Y ) in the frameworkof [9]. Without a suitable reference term, his definition is only valid for a restricted class offoliations at null infinity, and does not satisfy the supertranslation invariance property. III. SUPERTRANSLATION INVARIANCE OF TOTAL FLUXES
We assume I + extends from i ( u = −∞ ) to i + ( u = + ∞ ) and that there exists aconstant ε > N AB ( u, x ) = O ( | u | − − ε ) as u → ±∞ . (9)5 IG. 1: A supertranslation that maps ¯ u cuts to u = ¯ u + f ( x ) cuts. Y and ¯ Y are rotation BMSfields such that Y is tangent to u cuts and ¯ Y is tangent to ¯ u cuts. The total flux of the angular momentum (5) is defined to be δJ ( Y ) = lim u → + ∞ J ( u, Y ) − lim u →−∞ J ( u, Y ) . When two Bondi-Sachs coordinates are related by a supertranslation, one shows that (see(15) below) m (+) = lim u → + ∞ m ( u, x ) = lim ¯ u → + ∞ ¯ m (¯ u, x ) and m ( − ) = lim u →−∞ m ( u, x ) = lim ¯ u →−∞ ¯ m (¯ u, x ) (10)are two functions on S . Theorem 1- Under condition (9) , suppose two Bondi-Sachs coordinate systems are relatedby a supertranslation f , and Y and ¯ Y are rotation BMS fields related by f (8) . Then δJ ( ¯ Y ) − δJ ( Y ) = − Z S (cid:16) f ℓ ≤ ˆ Y A ∇ A ( m (+) − m ( − )) (cid:17) , (11) where f = f ℓ ≤ + f ℓ ≥ is the decomposition into the corresponding harmonic modes.Proof. The Einstein equation implies (see [10, 11]): ∂ u m = − N AB N AB + 14 ∇ A ∇ B N AB ∂ 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 . (12)6e calculate ∂ u ˜ J ( u, Y ) = 14 Z S Y A (cid:2) C AB ∇ D N BD − N AB ∇ D C BD − ∇ B ( C BD N DA ) (cid:3) (13)According to (5), the total flux δJ ( Y ) is thus δJ ( Y ) = δ ˜ J ( Y ) − (cid:20)Z S Y A c ∇ A m (cid:21) u =+ ∞ u = −∞ , (14)where δ ˜ J ( Y ) = R ∞−∞ R S Y A (cid:2) C AB ∇ D N BD − N AB ∇ D C BD − ∇ B ( C BD N DA ) (cid:3) ( u, · ) du .For a supertranslation (7), it is known that 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 ), the shear C AB ( u, x ), and the news N AB ( u, x ) in the ( u, x ) coordinate systemthrough: ¯ 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 ) (15)See [10, (C.117) and (C.119)] for example.The decay condition of the news (9) implies that the limits of the mass aspect and theshear satisfylim ¯ u →±∞ ¯ m (¯ u, x ) = lim u →±∞ m ( u, x ) , lim ¯ u →±∞ ¯ C AB (¯ u, x ) = lim u →±∞ C AB ( u, x ) − ∇ A ∇ B f + ∆ f σ AB . (16)In particular, the limit of the potential c satisfies:lim ¯ u →±∞ c (¯ u, x ) = lim u →±∞ c ( u, x ) − f ℓ ≥ . It follows that the contribution from the second term of (14) in the difference δJ ( ¯ Y ) − δJ ( Y )is R S [2 f ℓ ≥ ˆ Y A ∇ A ( m (+) − m ( − ))]. On the other hand, the contribution from the first term in(14), or δ ˜ J ( ¯ Y ) − δ ˜ J ( Y ), after substituting (15), integration by parts, and change of variables,is shown to be ( [6], see also [2])14 Z + ∞−∞ (cid:20)Z S f Y A ∇ A (cid:0) N BD N BD − ∇ B ∇ D N BD (cid:1)(cid:21) du.
7t this point, we invoke the Einstein equation (12) and rewrite the last integral as R S [ − f ˆ Y A ∇ A ( m (+) − m ( − ))]. Therefore δJ ( ¯ Y ) − δJ ( Y ) is given by (11). By (2), thetotal flux of linear momentum is δP k = 2 R S ( m (+) − m ( − )) ˜ X k . It follows that δJ ( ¯ Y ) = δJ ( Y ) + α i ε ikj δP j , if f = α + α i ˜ X i + f ℓ ≥ and ˆ Y A = ǫ AB ∇ B ˜ X k (17)In particular, if f is of harmonic mode ℓ ≥ δJ ( ¯ Y ) = δJ ( Y ) is invariant. IV. SPACETIME WITH VANISHING NEWS
We consider a non-radiative spacetime in the sense that the news vanishes. This includesall model spacetimes such as Minkowski and Kerr. If in a Bondi-Sachs coordinate system( u, x ), N AB ( u, x ) vanishes, by (12) the mass aspect m ( u, x ) = ˚ m ( x ) is a function on S . Theorem 2- Under the same assumption as Theorem 1, if in addition the news vanishes,then J (¯ u, ¯ Y ) ≡ J ( ¯ Y ) and J ( u, Y ) ≡ J ( Y ) are independent of ¯ u and u , respectively, and arerelated by J ( ¯ Y ) − J ( Y ) = − Z S (2 f ℓ ≤ ˆ Y A ∇ A ˚ m ) , (18)where ˚ m is the mass aspect. Proof.
The vanishing of news also implies C AB and thus c are independent of u . Theconstancy of J ( u, Y ) follows from (13). We denote C AB ( u, x ) = ˚ C AB ( x ) and c ( u, x ) = ˚ c ( x ).The exact formula for the angular aspect on a spacetime with vanishing news is obtainedby integrating (12) with respect to u : N A ( u, x ) = N A ( u , x ) + ( u − u )( ∇ A ˚ m − ∇ B ˚ P BA ) (19)for any u and fixed u , where ˚ P BA = ∇ B ∇ E ˚ C EA − ∇ A ∇ E ˚ C EB .Suppose (¯ u, ¯ x ) is related to ( u, x ) by a supertranslation f (7). By (15), ¯ N AB (¯ u, x ) ≡ J (¯ u, ¯ Y ) is independent of ¯ u . In addition, their mass aspects, shears, and shear potentialsare related by ˚¯ m = ˚ m, ˚¯ C AB = ˚ C AB − F AB , and ˚¯ c = ˚ c − f ℓ ≥ , (20)where F AB = 2 ∇ A ∇ B f − ∆ f σ AB .In this case, the angular aspect transforms according to [10, (C.123)] ¯ N A (¯ u, x ) = N A (¯ u + f, x ) + 3˚ m ∇ A f − ˚ P BA ∇ B f [16]. Combining this with (19) and set 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 (21)8or any ¯ u and fixed u .Let J (¯ u , ¯ Y ) (resp. J ( u , Y )) be the angular momentum of the ¯ u = ¯ u (resp. u = u ) cutin the (¯ u, ¯ x ) (resp. ( u, x )) coordinate system. Taking their difference and applying (20), J (¯ u , ¯ Y ) − J ( u , Y ) = 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 + Z S (2 f ℓ ≥ ˆ Y A ∇ A ˚ m ) . We then apply (21) to show that the sum of the first two integrals on the above right handside is R S ( − f ˆ Y A ∇ A ˚ m ) and J ( ¯ Y ) − J ( Y ) is of the desired expression (18), see [6] for details.In particular, J ( ¯ Y ) = J ( Y ) + α i ε ikj P j , if f = α + α i ˜ X i + f ℓ ≥ and ˆ Y A = ǫ AB ∇ B ˜ X k . (22) V. CONSERVED QUANTITIES ON NULL INFINITY
The limit of the quasilocal center of mass defined in [8] also gives a new definition ofcenter of mass at null infinity [7]. We say that a BMS field Y is a boost BMS field if in aBondi-Sachs coordinate system ( u, x ), Y = ˆ Y A ∂∂x A + u ˜ X ∂∂u , (23)where ˆ Y A = ∇ A ˜ X and ˜ X is a function on S of harmonic mode ℓ = 1. Definition of center of mass:
For a boost BMS field Y (23), the center of mass of a u cut isdefined to be: C ( u, Y ) = Z S ∇ A ˜ X (cid:18) N A − C AB ∇ D C DB − ∇ A (cid:0) C DE C DE (cid:1)(cid:19) ( u, · ) − u Z S ( ˜ Xm )( u, · )+ Z S c (cid:16) Xm − ∇ A ˜ X ∇ A m (cid:17) ( u, · )+ Z S (cid:18) ∇ A ˜ Xǫ AB ( ∇ B c ) m −
116 ˜ X ∇ A (∆ + 2) c ∇ A (∆ + 2) c (cid:19) ( u, · ) (24)Suppose (¯ u, ¯ x ) is related to ( u, x ) by a supertranslation f (7). Let ¯ Y be the boost BMSfield ¯ Y = ˆ Y A ∂∂ ¯ x A + ¯ u ˜ X ∂∂ ¯ u , Y A in (23). ¯ Y and Y (23) are said to be related by the supertrans-lation f and C (¯ u, ¯ Y ) is obtained by replacing m, C AB , N AB , N A , c, c, u in (24) with¯ m, ¯ C AB , ¯ N AB , ¯ N A , ¯ c, ¯ c, ¯ u . We show in [6] that their total fluxes are related by δC ( ¯ Y ) − δC ( Y ) = − Z S f ℓ ≤ ∇ A ˜ X ∇ A (cid:0) m (+) − m ( − ) (cid:1) + 6 Z S f ℓ ≤ ˜ X (cid:0) m (+) − m ( − ) (cid:1) , and therefore, δC ( ¯ Y ) − δC ( Y ) = α δP k + α k δE if f = α + α i ˜ X i + f ℓ ≥ and ˆ Y A = ∇ A ˜ X k . (25)When the spacetime has vanishing news, we show in [6] that C (¯ u, ¯ Y ) ≡ C ( ¯ Y ) and C ( u, Y ) ≡ C ( Y ) are independent of ¯ u and u , respectively, and are related by C ( ¯ Y ) − C ( Y ) = − Z S f ℓ ≤ ∇ A ˜ X ∇ A ˚ m + 6 Z S f ℓ ≤ ˜ X ˚ m. (26)Fixing ˜ X k , k = 1 , , J k ( u ) := J ( u, Y ) for Y = ǫ AB ∇ B ˜ X k ∂∂x A and C k ( u ) := C ( u, Y ) for Y = ∇ A ˜ X k ∂∂x A + u ˜ X k ∂∂u , we obtain the new definitions of angularmomentum J k ( u ) and center of mass C k ( u ). They complement the classical Bondi-Sachsenergy momentum E ( u ) , P k ( u ) and form a set of conserved quantities that correspond tothe Poincar´e symmetry. All of them can be derived from the limits of quasilocal conservedquantities defined in [8, 15]. The Poincar´e symmetry is due to the choice of Minkowski ref-erence and is acquired through the reference embedding into the Minkowski spacetime [15].The use of the Minkowski reference is essential. In the early days of the study of angularmomentum flux, there was confusion about nonzero flux in the Minkowski spacetime, whichwas eventually clarified in [4]. For our definitions of angular momentum and center of mass,not only that the fluxes are zero, but also that the angular momentum and center of massare zero in any Bondi-Sachs coordinate system of the Minkowski spacetime. VI. CONCLUSIONS
We obtain a complete set of ten conserved quantities (
E, P k , J k , C k ) at null infinity (allas functions of the retarded time u ) that satisfy the following properties:(1) ( E, P k , J k , C k ) all vanish for any Bondi-Sachs coordinate system of the Minkowskispacetime. 102) In a Bondi-Sachs coordinate system of the Kerr spacetime, P k and C k vanish, and E and J k recover the mass and angular momentum. ( E, P k , J k , C k ) are supertranslationinvariant.(3) If a spacetime admits a Bondi-Sachs coordinate system such that the news vanishes,then ( E, P k , J k , C k ) are constant (independent of the retarded time u ) and supertranslationinvariant.(4) On a general spacetime, the total fluxes of ( E, P k , J k , C k ) are supertranslation invari-ant.(5) ( E, P k , J k , C k ) and their fluxes transform according to basic physical laws (17), (25),(22), (26) under ordinary translations.Under additional assumptions and ∇ A m (+) = ∇ A m ( − ) = 0, the supertranslation in-variance holds by Remark 2 of the second paper in [2]. This manifests the importance ofthe correction term (the last term) in our definition of angular momentum (5) which issupertranslation invariant without any additional assumptions. We only consider super-translations which correspond to fixing the 2-metric σ AB at I + . The transformation ofangular momentum and fluxes under boosts, which change σ AB by conformal factors, willbe discussed in a forthcoming work. Acknowledgments
P.-N. Chen is supported by Simons Foundation collaboration grant [1] B. P. Abbott et al.
Phys. Rev. Lett. ravity 1 (1984), no. 1, 15–26, S. W. Hawking, M. J. Perry, and A. Strominger, Journal ofHigh Energy Physics, 2017(5):161, 2017.[4] A. Ashtekar and M. Streubel, Proc. Roy. Soc. Lond. A 376, 585 (1981).[5] H. Bondi, Nature, 186:535, May 1960, H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner,Proc. Roy. Soc. Ser. A 269 (1962) 21–52, R. K. Sachs, Proc. Roy. Soc. Ser. A 270 1962 103–126.[6] P.-N. Chen, J. Keller, M.-T. Wang, Y.-K. Wang, and S.-T. Yau, Evolution of angular momen-tum and center of mass at null infinity , preprint.[7] J. Keller, Y.-K. Wang, and S.-T. Yau, arXiv: 1811.02383, P.-N. Chen, M.-T. Wang, and S.-T.Yau, Comm. Math. Phys. 308 (2011), no.3, 845–863.[8] P.-N. Chen, M.-T. Wang, and S.-T. Yau, Comm. Math. Phys. 338 (2015), no.1, 31–80.[9] D. Christodoulou, Phys. Rev. Lett. 67 (1991), no. 12, 1486–1489.[10] P. T. Chru´sciel; J. Jezierski, J. Kijowski, Lecture Notes in Physics. Monographs, 70, Springer-Verlag, Berlin (2002).[11] T. M¨adler and J. Winicour, Scholarpedia, 11 (12): 33528, 2016.[12] R. Penrose, in Seminar on Differential Geometry, pp. 631–668, Ann. of Math. Stud., 102,Princeton Univ. Press, Princeton, N.J., 1982[13] A. Rizzi, Phys. Rev. Lett. 81 (1998), no. 6, 1150–1153.[14] R. Sachs, Phys. Rev. (2) 128 (1962), 2851–2864.[15] M.-T. Wang and S.-T. Yau, Phys. Rev. Lett. 102 021101 (2009).[16] Note that the convention of angular momentum aspect in [10] is − N A ..