Definitions of (super) angular momentum in asymptotically flat spacetimes: Properties and applications to compact-binary mergers
DDefinitions of (super) angular momentum in asymptotically flat spacetimes:Properties and applications to compact-binary mergers
Arwa Elhashash ∗ and David A. Nichols † Department of Physics, University of Virginia, P.O. Box 400714, Charlottesville, VA 22904-7414, USA (Dated: February 17, 2021)The symmetries of asymptotically flat spacetimes in general relativity are given by the Bondi-Metzner-Sachs (BMS) group, though there are proposed generalizations of its symmetry algebra.Associated with each symmetry is a charge and a flux, and the values of these charges and theirchanges can characterize a spacetime. The charges of the BMS group are relativistic angular mo-mentum and supermomentum (which includes 4-momentum); the extensions of the BMS algebraalso include generalizations of angular momentum called “super angular momentum.” Several dif-ferent formalisms have been used to define angular momentum, and they produce nonequivalentexpressions for the charge. It was shown recently that these definitions can be summarized in a two-parameter family of angular momenta, which we investigate in this paper. We find that requiringthat the angular momentum vanishes in flat spacetime restricts the two parameters to be equal. Ifwe do not require that the angular momentum agrees with a Hamiltonian definition, then there isno clear reason to fix the remaining free parameter to a particular value. We then also propose asimilar two-parameter family of super angular momentum. We examine the effect of the free pa-rameters on the values of the angular momentum and super angular momentum from nonprecessingbinary-black-hole mergers. The definitions of angular momentum differ at a high post-Newtonianorder for these systems, but only when the system is radiating gravitational waves (not before andafter). The different super-angular-momentum definitions occur at lower orders, and there is a dif-ference in the change of super angular momentum even after the gravitational waves pass, whicharises because of the gravitational-wave memory effect. We estimate the size of these effects usingnumerical-relativity surrogate waveforms and find they are small but resolvable.
CONTENTS
I. Introduction 1II. Bondi-Sachs framework, symmetries, andcharges 4A. Metric and Einstein’s equations 4B. Asymptotic symmetries 5C. Fluxes and charges 5D. Definitions of angular momentum and theirproperties 6E. Definitions of super angular momentum 7III. (Super) angular momentum in flat spacetime 7IV. Multipolar expansion of the (super) angularmomentum 9A. Spherical harmonics and multipolar expansionof the gravitational-wave data 9B. Multipolar expansion of the super angularmomentum 10C. Multipolar expansion of the intrinsic angularmomentum 12D. Multipolar expansion of the center-of-massangular momentum 12V. Standard and super angular momentum fornonprecessing BBH mergers 13 ∗ [email protected] † [email protected] A. Computing the leading GW memory effectand spin memory effect 141. (Displacement) GW memory effect 142. GW modes that produce the spin-memoryeffect 15B. Standard angular momentum 151. Post-Newtonian results 162. Results from NR surrogate models 17C. Super angular momentum 19VI. Conclusions 20Acknowledgments 21A. Conversion between STF tensors and sphericalharmonics 21References 23
I. INTRODUCTION
The LIGO, Virgo, and KAGRA collaborations havenow announced the detection of almost fifty binary-black-hole (BBH) mergers during the first three observing runsof the advanced-detector era beginning in 2015 [1, 2].There are a few ways in which these BBH mergers arecharacterized: for example, by the masses and spins ofthe individual black holes (BHs) plus the orbital elementsof the binary at a given reference frequency or by the fi-nal mass and spin of the BH formed after the merger a r X i v : . [ g r- q c ] F e b and ringdown (e.g., [1, 2]). An alternate way to char-acterize asymptotically flat systems is in terms of the“conserved” quantities conjugate to the symmetries ofasymptotically flat spacetimes and the net fluxes of theseconserved quantities. The symmetries of asymptoticallyflat spacetimes form the Bondi-Metzner-Sachs (BMS)group, which consists of transformations isomorphic tothe Lorentz group and supertranslations (of which thefour spacetime translations are a subgroup) [3–5]. Theradiated energy and linear momentum (often expressedas a recoil velocity) being the quantities conjugate to thetranslation symmetries are often quoted when describingBBH mergers (see, e.g., [6] and references therein).The flux of angular momentum (the quantity relatedto Lorentz symmetries) is somewhat more subtle. An-gular momentum must be computed about an origin inflat spacetime; in terms of the symmetries that form thePoincaré group, this implies that a translation must bespecified to identify the particular Lorentz transforma-tion under consideration. There is thus a four-parameterfamily of Lorentz transformations spanned by a basis ofthe spacetime translations in the Poincaré group. Inasymptotically flat spacetimes, this four-parameter fam-ily is enlarged to a countably infinite family of Lorentztransformations, each of which is associated with somebasis element of the infinite-dimensional supertranslationsubgroup in the BMS group. In stationary spacetimes,there is a natural way to choose a “preferred” set of su-pertranslations that reduces the dependence of the angu-lar momentum to a choice of origin as in flat spacetime(see [7, 8] or more recently [9]); however, in nonstation-ary solutions, there is no such natural choice, thoughthere are several different proposals to “fix” the super-translation freedom (see, e.g., [10] for a review). Theabsence of this preferred Poincaré group is referred to asthe “supertranslation ambigutity” of angular momentumin asymptotically flat spacetimes, which is, in essence,a statement that angular momentum in asymptoticallyflat spacetimes is different from its counterpart in flatspacetimes.This additional complexity in describing the value ofangular momentum for an asymptotically flat spacetimemay have contributed to it and its flux being less fre-quently quoted in the output of numerical-relativity (NR)simulations of merging black holes. The six degrees offreedom in the relativistic angular momentum are oftensplit into the three spin parts (corresponding to rota-tions) and three center-of-mass (CM) parts (correspond-ing to Lorentz boosts). Of these six components, themost commonly given from NR simulations of BBHs arethe magnitude of the final BH’s spin (though this spinis most often computed from quasilocal constructions onthe BH’s apparent horizon rather than in terms of quan-tities measured at or near future null infinity [11–13]);additional components of the angular momentum werecomputed in [14], for example.In addition to the supertranslation ambiguities, a num-ber of different definitions of the angular momentum of an asymptotically flat spacetime were (and continue to be)used. A nonexhaustive list of some of these definitions in-clude one based on the Landau-Lifshitz pseudotensor forthe intrinsic part of the angular momentum (in the CMframe of the source) [15], a definition based on construc-tions called “linkages” [16], ones inspired from twistor the-ory [17, 18], and those related to Hamiltonians conjugateto conserved quantities [19, 20]. When considered in theirrespective domains of validity, the different definitions ofthe angular momentum described above agree [20, 21].More recently, however, new definitions of angular mo-menta arose from revisiting the Landau-Lifshitz formal-ism when not restricted to the CM frame [22] and fromconsiderations about soft theorems [23] (particularly asubleading correction to Weinberg’s soft theorem [24];see [25] for a review).It was pointed out in [26] that these new definitions ofangular momentum differ from the Hamiltonian defini-tion of Wald and Zoupas [20]. Moreover, it was shownthat the discrepancies in these definitions can be writ-ten in terms of two functions that are quadratic in theshear related to the outgoing GWs in asymptotically flatspacetimes. The different definitions were parametrizedin terms of two real coefficients multiplying these twoquadratic functions, respectively, and when the coeffi-cients equal one, the Hamiltonian definition of [20] is re-covered. All members of this two-parameter family ofangular momenta satisfy flux balance laws, are covariantwith respect to quantities defined on 2-sphere cross sec-tions of null infinity, and lead to the same correspondencewith the subleading soft theorem [26]. This led Compère et al . in [26] to conclude that there was not a compellingphysical reason to prefer one definition over another andto suggest that there could be a two-parameter fam-ily of self-consistent definitions of angular momentum ofasymptotically flat spacetimes. Compère et al . later de-scribed in [27] how these different definitions can all beconsidered to be Hamiltonian definitions [which is whywe take care to describe which (or whose) Hamiltoniandefinition of the charge is being used].In this paper, we investigate this new two-parameterfamily of angular momenta in greater detail. We findthat if we require the angular momentum to vanish inflat spacetime, then two of the parameters must be equal,thereby reducing the two parameters to one. However, ifwe do not require that the angular momentum agree withthe Wald-Zoupas definition, we find that there is no rea-son to prefer one value of this parameter over another. Note that what we call the six-parameter (Lorentz-covariant) an-gular momentum, Compère et al . in [26] call the “Lorentz charge.”We also have different usages for how we describe the parts thatcorrespond to the rotations and the Lorentz boosts. We both callthe part corresponding to Lorentz boosts “center-of-mass angu-lar momentum,” but Compère et al . call the parts correspondingto rotations simply “angular momentum,” whereas we refer to itas “intrinsic” or “spin” angular momentum, because it reduces tothose quantities in the rest-frame of the source.
For this residual one-parameter family of angular mo-menta, we expand the difference of the angular momen-tum from the Wald-Zoupas definition in terms of spin-weighted spherical-harmonic moments of the GW strain.These difference terms involve only products of electric-and magnetic-type spherical-harmonic coefficients (un-like the flux of the Wald-Zoupas angular momentum),which is consistent with the results of [26]. This impliesthat the difference will vanish in stationary regions ofspacetimes and nonradiative regions of spacetime withvanishing magnetic shear, though more generally, it willnot vanish. We compute the time-dependent differenceterms for nonspinning BBH mergers, and we find thatthey are small compared to the total radiated angularmomentum.In addition to the BMS group, there are two differ-ent proposals for larger symmetry groups or algebras ofasymptotically flat spacetimes. The first, due to Bar-nich and Troessaert [28–30], considers all the conformalKilling vectors of the 2-sphere, rather than the glob-ally defined vectors, which are isomorphic to the Lorentzgroup. These vectors were dubbed “super-rotations,”and, analogously to the supertranslations, they are a kindof asymptotic angle-dependent rotations and Lorentzboosts. To maintain the algebra structure of theseasymptotic symmetries, the supertranslations must becorrespondingly modified. A second extended symmetrygroup, due to Campiglia and Laddha [31, 32], considersall the diffeomorphisms of the 2-sphere rather than thoseequal to the Lorentz transformations, but the supertrans-lations are the same as in the BMS group. The 2-spherediffeomorphisms are often referred to as super Lorentztransformations [33].Both the super-rotations and super Lorentz trans-formations have corresponding conserved charges. Thecharges for both algebras have been called “super angu-lar momentum,” but they have also been called simplysuper-rotation charges or super Lorentz charges, for therespective algebras. We shall primarily focus on the gen-eralized BMS algebra, and we shall refer to the chargesassociated with this algebra as the super angular mo-mentum (and will call those associated with the super-rotations the “super-rotation charges.”). Note that wewill call the split of the charges into their electric- andmagnetic-parity parts by super CM and superspin, re-spectively, in analogy with the convention used initiallyin [9] for the super-rotation charges, and subsequently forthe super angular momentum in [21, 34]. The super-rotation charges have a similar form to theangular momenta, but a super-rotation vector field entersinto the expression for the charge rather than a Lorentz This is a second discrepancy with the nomenclature used in [26].There, what we call superspin is called super angular momen-tum, and what we call super angular momentum is called a superLorentz charge. Our usages of super center-of-mass are equiva-lent, however. vector field (see, e.g., [9, 30]). The super Lorentz chargesconstructed defined in [33] also have a similar form to theangular momentum with the Lorentz vector field is re-placed by a super Lorentz transformation, but they havean additional term linear in the shear tensor needed tosatisfy a flux balance law [33]. Given that there is a one-parameter family of angular momentum that satisfies anumber of reasonable physical constraints, it is also nat-ural to ask whether there is such a parametrization forthe super angular momentum. We investigate this is-sue as well by allowing for a two-parameter family ofsuper angular momentum that generalizes the Hamilto-nian definition of [33] in a way completely analogous tothe two-parameter extension of the Wald-Zoupas angularmomentum given in [26]. In this case, setting the param-eters to be equal (thereby reducing it to a one-parameterfamily) does not seem to make the super Lorentz chargesvanish. This is consistent with a calculation performedby Compère and Long [35] for the Hamiltonian charges.There is a choice of parameters that makes the superangular momentum vanish, but this choice does not cor-respond to the Hamiltonian definition of [33]. Rather,this choice is the same as the one used in [27] to deter-mine a representation of the extended BMS algebra innonradiative regions of spacetime for the super Lorentzcharges in terms of the standard Poisson bracket. Thisalso leads to the intriguing possibility that properties ofthe generalized BMS algebra and charges could be usedto fix the value of this free parameter in the charges asso-ciated with the standard BMS symmetries, which is notclearly fixed by other physical arguments (though we willnot discuss this possibility further in this paper). We then compute the multipolar expansion of the dif-ference of the two-parameter family of super angular mo-mentum from the Hamiltonian super angular momentumof [33]. This allows us to see that unlike the angular mo-mentum, the change in the difference in the super angu-lar momentum will be nonvanishing even in stationaryregions. As a concrete example, we estimate the value ofthe change in the difference of the super angular momen-tum for nonspinning, quasicircular BBH mergers. Therelative size of the net change in Hamiltonian value ofthe super angular momentum and the net change in thedifference term is small for these BBH mergers (a roughlyone-percent effect). Although it is small, it can be re-solved given the current accuracy of numerical relativity(NR) simulations. For the standard BMS algebra and in nonradiative regions, theWald-Zoupas charges were shown in [30] to represent the algebrausing the standard Poisson bracket; thus, it does not seem thatthe charge algebra can be used to determine the value of thefree parameter for the angular momentum. The possibility offixing this additional freedom, therefore, relies upon imposingcertain desirable properties of the generalized BMS charges, suchas requiring the charges vanish in flat spacetime or that they havea nice representation of their charge algebra.
Overview
The outline of the rest of this paper is asfollows. Section II is mostly a review in which we intro-duce Bondi coordinates, the metric in these coordinates,the evolution equations for the Bondi mass and angular-momentum aspects, the (extended) BMS symmetries ofasymptotically flat spacetimes, and the expressions forthe various definitions of angular momentum in Bondicoordinates. We end this section, however, by introduc-ing the proposed two-parameter definition of the superangular momentum. In Sec. III, we compute the (super)angular momentum in flat spacetime (where we show twoof the parameters must be equal for the angular momen-tum to vanish). In the next section, Sec. IV, we performa multipolar expansion of the (super) angular momentumthat is valid for general asymptotically flat spacetimes.In Sec. V, we estimate the effect that the remaining freeparameter in the angular momentum and super angularmomentum has on BBH mergers of different mass ratios.We compute results in the post-Newtonian approxima-tion and using NR surrogate waveforms. We conclude inSec. VI. In Appendix A, we compare our multipolar ex-pansion of the angular momentum with a related expan-sion performed in [26]. In this paper, we use geometricunits G = c = 1 , and the conventions on the metric andcurvature tensors in [36]. II. BONDI-SACHS FRAMEWORK,SYMMETRIES, AND CHARGES
In this section, we review aspects of the Bondi-Sachsframework including the metric, some components ofEinstein’s equations, the asymptotic symmetries, and thecorresponding charges. We then discuss different defini-tions of angular momentum and super angular momen-tum.
A. Metric and Einstein’s equations
We will perform our calculations in Bondi coordi-nates [3, 5] ( u, r, θ A ) , where A = 1 , , and we reviewthe properties of these coordinates and the solutions ofEinstein’s equations below. We will use the notation andconventions given in [9]. The metric in these coordinatesis written in the form ds = − U e β du − e β dudr + r γ AB ( dθ A − U A du )( dθ B − U B du ) (2.1)where the functions and tensors U , β , γ AB , and U A de-pend on all four Bondi coordinates ( u, r, θ A ) . The met-ric by construction satisfies the Bondi gauge conditions g rr = 0 and g rA = 0 ; Bondi coordinates also are definedsuch that det( γ AB ) = γ ( θ A ) is independent of u and r .Some important properties of these coordinates are that u is a retarded time variable (i.e., u = const. are null hy-persurfaces), r is an areal radius, and θ A (with A = 1 , )are coordinates on 2-spheres of constant r and u . Near future null infinity (i.e., where r is large), themetric functions U , β , γ AB , and U A can be expanded asseries in /r . Asymptotically flat solutions postulate agiven form of the expansion of these Bondi metric func-tions. For the tensor γ AB the conditions of asymptoticflatness generally impose γ AB = h AB + 1 r C AB + O ( r − ) , (2.2)where h AB ( θ C ) is the metric on the unit 2-sphere, C AB is a function of ( u, θ A ) , and the determinant condition ofBondi gauge implies that C AB h AB = 0 . The remainingfunctions U , β , and U A are assumed to have the followinglimits as r approaches infinity lim r →∞ β = lim r →∞ U A = 0 , lim r →∞ U = 1 . (2.3)We will now specify to vacuum spacetimes to discussEinstein’s equations, for simplicity. The ru , rA , andtrace of the AB components of Einstein’s equations takethe form of hypersurface equations that can be solvedon surfaces of constant u by integrating radially out-ward. The form of these equations is summarized inthe review [37], for example. The results of substitut-ing Eq. (2.2) into these hypersurface equations, radi-ally integrating, and applying the boundary conditionsin Eq. (2.3) gives the following solutions for the remain-ing functions U , β , and U A : β = − r C AB C AB + O ( r − ) , (2.4a) U = 1 − mr + O ( r − ) , (2.4b) U A = − r D B C AB + 1 r (cid:104) − N A + 116 D A ( C BC C BC )+ 12 C AB D C C BC (cid:105) + O ( r − ) . (2.4c)We have introduced a number of new pieces of nota-tion in the above equation, which we will now explain:First, the function m ( u, θ A ) is the Bondi mass aspect and N A ( u, θ B ) is the angular momentum aspect. They arerelated to “functions of integration” that arise from inte-grating the hypersurface equations radially. Second, inthe above equation, we have raised and lowered indices oftensors and vectors on the 2-sphere using the metric h AB (respectively h AB ). Third, we have defined the deriva-tive operator D A as the torsion-free, metric-compatiblederivative associated with the metric h AB .The evolution equation for γ AB , when expanded toleading order in /r , shows that the u derivative of C AB is unconstrained by Einstein’s equations and is defined tobe the Bondi news tensor N AB = ∂ u C AB . The leading-order parts of the uu and uA components of Einsteinequations are the conservation equations, which look likeevolution equations for the Bondi mass aspect m and theangular momentum aspect N A at fixed radii: ˙ m = − N AB N AB + 14 D A D B N AB (2.5a) ˙ N A = D A m + 14 D B D A D C C BC − D B D B D C C CA + 14 D B ( N BC C CA ) + 12 D B N BC C CA (2.5b)These equations are important for establishing flux bal-ance laws for the charges conjugate to the asymptoticsymmetries that form the BMS group and its extensions;we turn to the subject of these symmetries in the nextsubsection. B. Asymptotic symmetries
The Bondi-Metzner-Sachs (BMS) group [3, 4] can beobtained from set of transformations that preserve theBondi gauge conditions of the metric (2.1) and theasymptotic form of the functions that appear in the met-ric [Eqs. (2.2) and (2.4)]. The BMS group is the semidi-rect product of the infinite-dimensional abelian group ofsupertranslations with a six-dimensional group of con-formal transformations of the 2-sphere (which is isomor-phic to the proper, isochronous Lorentz group). The fourspacetime translations are a subgroup of the supertrans-lation group. More recent generalizations of the BMSalgebra take two forms. (i) The first is the extendedBMS algebra proposed by Barnich and Troessaert [28–30] (see also [38]). In this proposal, all conformal Killingvectors of the 2-sphere are added to the algebra, includ-ing those with complex-analytic singularities on the 2-sphere. These additional symmetry vector fields weredubbed super-rotations, and the vectors that are iso-morphic to the Lorentz transformations are a subalge-bra of the super-rotations. The supertranslations alsoare extended to include functions that are not necessar-ily smooth. (ii) The second proposal has been called thegeneralized BMS algebra, and is due to Campiglia andLaddha [31, 32]. Here all smooth diffeomorphisms of the2-sphere are considered instead of those equivalent to theLorentz transformations, but the supertranslations arethe same as in the original BMS group (though it is nolonger possible to identify a preferred spacetime transla-tion subgroup [39]).The BMS symmetries and their generalizations are de-scribed by infinitesimal vector fields (cid:126)ξ that formally aredefined at future null infinity, the null boundary of anasymptotically flat spacetime in the covariant conformalapproach of Penrose [40, 41]. The form of the vectorfields at future null infinity can be written in Bondi co-ordinates by restricting the vector fields that preserve theBondi gauge conditions and the fall off rates of the met-ric to the tangent space of surfaces of constant r , andthen taking the limit as r goes to infinity. In this limit,the vector fields for the BMS group and its extensions all take the same form; they are parameterized by a scalarfunction T ( θ A ) and a vector on the 2-sphere Y A ( θ B ) : (cid:126)ξ = (cid:20) T ( θ A ) + 12 uD A Y A ( θ B ) (cid:21) (cid:126)∂ u + Y A ( θ B ) (cid:126)∂ A (2.6)The function T ( θ A ) parametrizes the supertranslationsin the BMS algebra and its generalizations (for the stan-dard and generalized BMS algebras, it is assumed to bea smooth function, whereas for the extended BMS alge-bra, it can have complex analytic singular points). Thevector field Y A ( θ B ) is a conformal Killing vector on the2-sphere for the standard and extended BMS algebras(it is spanned by a six-parameter basis for the standardBMS algebra, or an infinite dimensional basis for the ex-tended BMS algebra), or a smooth vector field for thegeneralized BMS group.The symmetries at future null infinity can also be ex-tended into the interior of the spacetime at large, butfinite r by requiring that the diffeomorphisms generatedby these vector fields preserve the Bondi gauge condi-tions and the asymptotic fall-off conditions imposed onthe metric. Under these transformations, the functions C AB , N AB , m , and N A transform in a nontrivial way.For the discussion that follows, we will only need thetransformation law for C AB , and we denote this trans-formation by C AB → C AB + δ ξ C AB , which was derived,e.g., in [29]. It is convenient to first define a quantity f = T + u D A Y A , (2.7)which appears in δ ξ C AB as follows: δ ξ C AB = f N AB − (2 D A D B − h AB D ) f + L Y C AB − D C Y C C AB . (2.8)This transformation of C AB is useful for defining fluxesof conserved quantities associated with the BMS symme-tries, which we will discuss in the next subsection. Beforewe do so, it is useful to introduce a decomposition of thetensor C AB into its electric and magnetic (parity) partsas follows: C AB = (cid:18) D A D B − h AB D (cid:19) Φ + (cid:15) C ( A D B ) D C Ψ . (2.9)The scalars Φ and Ψ are both smooth functions of thecoordinates ( u, θ A ) . From the transformation of C AB inEq. (2.8), it follows that a supertranslation affects theelectric part of C AB , but leaves the magnetic part invari-ant. This property of the shear has been understood forquite some time (see, e.g., [8]). C. Fluxes and charges
There are a few different prescriptions used to definethe charges and the fluxes of charges that are associatedwith BMS symmetries. We will describe here the proce-dure of Wald and Zoupas [20], in which the charges andfluxes are computed using a generalization of Noether’stheorem that allows for the charges to change from emit-ted fluxes of gravitational waves and other matter fields.We denote the charges by Q ξ [ C ] , where the charges de-pend linearly upon a BMS vector field (cid:126)ξ and are definedon a cross section of null infinity C (in Bondi coordinates,a surface of constant u at fixed r in the limit of r → ∞ ).We call the flux F ξ [∆ I ] . Like the charge, it has a lineardependence on a BMS vector field (cid:126)ξ , but the flux de-pends on a region of null infinity ∆ I between two cuts(in Bondi coordinates, the region between two surfacesof constant u at fixed r in the limit of r → ∞ ). The fluxbalance law for the charges requires that Q ξ [ C ] − Q ξ [ C ] = F ξ [∆ I ] . (2.10)The explicit expression for the flux has a simple form inBondi coordinates in vacuum (see, e.g., [9]) F ξ [∆ I ] = − π (cid:90) ∆ I du d Ω N AB δ ξ C AB , (2.11)where δ ξ C AB is given in Eq. (2.8) and d Ω is the area ele-ment on the 2-sphere cuts of constant u . Using Eq. (2.8)and the conservation equations for the Bondi mass andangular momentum aspects in Eq. (2.5), it is possible toshow that the charge is given by Q ξ = 18 π (cid:90) C d Ω (cid:26) T m + Y A (cid:20) N A − uD A m − D A ( C BC C BC ) − C AB D C C BC (cid:21)(cid:27) (2.12)(again, see, e.g., [9]). We dropped the dependence of thecharge on the cut C to simplify the notation, and becauseit is made explicit in the domain of the integral on theright-hand side of the equation.When the vector field (cid:126)ξ has Y A = 0 and T (cid:54) = 0 , thenit is a supertranslation, and the corresponding charge isthe supermomentum. The other case, a vector field with Y A (cid:54) = 0 and T = 0 , has as its corresponding charge theangular momentum, when Y A is equivalent to a Lorentztransformation for the standard BMS group. The angu-lar momentum is often split into its intrinsic (or spin)and center-of-mass (CM) parts, which correspond to therotation and boost symmetries in the Lorentz group, re-spectively. It was observed in [9] that the charge inEq. (2.12) does not satisfy the flux balance law (2.10) forthe extended or generalized BMS vector fields. A chargethat does satisfy a flux balance for the super Lorentzcharges was determined in [33]. It is the same as that inEq. (2.12), up to the addition of two new terms linear in the tensor C AB , and it is given below: Q ξ = 18 π (cid:90) C d Ω (cid:26) T m + Y A (cid:20) N A − uD A m − D A ( C BC C BC ) − C AB D C C BC + u D D B C AB − D B D A D C C BC ) (cid:21)(cid:27) . (2.14)Note that the integral of the two additional terms inthe final line Eq. (2.14) can be shown to vanish for the Y A corresponding to Lorentz vector fields; note also thatthese two terms in the integrand are proportional to a dif-ferential operator acting on the magnetic part Ψ of theshear in Eq. (2.9) (see, e.g., [9]). The super angular mo-mentum in (2.14) can be divided into a magnetic-paritypart called superspin and an electric-parity part calledsuper center-of-mass, in analogy to the standard angularmomentum. In the next subsection, we focus on the an-gular momentum and discuss a subtlety in its definition. D. Definitions of angular momentum and theirproperties
As discussed in the introduction, the angular momen-tum computed by Wald and Zoupas is not the only notionof the angular momentum of an isolated system that iscommonly used. While a number of the different angu-lar momenta are equivalent, not all the definitions agree.First, for convenience, let us specialize the general BMScharges in Eq. (2.14) to a vector field (cid:126)ξ with T = 0 and Y A being a generator of Lorentz transformations: Q Y = 18 π (cid:90) C d Ω Y A (cid:20) N A − uD A m − D A ( C BC C BC ) − C AB D C C BC (cid:21) . (2.15)We used the notation Q Y rather than Q ξ to emphasizethat it depends only on Y A . It has been shown in [20]that the flux of this angular momentum agrees with thatof Ashtekar and Streubel [19] and the charge defined byDray and Streubel [18] (which came from twistorial def-initions of the angular momentum [17]). The Landau-Lifshitz definition of angular momentum in [15] (which isrestricted to the center-of-mass frame of the source andaveraged over a few wavelengths of the emitted gravi-tational waves) also agrees with the flux of the angular The flux for which this charge satisfies the flux balance law differsfrom Eq. (2.11). It is necessary to add a term of the form π (cid:90) ∆ I du d Ω u ( D D B N AB − D B D A D C N BC ) (2.13)to the right-hand side of Eq. (2.11) to restore the balance lawwith the definition of the charge in Eq. (2.14) (see [26] for furtherdetails). momentum charge in Eq. (2.15), when the expression isrestricted to this context [21].There are a few notable examples of definitions of an-gular momentum that differ from the one in Eq. (2.15), afact that was recently pointed out in a paper by Compère et al. in [26]. First, in the context of conservation lawsof gravitational scattering, a definition of an angular mo-mentum involving just the mass and angular momentumaspects and the vector field on the 2-sphere, Y A , wasused in [23, 42] to define the (super) angular momentum:i.e., Q (0) Y = 18 π (cid:90) C d Ω Y A ( N A − uD A m ) . (2.16)Also recently, a more general definition of the Landau-Lifshitz angular momentum was proposed by by Bongaand Poisson [22], who no longer required that the resultbe defined in the CM frame or by averaging over a fewwavelengths of the gravitational waves. They specializedto the intrinsic (as opposed to CM) angular momentu,which they defined by using a collection of vector fieldson the 2-sphere, Y Ai = (cid:15) AD ∂ D n i . Here n i is a unit vec-tor normal to the 2-sphere in quasi-Cartesian coordinatesconstructed from the spatial Bondi coordinates ( r, θ A ) ,and (cid:15) AD is the Levi-Civita tensor on the unit 2-sphere.After converting their definition of the intrinsic angularmomentum into our notation, their result can be writtenas J i = 18 π (cid:90) C d Ω (cid:15) AD ∂ D n i (cid:20) N A − uD A m − C AB D C C BC (cid:21) . (2.17)There is a definition of the CM part of the angular mo-mentum in the Landau-Lifshitz formalism from Blanchetand Faye [43], but it was shown in [26] that it cannot eas-ily be written in terms of the 2-sphere-covariant Bondi-metric functions. As we discuss further below, the threedefinitions of the angular momentum in Eqs. (2.15)–(2.17) all vanish in flat spacetime, give the same angularmomentum of a Kerr black hole and satisfy flux balancelaws; they thus appear to be equally viable definitions ofthe angular momentum of an isolated source.Given that the angular momenta in Eqs. (2.15)–(2.17)differ in the factors in front of the two terms quadraticin C AB in Eq. (2.15), Compère et al. [26] observed that atwo-parameter family of charges could be defined by al-lowing the coefficients in front of these terms to be arbi-trary real numbers. When the coefficients are restrictedto specific values, the two-parameter family of chargesreduces to one of the specific definitions in Eqs. (2.15)–(2.17). Thus, the two-parameter family of angular mo-mentum of Compère et al. [26] is given by Q ( α,β ) Y = 18 π (cid:90) C d Ω Y A (cid:104) N A − uD A m − α C AB D C C BC − β D A ( C BC C BC ) (cid:21) , (2.18) where α and β are real constants. The Wald-Zoupasangular-momentum corresponds to the case α = β = 1 ;the angular momentum in Eq. (2.16) corresponds to α = β = 0 ; and the intrinsic angular momentum inEq. (2.17) corresponds to α = 3 (and β can take on anyreal value, because it does not contribute to the intrinsicpart). For all values of α and β , the angular momen-tum in Eq. (2.18) satisfies flux balance laws, but it isnot immediately apparent that they will vanish in flatspacetime. In the next section, we will derive the condi-tions under which the angular momentum in Eq. (2.18)vanishes in flat spacetime. E. Definitions of super angular momentum
The charge in Eq. (2.18) was definied specifically forthe angular momentum. There are also differing defini-tions of the super angular momentum, however, becauseseveral of the definitions of the super angular momen-tum were defined through promoting the vector field Y A that enters into the charge from a Lorentz vector fieldto a super Lorentz vector. The definition in Eq. (2.16)was also used for a super-rotation charge (where Y A is asuper-rotation vector field, for example), and this defini-tion differs from that in Eq. (2.15). The main differencebetween the two charges is are the terms quadratic in theshear tensor. It thus seems reasonable to define a two-parameter family of charges that satisfy a flux balancelaw by generalizing Eq. (2.14) (when T = 0 ) to includereal coefficients α and β in front of the terms quadratic in C AB . Thus, we will also consider a two-parameter familyof super angular momentum defined by Q ( α,β ) Y = 18 π (cid:90) C d Ω Y A (cid:20) N A − uD A m + u D D B C AB − D B D A D C C BC ) − α C AB D C C BC − β D A ( C BC C BC ) (cid:21) . (2.19)We will investigate the properties of this charge in flatspacetime next. III. (SUPER) ANGULAR MOMENTUM INFLAT SPACETIME
While the focus in this section will be determining thevalues of the coefficients α and β for which the angularmomentum vanishes, much of the calculation holds for The terms D A ( C BC C BC ) and C AB D C C BC form a kind of basisof vectors constructed from contractions of C AB and D A C BC ,in the sense that other possible contractions can be rewritten interms of these two quantities [26]. any smooth vector field on the 2-sphere Y A , and thusapplies to the super angular momentum of the general-ized BMS algebra. In the derivation that follows, it isstructured so that the first part applies to smooth gener-alized BMS vectors Y A , and the next part is specified to Y A that generate Lorentz transformations. Note that asimilar calculation was performed by Compère and Longin [44] for the Wald-Zoupas charges (i.e., α = β = 1 ).In flat spacetime, there is no radiation, and the newstensor vanishes [45]. In this case, the Bondi mass as-pect and the Bondi angular momentum are also pro-portional to components of the vacuum Riemann tensor(see, e.g., [9]) and thus they must also vanish. FromEq. (2.5b), one can then also show that Ψ , the scalarthat parametrizes the magnetic part of C AB must alsovanish. Because C AB is electric type, then by perform-ing a supertranslation it follows from Eq. (2.8) that itis possible to choose a frame in which the tensor C AB vanishes (note that from the transformation propertiesof m and N A given in, e.g., [9], the mass and angularmomentum aspects will remain zero under this transfor-mation). We will not work in the frame where C AB van-ishes, but rather we will choose a frame where it has anonzero electric part. Thus, the values of the relevantfunctions needed to compute the super angular momen-tum in Eq. (2.19) are given by m = 0 , (3.1a) N A = 0 , (3.1b) C AB = (cid:18) D A D B − h AB D (cid:19) Φ . (3.1c)In flat spacetime, therefore, the additional terms in thesecond line of Eq. (2.19) do not contribute, and the superangular momentum is given by Q ( α,β ) Y = − π (cid:90) C d Ω (cid:2) αY A C AB D C C BC + βY A D A ( C BC C BC ) (cid:3) . (3.2)We will now substitute in the expression in Eq. (3.1c)for C AB in Eq. (3.2) in several places, and begin simpli-fying the expression. Because we are assuming Y A is asmooth vector on the 2-sphere and Φ is a smooth func-tion, we can integrate the first term by parts and dropthe terms involving divergences of vector fields on the 2-sphere. For the second term, we use the fact that the co-variant derivative acting on the shear tensor in Eq. (3.1c) Note however that if Y A is a super-rotation vector field of the ex-tended BMS algebra, then the singular points of the vector fieldsmake integration by parts on the 2-sphere more challenging. Al-though the 2-sphere is a compact manifold without boundary,when integrating by parts one must carefully analyze the con-tributions that come from boundary-like terms at the singularpoints of the super-rotation vectors, which can contribute to theintegral (see, e.g., [44] for further details). is given by D B C AB = D B D A D B Φ − D A D Φ . (3.3)We can then use the definition of the Riemann tensor(associated with the derivative operator D A ) to commutethe first two covariant derivatives in the first term. Wefind that it can be written as D B C AB = D A D Φ + R AB D B Φ − D A D Φ , (3.4)where R AB is the Ricci tensor on the 2-sphere. Assumingthat the metric is that of a round 2-sphere, then the scalarcurvature of the sphere is given by R = 2 , the Ricci tensoris R AB = h AB , and the Riemann tensor can be writtenas R ABCD = h AC h BD − h AD h BC . (3.5)This implies that D B C AB simplifies to D B C AB = 12 D A ( D + 2)Φ . (3.6)Next, substituting Eqs. (3.6) and (3.1c) into Eq. (3.2),we can write the charge in terms of Y A , Φ , and deriva-tive operators D A (though we leave one term involving C AB ). If we integrate by parts once more for both theterms proportional to α and β , we find the super angularmomentum is given by Q ( α,β ) Y =1128 π (cid:90) C d Ω (cid:26) βD A Y A [ D B D C Φ D B D C Φ −
12 ( D Φ) ]+2 α (cid:20) D B Y A C AB + 12 Y A D A ( D + 2)Φ (cid:21) ( D + 2)Φ (cid:27) . (3.7)While for each Φ and Y A there should exist a choice of α and β that makes Q Y vanish, the most obvious choiceof α and β that makes the super angular momentumvanish for all Φ and Y A in flat spacetime is α = β =0 . However, it is not necessarily clear that one shouldrequire that the super angular momentum should vanish,as Compère and collaborators have argued that the superangular momentum can be used to distinguish vacuumstates that differ by a supertranslation [33, 44]. We thusonly identify α = β = 0 as a choice that makes the superangular momentum vanish in flat spacetime, but to notrequire this property for the charge. Angular momentum
We do require that the charge Q Y vanish for vectors Y A that generate Lorentz transfor-mations. We now continue our simplification of Eq. (3.7)by using the fact that Y A is a conformal Killing vec-tor on the 2-sphere; i.e., it satisfies the conformal Killingequation D ( A Y B ) − D C Y C h AB = 0 . (3.8)Because C AB is symmetric and trace free, then C AB D B Y A involves only the symmetric-trace-free partof D B Y A . By the conformal Killing equation (3.8), how-ever, D B Y A is proportional to h AB , so C AB D B Y A van-ishes. After performing a large number of integration byparts (so as to write the expression mostly in terms ofsquares of Φ and its derivatives) and using the followingidentity D D C Φ = D C D Φ + D C Φ , (3.9)we find that the angular momentum can be written as Q ( α,β ) Y = 1256 π (cid:90) C d Ω (cid:8) ( D A Y A ) (cid:2) ( β − α )( D Φ) − α Φ + 2(2 α − β ) D C Φ D C Φ (cid:3) − D ( D A Y A ) (cid:2) α Φ − βD C Φ D C Φ (cid:3) − βD B D C D A Y A D B Φ D C Φ (cid:9) . (3.10)Conformal Killing vectors also satisfy the property that ( D + 2)( D A Y A ) = 0 , (3.11)which leads to the cancellation of some terms propor-tional to α in Eq. (3.10). The globally defined conformalKilling vectors (the vector fields Y A that can be writtenas a superposition of the six l = 1 vector spherical har-monics on the 2-sphere) satisfy the additional property D B D C D A Y A = − h BC D A Y A . (3.12)After using the property in Eq. (3.12) in Eq (3.10), wesee that the angular momentum in flat spacetime can bewritten as Q ( α,β ) Y = 1256 π ( β − α ) (cid:90) C d Ω D A Y A [( D Φ) − D C Φ D C Φ] . (3.13)The intrinsic angular momentum (i.e., the charge Q ( α,β ) Y for vectors Y A with D A Y A = 0 ) vanishes for all valuesof α and β . For the center-of-mass angular momentum(i.e., the charge with Y A that has nonvanishing D A Y A ),the charge will typically be nonvanishing unless α = β .Having the physical requirement that the angular mo-mentum should vanish in flat spacetime thus reduces thetwo-parameter family of charges to a one-parameter fam-ily given by α . We will typically work with this reducedone-parameter family in the rest of the paper, unless wenote otherwise. IV. MULTIPOLAR EXPANSION OF THE(SUPER) ANGULAR MOMENTUM
We will first summarize our conventions for the spher-ical harmonics that we use in our multipolar expansion.We will then perform multipolar expansions of the super angular momentum, which we will subsequently special-ize to the standard angular momentum.Because the multipolar expansion of Hamiltoniancharges and fluxes had been computed previously (see,e.g., [21, 26, 34]), we will focus on the difference of thetwo-parameter family of charges from the charge definedin [33]. Thus, for a vector field Y A we will write Q ( α,β ) Y = Q ( α =1 ,β =1) Y + ( α − δQ ( α =1) Y + ( β − δQ ( β =1) Y , (4.1a)where Q ( α =1 ,β =1) Y is the charge with α = β = 1 and δQ ( α =1) Y and δQ ( β =1) Y are defined by δQ ( α =1) Y = − π (cid:90) C d Ω Y A C AB D C C BC , (4.1b) δQ ( β =1) Y = − π (cid:90) C d Ω Y A D A ( C BC C BC ) . (4.1c)In the special case of angular momentum, we will also usethe notation δJ ( α =1) Y and δk ( α =1) Y (and similarly for the β term) for the difference in the intrinsic and CM angu-lar momentum, respectively, associated with a vector Y A (which is a rotation or Lorentz boost, respectively). Asimilar calculation was performed in [26]; however, herewe also compute the α -dependent term in the CM angu-lar momentum, and we write the result in terms of themultipole moments U lm and V lm (defined below) ratherthan the rank- l symmetric-trace-free (STF) tensors U L and V L (discussed in Appendix A). The moments U lm and V lm are somewhat easier to relate to the moments ofthe gravitational-wave strain h lm that can be obtainedfrom numerical-relativity simulations or surrogate mod-els fit to simulations (the latter of which we will use laterin Sec. V).In the cases where we restrict to α = β (so that theangular momentum vanishes in flat spacetime), then wewill use the notation Q ( α = β ) Y = Q ( α = β =1) Y + ( α − δQ ( α = β =1) Y , (4.2a)where Q ( α = β =1) Y = Q ( α =1 ,β =1) Y is the charge with α = β = 1 and δQ ( α = β =1) Y is defined by δQ ( α = β =1) Y = − π (cid:90) C d Ω Y A [4 C AB D C C BC + D A ( C BC C BC )] . (4.2b)We will similarly use the notation δJ ( α = β =1) Y and δk ( α = β =1) Y for the intrinsic and CM angular momentum,respectively, when Y A is a rotation or Lorentz boost (alsorespectively). A. Spherical harmonics and multipolar expansionof the gravitational-wave data
In addition to the scalar spherical harmonics (with theusual Condon-Shortly phase convention), Y lm ( θ, φ ) , we0will use vector and tensor harmonics on the unit 2-sphere,which we define as in [21]. The vector harmonics aregiven by T A ( e ) ,lm = 1 (cid:112) l ( l + 1) D A Y lm , (4.3a) T A ( b ) ,lm = 1 (cid:112) l ( l + 1) (cid:15) AB D B Y lm , (4.3b)which are nonzero for l ≥ and the tensor harmonics T ( e ) ,lmAB = 12 (cid:115) l − l + 2)! (cid:0) D A D B − h AB D (cid:1) Y lm , (4.4a) T ( b ) ,lmAB = (cid:115) l − l + 2)! (cid:15) C ( A D B ) D C Y lm , (4.4b)which are nonzero for l ≥ .We use these harmonics to expand the shear tensor as C AB = (cid:88) l,m ( U lm T AB ( e ) ,lm + V lm T AB ( b ) ,lm ) . (4.5)Because the shear is real, the coefficients in this expan-sion obey the properties U l, − m = ( − m ¯ U lm , V l, − m = ( − m ¯ V lm , (4.6)where the overline means to take the complex conjugate.By using Eqs. (4.3a)–(4.4b) and (3.4), we can write thecovariant derivative of the shear tensor in terms of vectorharmonics as follows: D C C BC = (cid:88) l,m (cid:114) ( l − l + 2)2 ( U lm T B ( e ) ,lm − V lm T B ( b ) ,lm ) . (4.7)The vector and tensor harmonics are related to spin-weighted spherical harmonics s Y lm of spin weight s = ± and s = ± , respectively, and a complex null dual vectoron the 2-sphere m A ∂ A = 1 √ ∂ θ + i csc θ∂ φ ) . (4.8)and its complex conjugate ¯ m A . The relationships for thevector harmonics are T ( e ) ,lmA = 1 √ − Y lm m A − Y lm ¯ m A ) , (4.9a) T ( b ) ,lmA = i √ − Y lm m A + Y lm ¯ m A ) , (4.9b)and for the tensor harmonics are T ( e ) ,lmAB = 1 √ − Y lm m A m B + Y lm ¯ m A ¯ m B ) , (4.10a) T ( b ) ,lmAB = − i √ − Y lm m A m B − Y lm ¯ m A ¯ m B ) . (4.10b) The spin-weighted spherical harmonics sat-isfy the well-known complex-conjugate property s ¯ Y lm = ( − s + m − s Y l − m .The charges are quadratic in C AB and involve a vectorfield Y A , and we will expand all three quantities in termsof spin-weighted spherical harmonics using Eqs. (4.3a)–(4.10b). When evaluating the charges, we will frequentlyencounter integrals of three spin-weighted spherical har-monics over S . We use the notation of [21] to describethese integrals, which we denote by C l ( s (cid:48) , l (cid:48) , m (cid:48) ; s (cid:48)(cid:48) , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) ≡ (cid:90) d Ω ( s (cid:48) + s (cid:48)(cid:48) ¯ Y lm (cid:48) + m (cid:48)(cid:48) )( s (cid:48) Y l (cid:48) m (cid:48) )( s (cid:48)(cid:48) Y l (cid:48)(cid:48) m (cid:48)(cid:48) ) . (4.11)The complex-conjugated spherical harmonic s (cid:48) + s (cid:48)(cid:48) ¯ Y lm (cid:48) + m (cid:48)(cid:48) has spin-weight s = s (cid:48) + s (cid:48)(cid:48) and az-imuthal number m = m (cid:48) + m (cid:48)(cid:48) , because for all othervalues of s and m , the integral vanishes. It can beshown that the coefficients C l ( s (cid:48) , l (cid:48) , m (cid:48) ; s (cid:48)(cid:48) , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) canbe written in terms of Clebsch-Gordon coefficients (cid:104) l (cid:48) , m (cid:48) ; l (cid:48)(cid:48) , m (cid:48)(cid:48) | l, m (cid:48) + m (cid:48)(cid:48) (cid:105) as follows: C l ( s (cid:48) , l (cid:48) , m (cid:48) ; s (cid:48)(cid:48) , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) = ( − l + l (cid:48) + l (cid:48)(cid:48) (cid:115) (2 l (cid:48) + 1)(2 l (cid:48)(cid:48) + 1)4 π (2 l + 1) × (cid:104) l (cid:48) , s (cid:48) ; l (cid:48)(cid:48) , s (cid:48)(cid:48) | l, s (cid:48) + s (cid:48)(cid:48) (cid:105) (cid:104) l (cid:48) , m (cid:48) ; l (cid:48)(cid:48) , m (cid:48)(cid:48) | l, m (cid:48) + m (cid:48)(cid:48) (cid:105) . (4.12)The coefficients are also nonvanishing only when the l index is in the range { max( | l (cid:48) − l (cid:48)(cid:48) | , | m (cid:48) + m (cid:48)(cid:48) | , | s (cid:48) + s (cid:48)(cid:48) | ) , ..., l (cid:48) + l (cid:48)(cid:48) − , l (cid:48) + l (cid:48)(cid:48) } . There are two additionaluseful identities under sign flips of the spin weight andazimuthal numbers that we will need in the discussionbelow C l ( s (cid:48) , l (cid:48) , m (cid:48) ; s (cid:48)(cid:48) , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) =( − l + l (cid:48) + l (cid:48)(cid:48) × C l ( − s (cid:48) , l (cid:48) , m (cid:48) ; − s (cid:48)(cid:48) , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) , (4.13a) C l ( s (cid:48) , l (cid:48) , m (cid:48) ; s (cid:48)(cid:48) , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) =( − l + l (cid:48) + l (cid:48)(cid:48) × C l ( s (cid:48) , l (cid:48) , − m (cid:48) ; s (cid:48)(cid:48) , l (cid:48)(cid:48) , − m (cid:48)(cid:48) ) . (4.13b)We can now turn to the evaluation of the terms δQ ( α =1) Y and δQ ( β =1) Y in a few specific cases of interest next. B. Multipolar expansion of the super angularmomentum
In this part, we will compute the multipolar expan-sion of the α and β “difference terms” in Eqs. (4.1b)and (4.1c) from the super angular momentum of [33]. Wewill consider two types of vector fields Y A to computethe charges: namely, the electric- and magnetic-parityvectors harmonics defined in Eqs. (4.3a) and (4.3b). Wewill thus denote these terms by δQ ( α =1)( e ) ,lm and δQ ( α =1)( b ) ,lm ,1respectively, for Eq. (4.1b) and δQ ( β =1)( e ) ,lm and δQ ( β =1)( b ) ,lm ,respectively, for Eq. (4.1c). The results here hold forboth the standard BMS charges (CM and intrinsic angu-lar momentum) and the generalized BMS charges (superangular momentum). There are a number of additionalsimplifications that occur for the intrinsic and CM angu-lar momentum, and we will therefore treat these simplercases separately afterwards.In this calculation, we will not require initially that thetwo parameters α and β be equal, because this choice wasmade to require that the standard (rather than the super)angular momentum vanishes in flat spacetimes. For thesuper angular momentum, the choice of α = β does notguarantee that these charges vanish in flat spacetimes,and it is not agreed upon universally that these chargesshould vanish in flat spacetime (see, e.g., [44]).Before we begin the calculations, note that because D A T ( b ) ,lmA = 0 , then by performing an integration byparts of Eq. (4.1c), one can show that δQ ( β =1)( b ) ,lm = 0 ; (4.14)we will thus focus on the three quantities δQ ( α =1)( e ) ,lm , δQ ( α =1)( b ) ,lm , and δQ ( β =1)( e ) ,lm . The calculation of these threequantities is quite similar, so we will describe in detailthe procedure for just δQ ( α =1)( e ) ,lm (and the other two quan-tities can be determined through a nearly identical cal-culation).Starting from Eq. (4.1b), we then substitute in themultipolar expansion of C AB and D A C AB given inEqs. (4.5) and (4.7) and the vector spherical harmonicin Eq. (4.3a). We then use the relationships betweenthe vector and tensor spherical harmonics and the spin-weighted spherical harmonics in Eqs. (4.9a)–(4.10b) towrite δQ ( α =1)( e ) ,lm in terms of the multipole moments U lm and V lm as well as the integrals of three spin-weightedspherical harmonics in Eq. (4.11). We then make use ofthe identities for the coefficients C l ( s (cid:48) , l (cid:48) , m (cid:48) ; s (cid:48)(cid:48) , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) in Eq. (4.13) and the complex conjugate properties of U lm and V lm in Eq. (4.6) to simplify the expression. It isuseful to make the definitions (similar to those in [34]) s l, ( ± ) l (cid:48) ; l (cid:48)(cid:48) = 1 ± ( − l + l (cid:48) + l (cid:48)(cid:48) , (4.15a) f ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) = (cid:112) ( l (cid:48) + 2)( l (cid:48) − C l ( − , l (cid:48) , m (cid:48) ; 2 , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) , (4.15b) g ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) = (cid:112) l ( l + 1) C l ( − , l (cid:48) , m (cid:48) ; 2 , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) . (4.15c)The result can then be written as is δQ ( α =1)( e ) ,lm = − π (cid:88) l (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) f ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) × [ s l, (+) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) + V l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) )+ is l, ( − ) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) − V l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) )] , (4.16a) where the indices on the charges should be integers inthe ranges l ≥ and − l ≤ m ≤ l , and where the sumsrun over integers in the ranges l (cid:48) ≥ , − l (cid:48) ≤ m (cid:48) ≤ l (cid:48) , l (cid:48)(cid:48) ≥ , and − l (cid:48)(cid:48) ≤ m (cid:48)(cid:48) ≤ l (cid:48)(cid:48) This gives the α -dependentdifference from the super-CM charge of [33]. A similarcalculation shows that the α -dependent correction to thesuperspin can be written as δQ ( α =1)( b ) ,lm = i π (cid:88) l (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) f ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) × [ s l, ( − ) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) + V l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) )+ is l, (+) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) − V l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) )] . (4.16b)Finally, the β -dependent correction to the super-CMcharge is given by δQ ( β =1)( e ) ,lm = − π (cid:88) l (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) g ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) × [ s l, (+) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) + V l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) )+ is l, ( − ) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) − V l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) )] . (4.16c)The values of l , l (cid:48) , l (cid:48)(cid:48) , m , m (cid:48) , and m (cid:48)(cid:48) in Eqs. (4.16b)and (4.16c) are the same as in Eq. (4.16a). From thesedifference terms and the super-CM and superspin chargeswith α = 1 and β = 1 (i.e., Q ( α =1 ,β =1)( e ) ,lm and Q ( α =1 ,β =1)( b ) ,lm )one can then construct the full α and β dependent superCM and superspin (i.e., Q ( α,β )( e ) ,lm and Q ( α,β )( b ) ,lm ).Although we do not require that the superspin and su-per CM vanish in flat spacetime, it is still useful to writedown the expressions for the α - and β -dependent differ-ence terms in this case: namely, the quantities δQ ( α = β =1)( e ) ,lm and δQ ( α = β =1)( b ) ,lm . It is then straightforward to specializeour previous results to find that δQ ( α = β =1)( e ) ,lm = − π (cid:88) l (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) (2 f ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) + g ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) ) × [ s l, (+) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) + V l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) )+ is l, ( − ) l (cid:48) ; l (cid:48)(cid:48) ( U l (cid:48) m (cid:48) V l (cid:48)(cid:48) m (cid:48)(cid:48) − V l (cid:48) m (cid:48) U l (cid:48)(cid:48) m (cid:48)(cid:48) )] . (4.17a)The superspin is the same, because the term δQ ( β =1)( b ) ,lm vanishes: i.e., δQ ( α = β =1)( b ) ,lm = δQ ( α =1)( b ) ,lm . (4.17b)In the next subsections, we will further specializeEqs. (4.17a) and (4.17b) to l = 1 spherical harmonicsto compute the CM and intrinsic angular momentum.2 C. Multipolar expansion of the intrinsic angularmomentum
We begin by simplifying the expression in Eq. (4.16b)in the case where l = 1 (which corresponds to the correc-tion to the intrinsic angular momentum). When l = 1 ,the coefficients f l (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) are nonvanishing for l (cid:48)(cid:48) = l (cid:48) or l (cid:48)(cid:48) = l (cid:48) ± . Thus, the coefficient s , ( − ) l (cid:48) ; l (cid:48)(cid:48) is nonvanishingonly when l (cid:48)(cid:48) = l (cid:48) and the coefficient s , (+) l (cid:48) ; l (cid:48)(cid:48) is nonvanish-ing for l (cid:48)(cid:48) = l (cid:48) ± . Because the index m satisfies m = 0 or m = ± , then for the first set of terms in Eq. (4.16b)proportional to s , ( − ) l (cid:48) ; l (cid:48)(cid:48) the nonzero terms in the doublesum will be one of the terms of the form f l (cid:48) ,m (cid:48) ; l (cid:48) , − m (cid:48) or f l (cid:48) ,m (cid:48) ; l (cid:48) , − m (cid:48) ± . Given the complex-conjugate relation-ships for the U lm and V lm moments in Eq. (4.6) and thesymmetries of the coefficients C ( − , l (cid:48) , m (cid:48) ; 2 , − l (cid:48) , m (cid:48)(cid:48) ) under the change of sign of m (cid:48) in Eq. (4.13), then onecan show that the terms proportional to s , ( − ) l (cid:48) ; l (cid:48)(cid:48) vanish.The difference term from the Wald-Zoupas angular mo-mentum is then given by δJ ( α =1)1 ,m ≡ δQ ( α =1)( b ) , ,m = 1128 π (cid:88) l (cid:48) ,m (cid:48) ,l (cid:48)(cid:48) ,m (cid:48)(cid:48) s , (+) l (cid:48) ; l (cid:48)(cid:48) f l (cid:48) ,m (cid:48) ; l (cid:48) ,m (cid:48)(cid:48) × ( U l (cid:48) m (cid:48) V l (cid:48)(cid:48) ,m (cid:48)(cid:48) − V l (cid:48) m (cid:48) U l (cid:48)(cid:48) ,m (cid:48)(cid:48) ) . (4.18)Note that although we left the expression as a doublesum over l (cid:48) and l (cid:48)(cid:48) , the l (cid:48)(cid:48) sum is restricted to l (cid:48)(cid:48) = l (cid:48) − or l (cid:48)(cid:48) = l (cid:48) + 1 ; similarly, the m (cid:48)(cid:48) sum is restricted to thevalues m (cid:48)(cid:48) = m − m (cid:48) , where m = 0 or m = ± . If weevaluate the coefficients f l (cid:48) ,m (cid:48) ; l (cid:48) ± , − m (cid:48) , f l (cid:48) ,m (cid:48) ; l (cid:48) ± , − m (cid:48) − ,and f l (cid:48) ,m (cid:48) ; l (cid:48) ± , − m (cid:48) +1 in the sum using the expression inEq. (4.12), then the expressions can be simplified tosquare roots of rational functions in these cases. We fol-low [34] and define coefficients a l , b ( ± ) lm , c lm and d ( ± ) lm by a l = (cid:115) ( l − l + 3)(2 l + 1)(2 l + 3) , (4.19a) b ( ± ) lm = (cid:112) ( l ± m + 1)( l ± m + 2) , (4.19b) c lm = (cid:112) ( l − m + 1)( l + m + 1) , (4.19c) d ( ± ) lm = (cid:112) ( l ± m + 1)( l ∓ m ) (4.19d)(though we do not use d ( ± ) lm until the next subsection).In terms of these quantities, and after relabelling l (cid:48) with l and m (cid:48) with m in the sum, we can write the difference term from the Wald-Zoupas angular momentum as δJ ( α =1)1 , = 116 (cid:114) π (cid:88) l ≥ ,m a l c lm l + 1 × ( ¯ U lm V l +1 ,m − ¯ V lm U l +1 ,m ) , (4.20a) δJ ( α =1)1 , ± = 132 (cid:114) π (cid:88) l ≥ ,m a l b ( ± ) lm l + 1 × ( ¯ U lm V l +1 ,m ± − ¯ V lm U l +1 ,m ± ) . (4.20b)The calculation to arrive at these simplified expressionsrequires some relabelling of indices in the sum so thatonly terms with l + 1 appear rather than l − .A similar calculation was performed in [26] using STF l -index tensors rather than expanding C AB in the har-monics in Eq. (4.5). The two formalisms can be related,and we compared the result of the difference term in [26]for the intrinsic angular momentum to our expressionsin Eqs. (4.20a) and (4.20b). We found that our resultdiffers from Eq. (4.16) of [26] by an additional factor of / ( l + 1) , and we could not identify from where this dis-crepancy was arising. We give a detailed calculation ofthis comparison in Appendix A. Given our results in thenext subsection, we believe our result to be correct, sowe suspect that the error lies in the conversion betweenthe two formalisms. D. Multipolar expansion of the center-of-massangular momentum
We now derive a similar expression for the differenceterms from the Wald-Zoupas center-of-mass angular mo-mentum when expanded in terms of the the mass andcurrent multipole moments of C AB in Eq. (4.5). We firstgive a result for general real coefficients α and β , and wethen specify to the α = β choice. The calculation is quitesimilar to that in the previous subsection for the intrinsicangular momentum. When the expression in Eq. (4.16a)is restricted to l = 1 , then there is again a similar can-cellation of the terms proportional to s , ( − ) l (cid:48) ; l (cid:48)(cid:48) leaving justthe terms proportional to s , (+) l (cid:48) ; l (cid:48)(cid:48) . Again, because the al-lowed values of l (cid:48)(cid:48) are given by l (cid:48)(cid:48) = l (cid:48) ± , the coefficients f l (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) simplify to square roots of rational functions.3The α -dependent difference terms are then given by δQ ( α =1)( e ) , , ≡ δk ( α =1)1 , = 116 (cid:114) π (cid:88) l ≥ ,m a l c lm l + 1 × ( ¯ U lm U l +1 ,m + ¯ V lm V l +1 ,m ) , (4.21a) δQ ( α =1)( e ) , , ± ≡ δk ( α =1)1 , ± = 132 (cid:114) π (cid:88) l ≥ ,m a l b ( ± ) lm l + 1 × ( ¯ U lm U l +1 ,m ± + ¯ V lm V l +1 ,m ± ) , (4.21b)for the m = 0 and m = ± modes, respectively.For the β -dependent difference term in Eq. (4.16c), it isno longer the case that the s , ( − ) l (cid:48) ; l (cid:48)(cid:48) terms vanish. However,because the coefficients g l (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) also have the propertythat they vanish except when l (cid:48)(cid:48) = l (cid:48) or l (cid:48)(cid:48) = l (cid:48) ± andwhen m (cid:48)(cid:48) = m − m (cid:48) for m = 0 or m = ± , then thecoefficients can similarly be evaluated in terms of ratio-nal functions and their square roots. The result of thiscalculation is as follows: δQ ( β =1)( e ) , , ≡ δk ( β =1)1 , = − (cid:114) π (cid:88) l ≥ ,m l + 1 × (cid:20) a l c lm ( ¯ U lm U l +1 ,m + ¯ V lm V l +1 ,m ) − iml ¯ U lm V lm (cid:21) , (4.22a) δQ ( β =1)( e ) , , ± ≡ δk ( β =1)1 , ± = − (cid:114) π (cid:88) l ≥ ,m l + 1 × (cid:20) a l b ( ± ) lm ( ¯ U lm U l +1 ,m ± + ¯ V lm V l +1 ,m ± ) ± il d ( ± ) lm ¯ U lm V l,m ± (cid:21) . (4.22b)The coefficients d ( ± ) lm are defined in Eq. (4.19).A significant simplification occurs when the two pa-rameters are equal; only the terms involving products of U lm and V lm moments remain. We find that the result isgiven by δQ ( α = β =1)( e ) , , ≡ δk ( α = β =1)1 , = i (cid:114) π (cid:88) l ≥ ,m ml ( l + 1) ¯ U lm V lm , (4.23a) δQ ( α = β =1)( e ) , , ± ≡ δk ( α = β =1)1 , ± = ∓ i (cid:114) π (cid:88) l ≥ ,m d ( ± ) lm l ( l + 1) ¯ U lm V l,m ± . (4.23b)This result is consistent with our calculation in flat space-time in Sec. III. In that section, we showed that when α = β , the angular momentum should vanish in flat spacetime. Because the tensor C AB can be decomposedusing just electric-type tensor harmonics (i.e., the U lm modes can be nonvanishing but all V lm modes must van-ish), then the multipolar expansion should not involveproducts of U lm moments with other U lm moments, be-cause these terms would be nonvanishing in flat space-time.Our result for the β -dependent term in Eqs. (4.22)agrees with Eq. (4.17) of [26] after performing the sameconversion between their STF l -index tensors and ourmass and current multipoles U lm and V lm . This compar-ison is given in detail in Appendix A. The α -dependentterms in Eq. (4.21) was not computed in [26]. Note,however, that the coefficients in δk ( α =1)1 m in Eq. (4.21)that multiply the products of U lm and V lm moments areprecisely the same ones that appear in Eq. (4.20) for δJ ( α =1)1 m . Since the coefficients are the same in Eqs. (4.20)and (4.21), and since these coefficients are needed to havethe angular momentum vanish in flat spacetime, then thisprovides a consistency check on the result in Eq. (4.20).Now that we have the multipolar expressions for thedifference terms from the Wald-Zoupas definition of theangular momentum, it is possible to assess how largethese terms are for different systems of interest. We willfocus on nonspinning compact binaries in the next sec-tion. V. STANDARD AND SUPER ANGULARMOMENTUM FOR NONPRECESSING BBHMERGERS
In this part, we compute the effect of the remainingfree parameter α on the standard and super angular mo-mentum from nonprecessing binary-black-hole mergers.As discussed in the introduction, the value of the (su-per) angular momentum depends on a choice of Bondiframe. For the explicit calculations using PN theory andNR surrogate models in this section, we will work in thecanonical frame (e.g., [9]) associated with the binary as u → −∞ . This frame is a type of asymptotic rest framein which C AB = 0 and the system has vanishing massdipole moment (i.e., a CM frame).For the difference of the angular momentum from theWald-Zoupas values [i.e., Eqs. (4.20) and (4.23)], this dif-ference depends on products of both the U lm and the V lm modes. As we discuss in the first subsection in this part,the U lm modes can be nonvanishing after the passage ofGWs for these BBH mergers, because of the GW mem-ory effect. The V lm modes vanish after the radiationpasses for these BBH systems (see, e.g., [46]; thus, thedifference terms in Eqs. (4.20) and (4.23) will vanish af-ter the passage of the GWs. This implies that the netchange in the angular momentum between two nonradia-tive regions for these binaries will be the same. Neverthe-less, while the binary is emitting GWs, the instantaneousvalue of the angular momentum will differ from the Wald-Zoupas value. We compute the size of this effect in the4post-Newtonian (PN) approximation and using surrogatemodels fit to numerical-relativity (NR) simulations in thefollowing subsections.We then perform similar calculations involving thedifference terms from the super angular momentumof [33]. Because the super angular momentum terms inEq. (4.16a) involve products of U lm moments, then thesuper angular momentum can differ from the α = β = 1 values when there is the GW memory effect. We thus es-timate the magnitude of this difference in the PN approx-imation and from the dominant waveform modes fromNR simulations. As we will discuss further below, the ef-fect is small compared to the change in the super angularmomentum, but is within the numerical accuracy of thesimulations.Because we are interested in investigating the order-of-magnitudes of the effects rather than their precise values,we will generally work with the leading-order approxima-tions to the results in this section, as we will describe inmore detail in the relevant parts below. A. Computing the leading GW memory effect andspin memory effect
In post-Newtonian theory, the GW memory effectand the spin memory effect have been computed, andthe relevant results can be obtained from, e.g., [47]or [21], respectively. For NR simulations, GW mem-ory effects are not captured in most Cauchy simulations(see, e.g., [48]) and the additional post-processing stepof Cauchy-characteristic extraction [49] needs to be per-formed [46, 50] to get the memory effects directly fromsimulations. However, by enforcing the flux balance lawsin Eq. (2.10), one can determine constraints on the GWmemory effects from waveforms that do not contain thememory (e.g., [21, 51]). This approximate procedure isquite accurate [46]. We summarize our procedure forcomputing GW memory effects below.
1. (Displacement) GW memory effect
The GW memory effect can be computed by integrat-ing the conservation equation for the Bondi mass aspectin Eq. (2.5a) with respect to u [this equation containsequivalent information to the flux balance law (2.10)for a basis of supertranslation vectors]. Integrating theterm D A D B N AB in Eq. (2.5a) with respect to u givesrise to a change in the shear, which we will denoteby D A D B ∆ C AB . This quantity D A D B ∆ C AB is con-strained by changes in the mass aspect ∆ m and the inte-grated flux of energy per solid angle (a term proportionalto (cid:82) duN AB N AB ; see, e.g., [9] and references therein forfurther discussion). This equation constrains only theelectric part of ∆ C AB , and for this reason it is conve-nient to write the memory using a single scalar function ∆Φ as ∆ C AB = (cid:18) D A D B − h AB D (cid:19) ∆Φ . (5.1)It is then useful to expand ∆Φ in scalar spherical har-monics Y lm . Once this is done, when the the opera-tor (2 D A D B − h AB D ) acts on these scalar harmonics,Eq. (5.1) can be written in terms of the electric-paritytensor harmonics in Eq. (4.4a) as ∆ C AB = (cid:88) l,m (cid:115) ( l + 2)!2( l − T ( e ) ,lmAB ∆Φ lm . (5.2)By comparing Eq. (5.2) with Eq. (4.5), it is straightfor-ward to see that the change in the U lm moments can berelated to the ∆Φ lm modes via the relationship ∆ U lm = (cid:115) ( l + 2)!2( l − lm . (5.3)Although both changes in the Bondi mass aspect andthe flux of energy per solid angle produce GW memoryeffects, for nonprecessing BBH mergers, the flux termproduces the much larger memory effect (i.e., the nonlin-ear memory is much larger than the linear memory; thisis true in both the post-Newtonian approximation [52]and in NR simulations [46]). For this reason, just thecontributions from the nonlinear memory to ∆Φ werecomputed in [21], and the result is given in terms of themass and current multipole moments by ∆Φ lm = 12 ( l − l + 2)! (cid:88) l (cid:48) ,l (cid:48)(cid:48) ,m (cid:48) ,m (cid:48)(cid:48) C l ( − , l (cid:48) , m (cid:48) ; 2 , l (cid:48)(cid:48) , m (cid:48)(cid:48) ) × (cid:90) ∞−∞ du { is l, ( − ) l (cid:48) ; l (cid:48)(cid:48) ˙ U l (cid:48) m (cid:48) ˙ V l (cid:48)(cid:48) m (cid:48)(cid:48) + s l, (+) l (cid:48) ; l (cid:48)(cid:48) ( ˙ U l (cid:48) m (cid:48) ˙ U l (cid:48)(cid:48) ,m (cid:48)(cid:48) + ˙ V l (cid:48) m (cid:48) ˙ V l (cid:48)(cid:48) ,m (cid:48)(cid:48) ) } . (5.4)Both in the PN approximation and in NR simulations,the largest contribution to the GW memory effect fromnonprecessing BBH mergers comes from terms involvingproducts of U and U − = ¯ U modes in Eq. (5.4).The dominant memory effect produced by the U modeappears in the ∆Φ and ∆Φ modes. Evaluating theappropriate coefficients in Eq. (5.4) and using Eq. (5.3),we find that the leading GW memory effect in the mode U is given by ∆ U = 142 (cid:114) π (cid:90) ∞−∞ du | ˙ U | . (5.5a)The expression for the U mode is given by ∆ U = 1504 √ π (cid:90) ∞−∞ du | ˙ U | = 160 √ U . (5.5b)We will also consider quantities U and U which areobtained by integrating Eq. (5.5) from −∞ up to a finiteretarded time u rather than than taking the limit u → ∞ .5
2. GW modes that produce the spin-memory effect
The other type of GW memory that we will need toconsider in this paper is the GW spin memory effect.Like the GW memory effect in the previous subsection,the spin memory effect can also be determined from theflux balance law in Eq. (2.10). Unlike the displacementmemory, the spin memory is constrained by changes inthe super angular momentum (rather than the supermo-mentum) and the flux of angular momentum per solidangle (rather than the flux of energy per solid angle). Inaddition, the spin memory effect appears in the magnetic-parity part of the retarded-time integral of the shear ten-sor, rather than the electric part of the change in theshear. We will not need the spin memory itself, but wedo need the GW modes that produce the spin memory ef-fect. Nevertheless, it is easiest to describe the calculationof these modes by summarizing the calculation of the spinmemory. We thus begin by writing the shear tensor C AB as a sum of two terms of electric- and magnetic-parityparts C AB = 12 (cid:0) D A D B − h AB D (cid:1) Φ + (cid:15) C ( A D B ) D C Ψ , (5.6)where Φ and Ψ are smooth functions of the coordinates ( u, θ A ) . The spin memory is related to the retarded timeintegral of the function Ψ [21] ∆Σ ≡ (cid:90) + ∞−∞ du Ψ . (5.7)The full multipolar expansion of the spin memory is asomewhat lengthy expression, so we do not reproduce ithere (although it is given in [21]). Analogously to thedisplacement GW memory effect, there are two contri-butions to the spin memory effect from the linear andnonlinear terms. However, the linear terms are smallerthan the nonlinear terms for nonprecessing compact bi-naries (see, e.g., [46]), so we focus on just the nonlinearterms. We will also give just the largest terms that arecomputed from the mode U (which is the dominantterm in the PN approximation, and also the most signifi-cant term in NR simulations). The U mode produces aspin memory effect that appears in the u integral of the l = 3 , m = 0 mode of the waveform; it was computedin [21] to be ∆Σ = 180 √ π Y (cid:90) du (cid:61) ( ¯ U ˙ U ) . (5.8)Acting on ∆Σ with the operator (cid:15) C ( A D B ) D C gives theretarded-time integral of the magnetic-parity part of theshear tensor C AB : (cid:15) C ( A D B ) D C ∆Σ = 140 (cid:114) π T ( b ) , AB (cid:90) du (cid:61) ( ¯ U ˙ U ) . (5.9) By differentiating Eq. (5.9) with respect to u , we can ob-tain the magnetic part of the shear that produces the spinmemory effect. Because Eq. (5.9) is already expanded inmagnetic-parity tensor harmonics, we can immediatelydetermine that the relevant spin-memory mode is V ,which is given by V = 140 (cid:114) π (cid:61) ( ¯ U ˙ U ) . (5.10)We will use Eqs. (5.5) and (5.10) to add in the contri-butions of the memory and spin memory effects that arenot included in the NR surrogate waveform model thatwe use to compute the difference terms from the respec-tive Hamiltonian definitions of [20] and of [33] for theangular momentum and super angular momentum in thenext subsections. B. Standard angular momentum
We noted above that the different definitions of theangular momentum for nonprecessing BBH mergers willagree after the gravitational waves pass, but they will dif-fer while these systems are radiating gravitational waves.We will calculate the size of this difference first in thepost-Newtonian (PN) approximation and second in fullgeneral relativity using numerical-relativity waveformsfrom BBH mergers. The NR waveforms are usually givenin terms of the multipole moments of the strain h , whichis related to the tensor C AB by the relation h ≡ h + − ih × = 1 r C AB ¯ m A ¯ m B . (5.11)This expression defines the two polarizations h + and h × and ¯ m A is the complex conjugate of the dyad definedin Eq. (4.8). The strain h can be expanded in terms ofspin-weighted spherical harmonics − Y lm as h = (cid:88) lm h lm ( − Y lm ) . (5.12)It then follows that the moments h lm are related to U lm and V lm by h lm = 1 r √ U lm − iV lm ) (5.13)(see, e.g., [21] and references therein).Because of the symmetries of nonprecessing binaries,the relationship between the h lm mode and the U lm and V lm modes simplifies. Specifically, the mass multipolemoments U lm are nonzero only when l + m is even, and thecurrent multipole moments V lm are nonzero only when l + m is odd (see, e.g., [53]). Therefore, the mass andcurrent multipole moments can be written in terms ofthe strain modes for these systems as U lm = r √ h lm , for l + m even , (5.14a) V lm = ir √ h lm , for l + m odd . (5.14b)6Note that our definition of the polarizations h + and h × (and hence h lm ) have a relative minus sign to thosein [53], though the U lm and V lm moments agree in sign.Combining these properties of the U lm and V lm mo-ments with the expressions for the difference terms inEqs. (4.20) and (4.23), we find that multipole moments δJ ( α = β =1)1 ± and δk ( α = β =1)10 vanish. Thus, we focus on the δJ ( α = β =1)10 and δk ( α = β =1)1 ± modes below.The waveforms from PN calculations and surrogatemodels from NR simulations contain a finite number of ( l, m ) modes [in the PN context, the waveform has onlybeen computed up to a finite PN order, whereas for sur-rogate models, the NR simulations extract only a subsetof all ( l, m ) modes, and the surrogate models only fit toa further subset of the extracted modes]. The number ofmodes that we use in the calculations of the quantities δJ ( α = β =1)10 and δk ( α = β =1)1 ± will differ, but it is chosen suchthat we capture the leading nonvanishing effect in the PNapproximation. We will then use the same set of modesfor the calculations with the NR surrogate waveform (ab-sent any modes that the surrogate model does not con-tain). As we will discuss in more detail below, we willuse waveform modes that go up to 2.5PN orders abovethe leading part of the U mode to compute δJ ( α = β =1)10 ,whereas for δk ( α = β =1)1 ± , we can capture the leading effectusing just the leading U mode and the V mode. Thus,to compute δJ ( α = β =1)10 we use the expression δJ ( α = β =1)10 = 18 (cid:114) π (cid:60) (cid:20) a c U V + a c U V + a c U V + a c U V − a c V U − a c V U − a c V U − a c V U (cid:21) , (5.15)Note that the real part of the quantity in parenthesesis being taken, which arises from using the complex-conjugate properties of the modes U lm and V lm inEq. (4.6). For δk ( α = β =1)1 ± , we use the expressions δk ( α = β =1)11 = i (cid:114) π (cid:16) d (+)2 − U ¯ V − d (+)20 ¯ U V (cid:17) , (5.16a) δk ( α = β =1)1 − = i (cid:114) π (cid:16) d ( − )22 ¯ U V − d ( − )20 U ¯ V (cid:17) . (5.16b)Here note that δk ( α = β =1)11 = − δ ¯ k ( α = β =1)1 − , since δk ( α = β =1)11 and δk ( α = β =1)1 − can both be related to the real differenceterms from the x and y components of the Wald-ZoupasCM angular momentum (see Appendix A).
1. Post-Newtonian results
For nonprecessing binaries, the mass and current mul-tipole moments U lm and V lm are expressed convenientlyin terms of several different mass parameters and massratios. Here we denote the individual masses by m and m with m > m . We then denote the totalmass by M = m + m , the relative mass difference by m = ( m − m ) /M , the mass ratio by q = m /m ≥ ,and the symmetric mass ratio ν = m m /M . We alsouse the notation Ω for the orbital frequency, ψ for theorbital phase, and x = ( M Ω) / for the PN parameter,as in [53]. It is shown in [53] that all the waveform modes h lm can be written in the form h lm = − M νxr (cid:114) π H lm e − imψ , (5.17)where the terms H lm are given in Eqs. (328)–(329) of [53]and can be written as polynomials in the square root ofthe PN parameter (i.e., √ x ). We do not use the fullexpressions for H lm in Eqs. (328)–(329) of [53]; rather weonly go up to 2.5PN order (i.e., x / ) in these equations.After substituting these expressions into Eq. (5.15), wefind that the result for δJ α = β =11 , is given by δJ α = β =110 = 85 (cid:114) π M ν (cid:18) − − m
210 + 93294410 ν (cid:19) x / + O ( x ) . (5.18)The angular momentum in the Newtonian limit goesas x − / , so the correction term in Eq. (5.18) appearsat 5PN order with respect to the leading-order effect.During the inspiral when the PN parameter x is small, δJ α = β =110 is not expected to be very large. Given the factthat the product ¯ U V scales with the PN parameteras x , it might initially seem unusual that the net effect δJ α = β =110 goes like x / . Because there is a real part inEq. (5.15), there are a number of cancellations that oc-cur between different modes. These cancellations in the U lm and V lm moments occur in the conservative part ofthe dynamics, but not the dissipative part from GW ra-diation reaction. These dissipative dynamics appear as arelative 1.5PN correction to V , which explains why theleading order part of δJ α = β =110 goes like x / . Analogousarguments can be made for the other terms in Eq. (5.15).There is another feature of Eq. (5.18) worth describingthat relates to the dependence of δJ α = β =110 on the massratio q (and which is a feature that also appears in theNR simulations, which we discuss later). Specifically, thesign of δJ α = β =110 changes, and there is a specific mass ratioat which the leading PN expression vanishes. The valueof the mass ratio can be computed from Eq. (5.18) to be q ≈ . . The physical reason for this value was less clearto us, though it arises from the change in amplitudes ofthe multipole moments U lm and V lm as a function of massratio q .7The leading-order contribution to δk ( α = β =1)1 ± turnsout to require fewer terms to compute, as indicated inEq. (5.16), and it only requires the leading-order parts ofthe moments U and V . It is reasonably straightfor-ward to show that δk ( α = β =1)1 ± is given by δk , ± = − i (cid:114) π M ν m x / e ∓ iψ + O ( x ) . (5.19)The difference term from the Wald-Zoupas definition ofthe CM angular momentum scales as x / , which is twoPN orders lower than the correction term to the intrin-sic angular momentum. However, this effect also goesas e ∓ iψ , so the average over an orbital period vanishes.As was discussed in [34], while the change in the Wald-Zoupas definition of the CM angular momentum scaleswith the PN parameter as x = O (1) , there is a choice ofreference time u that can set the change in the CM angu-lar momentum to zero through 2PN order (i.e., through x ). At 2.5PN order ( x / ), there is no longer just achoice of reference time that allows the effect to be setto zero, which also preserves the fact that the binarywas initially chosen to be in the CM frame and rest-frame of the source with the supertranslations chosensuch that C AB = 0 initially. Thus, the terms δk ( α = β =1)1 ± in Eq. (5.19) are of the same PN order as the nontriv-ial (in the sense discussed here) Wald-Zoupas CM angu-lar momentum. The impact of the different definitionsof angular momentum is thus largest for the CM angu-lar momentum (although the impact of the CM angularmomentum on the evolution of compact binaries has notbeen discussed as extensively as that of the other chargesassociated with the Poincaré group).Finally, we also point out that from Eq. (5.19) it can beshown that the maximum effect happens approximatelyat q = 2 . . This is comparable to the value of the massratio that results in the maximum kick velocity for non-spinning binaries ( q = 2 . ± . ) [54].
2. Results from NR surrogate models
While the PN approximation gives useful intuitionabout the effect of the remaining free parameter α on theintrinsic and CM angular momentum during the inspiralphase of a compact binary, it is not expected to be accu-rate during the merger and ringdown phases. Instead, itis preferable to use the results of NR simulations duringthese late stages of a BBH merger. In particular, we willuse the hybrid NR surrogate model NRHyb3dq8 [55] togenerate the waveform modes that enter into Eqs. (5.15)and (5.16). The surrogate produces the waveform modes rh lm /M , which we convert to the U lm and V lm momentsusing Eq. (5.14). Because the surrogate does not modelthe modes h , h and h , we cannot include the surro-gate model’s contribution to these modes in Eq. (5.15).Also, because the surrogate does not have the memoryor spin memory contributions to the modes h , h , and h , we add these contributions to those of the surrogatemodel. The procedure we use to compute these memorymodes is reviewed in Sec. V A.For presenting our results from the surrogate wave-forms, we opt to show the Cartesian components of theintrinsic or CM angular momentum instead of the multi-pole moments that were described in the previous parts.The conversion between these two descriptions is reason-ably straightforward and is described in further detailin Appendix A. We thus quote the results here. First,the z component for the intrinsic angular momentum δJ ( α = β =1) z can be related to δJ ( α = β =1)10 by δJ ( α = β =1) z = − (cid:114) π δJ ( α = β =1)10 . (5.20)Similarly, δk ( α = β =1) x and δk ( α = β =1) y can be related to δk ( α = β =1)1 ± by δk ( α = β =1) x = − (cid:114) π (cid:60) (cid:104) δk ( α = β =1)11 (cid:105) , (5.21a) δk ( α = β =1) y = 4 (cid:114) π (cid:61) (cid:104) δk ( α = β =1)11 (cid:105) (5.21b)(see also [34]). Because δk ( α = β =1) z is proportional to δk ( α = β =1)10 = 0 for nonspinning BBHs, then the magni-tude of the difference of the CM angular momentum isgiven by | δ k ( α = β =1) | = (cid:114)(cid:16) δk ( α = β =1) x (cid:17) + (cid:16) δk ( α = β =1) y (cid:17) . (5.22)We first show the difference of the intrinsic angularmomentum from the Wald-Zoupas value, δJ ( α = β =1) z , forBBHs with different mass ratios. The top panel of Fig. 1displays δJ ( α = β =1) z as a function of retarded time forthree different mass ratios, q = 1 , 2, and 4 as solid blue,orange dashed, and green dotted curves, respectively.The extreme values of the time series for δJ ( α = β =1) z ap-proach the largest positive, the closest to zero, and themost negative value for these three mass ratios, respec-tively. The dependence of the extreme value of δJ ( α = β =1) z as a function of mass ratio is illustrated in more detail inthe bottom panel of Fig. 1. As was noted in the discussionof δJ ( α = β =1) z in the PN approximation, the extreme valueof this quantity changes sign as a function of mass ratio.The value at which it undergoes this sign change for thesurrogate model is q ≈ . , which is close to the valuepredicted by the leading PN result of q ≈ . . There isa sharp feature in the curve near the mass ratio where δJ ( α = β =1) z goes to zero, because (what is for most massratios) the primary peak (which changes smoothly withmass ratio) becomes smaller than (what is for most massratios) the secondary peak (which also varies smoothlywith mass ratio, but at a different rate from the primarypeak). When the roles of primary and secondary peak re-verse for a small range of mass ratios, the slope changesabruptly, and this leads to this slight sharp feature.8 − − −
200 0 u/M − δ J ( α = β = ) z / M × − q=1q=2q=4 q − E x tr e m ao f δ J ( α = β = ) z / M × − FIG. 1.
Top : The z component of the difference of the intrin-sic angular momentum from the Wald-Zoupas values (denotedby δJ ( α = β =1) z ) as a function of retarded time for nonspinningBBH mergers of three mass ratios, q = 1 , 2, and 4. Note thatthe extreme value switches from a maximum to a minimumas a function of mass ratio. As discussed further in the text, δJ ( α = β =1) z was computed using a NR surrogate model (wherethe peak of the magnitude of the waveform is at retarded timeequal to zero) using Eqs. (5.15) and (5.20). Bottom : The ex-treme value of the z component of δJ ( α = β =1) z as a functionthe mass ratio. Consistent with the PN predictions, there is achange in the sign of the quantity δJ ( α = β =1) z that occurs nearthe mass ratio q = 2 . We also mention a few implications of the results pre-sented in Fig. 1. During the inspiral, the Newtonian valueof the orbital angular momentum is given by M νx − / .For an equal mass binary separated by a distance of M , the angular momentum will initially be of order ∼ . M . The final black hole is a Kerr black hole withspin of order ∼ . M , where M f is the final mass ofthe black hole (which is typically at least ninety percentof the total mass M ). Thus, the fact that δJ ( α = β =1) z is oforder a few times − M at its largest implies that thediscrepancies in the definitions of angular momentum willbe small for definitions where α is of order unity. How-ever, the final spin parameter of the black hole formed − − −
200 0 u/M − − δ k ( α = β = ) / M × − | δ k ( α = β =1) | /M δk ( α = β =1) x /M q . . . . . M a x i m ao f | δ k ( α = β = ) | / M × − FIG. 2.
Top : The magnitude and the x component of the dif-ference of the CM angular momentum from the Wald-Zoupasdefintion, | δ k ( α = β =1) | and δk ( α = β =1) x , respectively, as func-tions of retarded time. The system shown is a BBH mergerwith mass ratio q = 3 , and the waveform modes used inEqs. (5.19) and (5.21) were generated from a NR surrogate,where the peak magnitude of the waveform occurs at a timeequal to zero. The vector δ k ( α = β =1) is in phase with the or-bital motion of the binary during inspiral, and it grows inmagnitude until the merger, after which it settles to zero. Bottom : The maximum of the magnitude of the difference ofthe CM angular momentum from the Wald-Zoupas value asa function of the mass ratio of a BBH system. Note that themaximum value as a function of q occurs at roughly the samemass ratio that produces the maximum kick velocity of thefinal black hole (see the text for further discussion). from a BBH merger is often quoted to an accuracy whichis smaller than the values of δJ ( α = β =1) z described here(see, e.g., [48]). Thus, for completeness, NR simulationsshould specify which definition of angular momentum isbeing used.We now turn to the difference of the CM angular mo-mentum from the Wald-Zoupas value. We use the samesurrogate model to compute δk ( α = β =1) x and | δ k ( α = β =1) | as functions of retarded time. We plot these quantitiesin the top panel of Fig. 2 for q = 3 . The bottom panel of9Fig. 2 shows the peak value of the time series | δ k ( α = β =1) | as a function of the binary’s mass ratio, q . For an equalmass black-hole binary, q = 1 , the change in the CM an-gular momentum vanishes. This occurs because there isno linear momentum radiated from such a system, so theinitial and final rest frames are the same (and we havechosen the initial rest frame to be the CM frame). Thepeak value of | δ k ( α = β =1) | is reached at a mass ratio ofroughly q ≈ . . This is similar to the PN prediction of q ≈ . computed earlier. It is also near the peak value ofthe gravitational recoil computed in [54] of q ≈ . . Thedecrease in the magnitude of | δ k ( α = β =1) | at mass ratiosgreater than q ∼ . is likely related to the fact that thegravitational recoil also decreases at these larger massratios.As far as we are aware, there has not been a system-atic study of the size Wald-Zoupas CM angular momen-tum from numerical relativity simulations. In the PNapproximation, the calculations in [34], which were re-viewed in this subsection, suggest that the magnitudeof the Wald-Zoupas CM angular momentum, | k ( α = β =1) | ,goes as M x / . Thus, the magnitude of the CM an-gular momentum could be as large as order M nearthe merger (thereby making the difference | δ k ( α = β =1) | asmall effect). Further investigation is needed to have amore definitive statement about the possible importanceof the term | δ k ( α = β =1) | . C. Super angular momentum
We now turn to understanding effect of the free pa-rameter α ( = β ) on the difference of the super angularmomentum from the charge of [33] for nonspinning BBHmergers. Unlike the angular momentum, the super an-gular momentum can have a nontrivial net change be-tween the early- and late-time nonradiative regions of aspacetime for these systems. We thus focus on the netchange in the charges ∆ Q α = βY : namely, the differenceof Eq. (4.2a) between two nonradiative regions at earlyand late times. Thus, we will similarly be interested inthe change in the difference term from the α = β = 1 value of the charges; i.e., the quantity ∆ δQ α = β =1 Y , where δQ α = β =1 Y is defined in Eq. (4.2b).We now calculate the change in the largest (in mag-nitude) nonvanishing part of the super angular momen-tum, which appears in the l = 2 , m = 0 moments of thesuper-CM part (in both the PN approximation and fromNR simulations). First, we write the expression for thischange in the charges as ∆ Q ( α = β )( e ) , = ∆ Q ( α = β =1)( e ) , + ( α − δQ ( α = β =1)( e ) , . (5.23)The change in the term δQ ( α = β =1)( e ) , can be obtained bytaking the difference of Eq. (4.17a) evaluated at earlyand late times. For nonspinning binaries, all the V lm moments vanish in nonradiative regions; the change in the moments U lm can be nonvanishing in nonradiativeregions when there is a nontrivial GW memory effect.The largest moments are U and U , as described inSec. V A; however, because the mode U is a factor of √ times smaller than the U mode, we focus here onthe contribution from just U . We find that the leadingchange in the difference term is given by ∆ δQ ( α = β =1)( e ) , = 3448 π (cid:114) π ∆( U ) . (5.24)Finally, we will compute ∆ Q ( α = β =1)( e ) , . The term quadraticin C AB in Eq. (2.19) gives rise to a term quadratic in ∆ U which is identical to the expression for ∆ δQ ( α = β =1)( e ) , in Eq. (5.24). The term linear in the shear does notcontribute (because it involves only V lm modes) and theterm − uD A m does not have a contribution from non-spinning BBH mergers to this part of the charge. How-ever, the term involving N A in Eq. (2.19) does contributeto ∆ Q ( α = β =1)( e ) , . The form of N A is known in station-ary regions that are supertranslated from the canonicalframe in which C AB = 0 . It was shown in [9] that N A = − mD A Φ / , where Φ is the “potential” for theelectric part of the shear [as in Eq. (3.1c)], and the Bondimass aspect m is a constant in this frame. Using the factthat ∆ U = √ , we then find that the leading α = β = 1 super CM is given by ∆ Q ( α = β )( e ) , = − π M √ U + 3448 π (cid:114) π ∆( U ) . (5.25)The lowest multipole moment (consistent with the sym-metries of nonprecessing BBHs) in which the change inthe superspin part could appear is the l = 3 , m = 0 mode.When we evaluate the contribution of the U modes inEq. (4.16b) for l = 3 , m = 0 , we find it and the differencefrom the Hamiltonian charge of [33] both vanish: ∆ Q ( α = β =1)( b ) , = ∆ δQ ( α = β =1)( b ) , = 0 . (5.26)Note, however, that the instantaneous value of thecharges (not the change in a nonradiative-to-nonradiativetransition) can be nonvanishing, though we do not com-pute that quantity here. We next turn to the computa-tion of the super CM using the PN approximation andthe NR surrogate model discussed in the previous sub-section. PN approximation
We calculate the U waveformmodes associated with the GW memory effect as wasdescribed in Sec. V A. Because the PN approximationcovers only the inspiral, we truncate the calculation of ∆ U ( α = β =1)20 at a finite retarded time u , at which the bi-nary is at a PN parameter x . We thus denote the changein the PN parameter by ∆ x . This gives an expressionfor the U moment that is equivalent to the one givenin [53]. We thus find that the change in the super-CMangular momentum in Eq. (5.25) and the change in the0 q − − − − ∆ Q ( α = β = ) / M × − . . . . . . . ∆ δ Q ( α = β = ) / M × − ∆ Q ( α = β =1)20 /M ∆ δQ ( α = β =1)20 /M FIG. 3. The change in the Hamiltonian super-CM angularmomentum of [33], ∆ Q ( α = β =1)2 , (scale on the left), and thechange in the difference of the super-CM angular momentumfrom the Hamiltonian super-CM angular momentum of [33], ∆ δQ ( α = β =1)2 , (scale on the right), both as a function of themass ratio of the binary q . The difference term is about twoorders of magnitude smaller that the change in the super CM. difference in Eq. (5.24) are given by ∆ Q ( α = β )( e ) , = − (cid:114) π M ν ∆ x + 51372 (cid:114) π M ν ∆( x ) , (5.27a) ∆ δQ ( α = β =1)( e ) , = 51372 (cid:114) π M ν ∆( x ) . (5.27b)Thus, the different definitions of the super-CM angularmomentum causes a relative 1PN-order correction to theleading-order super-CM angular momentum. Numerical-relativity results
The GW memory effect islargest not during the inspiral, but after the merger andringdown of a BBH collision. To better understand thesize of the change in the super-CM angular momentum ofa BBH merger, we compute the full memory effect in the U mode as in Eq. (5.5a), and we substitute the resultinto Eqs. (5.24) and (5.25). We again consider nonspin-ning BBH mergers of different mass ratios, and we usethe same hybrid surrogate model NRHybSur3dq8 [55] tocompute ∆ U . We take the mass M that enters intoEq. (5.25) to be the final mass, which we compute usingthe NR fits computed in [6].In Fig. 3, we show the net change in difference inthe super-CM angular momentum from the Hamiltoniansuper-CM angular momentum of [33], as a function ofthe mass ratio of nonspinning BBH mergers of differentmass ratios between ≤ q ≤ . The maximum dif-ference occurs for equal-mass BBHs and decreases withhigher mass ratios, which is consistent with the ampli-tude of the memory effect computed from the dominantquadrupole modes, as in Eq. (5.5a). This figure illus-trates that the change in the difference terms of the lead-ing super-CM angular momentum are about one hun-dredth of the change in the super-CM of [33], which is itself a small effect in units of M . Nevertheless, thewaveform modes used to compute the result are suffi-ciently accurate that this difference can be resolved. VI. CONCLUSIONS
In this paper, we investigated the freedoms in defin-ing angular momentum and super angular momentumin asymptotically flat spacetimes and the implicationsof these freedoms on the values of the (super) angu-lar momentum of nonspinning binary-black-hole mergers.The fact that such freedoms exist was recently discussedin [26], which demonstrated that there can be a two (real)parameter family of angular momenta, which encompassa few commonly used definitions of angular momentumin asymptotically flat spacetimes. All members of thistwo-parameter family satisfy flux balance laws and areconstructed from quantities that are covariant with re-spect to 2-sphere cross sections of null infinity. We found,however, that for the angular momentum to vanish inflat spacetime, the two parameters must be equal; thisleads to a natural requirement that the family of an-gular momenta should depend upon only a single realparameter. If we do not require that the angular mo-mentum agree with the Hamiltonian definition of Waldand Zoupas, then we did not have a compelling reason tofix the remaining free parameter to a particular value.We further investigated the effect of this one free pa-rameter on the values of the angular momentum. To doso, we first derived a multipolar expansion (in terms ofthe radiative multipole moments of the GW strain) ofthe difference of the angular momentum from the Wald-Zoupas definition. The difference is constructed fromthe products of mass moments with current moments,unlike the flux of the Wald-Zoupas definition of angu-lar momentum, which is written in terms of productsof mass moments with themselves and current momentswith themselves. This fact has an important implica-tion for spacetimes that transition between nonradiativeregions at early times and at late times, the context inwhich the GW memory effect is usually computed. Forseveral types of systems of astrophysical interest, such ascompact-object mergers, the GW memory effect appearsin just the mass-type moments. Thus, the differenceterms that arise from products of mass and current mo-ments will vanish in these nonradiative-to-nonradiativetransitions, and the net change in the angular momen-tum will be independent of this remaining free parameter.There will, however, be a difference in the instantaneousvalue of the angular momentum while the system is ra-diating gravitational waves.We also proposed considering a two-parameter familyof super angular momentum in analogy with the two-parameter family of angular momentum given in [26].Choosing the two parameters to be equal does not gener-ically make the super angular momentum vanish in flatspacetime (and it has also been argued that the super1angular momentum should not necessarily vanish in thiscontext). There is a choice of the two parameters thatdoes manifestly make the super angular momentum van-ish in flat spacetime, but it does not correspond to theanalog of the Wald-Zoupas charge. We, therefore, de-rived a multipolar expansion of the difference in the su-per angular momentum from the Hamiltonian definitionof [33] that involved two real parameters. We also spe-cialized the result to have one free parameter, so thatthe charge reduces to the angular momentum when thesymmetry vector field reduces from an infinitesimal su-per Lorentz transformation to a standard infinitesimalLorentz transformation.Next, we investigated the magnitude of the differenceof the (super) angular momentum from the Wald-Zoupascharges for nonspinning, quasicircular binary-black-holemergers. For the standard angular momentum the differ-ence occurs only while the system is radiating GWs. Inthe post-Newtonian approximation, we found the differ-ence in the intrinsic angular momentum enters at a rel-ative 5PN-order to the Newtonian angular momentum,while the difference in the CM angular momentum, it ap-pears at the same PN order as the effect that cannot beset to zero through a particular choice of reference time(at 2.5PN order beyond the leading Newtonian expres-sion). Given the high PN orders, the effects will generallybe small, although they could become large near the bi-nary’s merger, when the PN approximation becomes in-accurate. During the inspiral, however, the difference inthe CM angular momentum from the Wald-Zoupas valuewill be larger than that of the intrinsic angular momen-tum, because of its lower PN order. For the super angularmomentum, the difference terms need not vanish after theradiation passes; thus, we focused on the net change ofthe charges between early times and late times. We foundthat the leading difference in the superspin vanishes forBBH mergers, while differences in the super-CM angu-lar momentum cause a relative 1PN difference from theHamiltonian super-CM angular momentum of [33].Finally, we estimated the difference terms for the (su-per) angular momentum using inspiral-merger-ringdownsurrogate waveforms of nonspinning BBH mergers thatwere fit to numerical-relativity simulation data. Theintrinsic angular momentum terms are largest at equalmass, change sign at a mass ratio near two, and thentake on the most negative value near a mass ratio offour before approaching closer to zero. The amplitudeof the effect is small compared to the Newtonian valueof the angular momentum. The maximum difference inthe CM angular momentum was found to happen ap-proximately at the mass ratio that produces the max-imum kick velocity of the final black hole. The differ-ence in the change of the super-CM angular momentumfrom the corresponding Hamiltonian expression of [33] ina nonradiative-to-nonradiative transition was only to afew percent correction. Although these differences in the(super) angular momentum are small compared to thevalues of the (super) angular momentum itself, they are able to be resolved for these systems. Thus, which defi-nition is being used should be specified when describingthe (super) angular momentum of nonspinning binary-black-hole mergers.
ACKNOWLEDGMENTS
A.E. and D.A.N. acknowledge support from the NSFgrant PHY-2011784. We thank Alex Grant for helpfuldiscussions about the Wald-Zoupas definition of angularmomentum in the covariant conformal approach to nullinfinity and for comments on the manuscript. We thankGeoffrey Compère and Ali Seraj for correspondence re-lated to their work [26]; we also thank Geoffrey Compère,Adrien Fiorruci, and Romain Ruzziconi for correspon-dence related to their work [27].
Appendix A: Conversion between STF tensors andspherical harmonics
In this section, we compare our expressions for the dif-ference in the intrinsic and center-of-mass angular mo-mentum from the Wald-Zoupas values in Eqs. (4.20)and (4.23) to a related result obtained by Compère etal. in [26]. We start with the intrinsic angular momen-tum terms, and we make this comparison by convertingthe u integral of the expression in Eq. (4.16) of [26] forthe intrinsic angular momentum in terms of STF l -indextensors U L ≡ U (cid:104) i ...i l (cid:105) and V L ≡ V (cid:104) i ...i l (cid:105) to the multi-pole moments U lm and V lm used in this paper (the an-gle brackets around indices mean that the symmetric,trace-free part of the tensor should be taken). We focuson the second term in Eq. (4.16) of [26] which repre-sents the difference from the Wald-Zoupas value of theangular momentum. We denote this correction term by δJ ( α = β =1) i , where the index i means the angular momen-tum was computed with respect to a vector on the 2-sphere Y Ai = (cid:15) AB D B n i . The quantity n i is a unit vectorin quasi-Cartesian coordinates that is constructed fromspherical polar coordinates ( θ, φ ) as follows n i = (sin θ cos φ, sin θ sin φ, cos θ ) . (A1)The expression for δJ ( α = β =1) i from [26] is given by δJ ( α = β =1) i = − (cid:88) l ≥ ( l + 1) µ l +1 ( b l U iL V L − b l +1 U L V iL ) . (A2)The coefficients b l (not to be confused with b ( ± ) lm definedin the main text) and µ l were defined in [26] to be b l = 2 ll + 1 , (A3a) µ l = ( l + 1)( l + 2)( l − ll !(2 l + 1)!! . (A3b)2To rewrite Eq. (A2) in terms of U lm and V lm modes,we relate the spherical harmonics Y lm to the symmet-ric trace-free tensors of rank- l (STF- l tensors) N L = n (cid:104) i . . . n i l (cid:105) using the result in [15] Y lm = Y lmL N L . (A4)The tensors Y lmL with − l ≤ m ≤ are a basis for the vectorspace of l -index STF tensors and are defined in [15] (wedo not need their explicit form here). They transformunder complex conjugation in the same way as the scalarspherical harmonics: ¯ Y lmL = ( − m Y l, − mL . (A5)The STF mass and current moments U L and V L arerelated to U lm , V lm , and Y lmL by U L = l !4 (cid:115) l ( l − l + 1)( l + 2) l (cid:88) m = − l U lm Y lmL , (A6a) V L = − ( l + 1)!8 l (cid:115) l ( l − l + 1)( l + 2) l (cid:88) m = − l V lm Y lmL ; (A6b)see, e.g., Eq. (2.10) of Ref. [52]. It is useful to make thedefinitions s l ≡ l !4 (cid:115) l ( l − l + 1)( l + 2) , (A7a) g l ≡ − ( l + 1)!8 l (cid:115) l ( l − l + 1)( l + 2) , (A7b)though note that s l and g l should not be confused with s l, ( ± ) l (cid:48) ; l (cid:48)(cid:48) or g ll (cid:48) ,m (cid:48) ; l (cid:48)(cid:48) ,m (cid:48)(cid:48) defined in the main text. By sub-stituting the STF moments into Eq. (A2), we can write δJ ( α = β =1) i as δJ ( α = β =1) i = (cid:88) l ≥ ( l + 1) µ l +1 (cid:88) m,m (cid:48) (cid:0) b l s l +1 g l U l +1 ,m (cid:48) ¯ V lm − b l +1 s l g l +1 ¯ U lm V l +1 ,m (cid:48) (cid:1) ¯ Y lmL Y l +1 ,m (cid:48) iL . (A8)We used the properties in Eqs. (4.6) and (A5) to simplifythe result. The quantity ¯ Y lmL Y l +1 ,m (cid:48) iL can be written interms of Clebsch-Gordan coefficients using Eq. (2.26b)of [15], and it is only non-zero only when m (cid:48) satisfies m (cid:48) = m or m (cid:48) = m ± (though note that we need tomultiply the result in [15] by a factor of π to accountfor the different normalization of the spherical harmonicsused in [26]). Evaluating the relevant Clebsch-Gordon coefficients gives δJ ( α = β =1) i = (cid:88) l ≥ ,m µ l +1 ( l + 1)(2 l − l ! (cid:112) (2 l + 3)(2 l + 1) × (cid:104) (cid:0) b l s l +1 g l U l +1 ,m ¯ V lm − b l +1 s l g l +1 ¯ U lm V l +1 ,m (cid:1) c lm ξ i + (cid:0) b l s l +1 g l U l +1 ,m +1 ¯ V lm − b l +1 s l g l +1 ¯ U lm V l +1 ,m +1 (cid:1) b (+) lm ξ i + (cid:0) b l s l +1 g l U l +1 ,m − ¯ V lm − b l +1 s l g l +1 ¯ U lm V l +1 ,m − (cid:1) b ( − ) lm ξ − i (cid:105) , (A9)where the basis vectors ξ i and ξ ± i are defined inEq. (2.15) of [15]: ξ i = δ zi , ξ ± i = 1 √ ∓ δ xi − iδ ii ) . (A10)To relate the multipole moments of the angular mo-mentum to the components of the angular momentumin inertial Minkowski coordinates, we follow a proceduresimilar to that described in [9, 21]. First we note thatone can write the magnetic-parity vector harmonics as ¯ T A ( b ) , m = ω i m (cid:15) AB D B n i , (A11)where the ω i m are then given by ω x = 0 , ω y = 0 , ω z = 12 (cid:114) π , (A12a) ω x ± = ∓ (cid:114) π , ω y ± = i (cid:114) π , ω z ± = 0 . (A12b)Because the angular momentum is a linear functionalof the vector field Y A , then the relationship between δJ ( α = β =1)1 m and δJ ( α = β =1) i is given by δJ ( α = β =1)1 m = ω i m δJ ( α = β =1) i . (A13)After substituting Eq. (A9) into Eq. (A13), we findthat δJ ( α = β =1)10 = 116 (cid:114) π (cid:88) l ≥ ,m a l c lm ( ¯ U lm V l +1 ,m − ¯ V lm U l +1 ,m ) , , (A14a) δJ ( α = β =1)1 ± = 132 (cid:114) π (cid:88) l ≥ ,m a l b ( ± ) lm ( ¯ U lm V l +1 ,m ± − ¯ V lm U l +1 ,m ± ) , (A14b)where each term in the sum is a factor of l +1 larger thanin Eq. (4.20) as noted in the text after that equation.We next perform a similar check for the center-of-massangular momentum. Since only the β -dependent termwas computed in [26], we convert their expression interms of STF tensors and compare it to the β -dependent3term in Eq. (4.22). We start from Eq. (4.17) of [26], andwe denote the second term by δk ( β =1) i , which is given by δk ( β =1) i = (cid:88) l ≥ (cid:104) ( l + 1) µ l +1 ( U iL U L + b l b l +1 V iL V L )+ 12 σ l (cid:15) ijk U jL − V kL − (cid:105) . (A15)The coefficient σ l is defined in [26] by σ l = 8( l + 2)( l − l + 1)!(2 l + 1)!! . (A16)We perform the same procedure of converting the l -indexSTF mass and current moments into the U lm and V lm .The β -dependent difference term in the CM can then be written as follows: δk ( β =1) i = (cid:88) l ≥ ,m (2 l + 1)!! l ! (cid:110) µ l +1 s l s l +1 (cid:115) (2 l + 3)(2 l + 1) × (cid:104) ( ¯ U lm U lm + ¯ V lm V lm ) c lm ξ i + ( ¯ U lm U l,m +1 + ¯ V lm V l,m +1 ) b (+) lm √ ξ i + ( ¯ U lm U l,m − + ¯ V lm V l,m − ) b ( − ) lm √ ξ − i (cid:105) + im l σ l s l g l ¯ U lm V lm ξ i − d (+) lm √ U lm V l,m +1 ξ i + d ( − ) lm √ U lm V l,m − ξ − i (cid:111) (A17)To relate the multipole moments of the CM angular mo-mentum to its components in inertial Minkowski coordi-nates, we follow the same procedure as with the intrinsicangular momentum. We first write the electric-type vec-tor harmonics as ¯ T A ( e ) , m = ω i m D A n i , (A18)where the coefficients ω i m are given in Eq. (A12). We canthen solve for the multipole moments of the CM angularmomentum given the relation δk ( β =1)1 m = ω i m δk ( β =1) i . (A19)Using Eqs. (A12) and (A19) with Eq. 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