Center of Mass and spin for isolated sources of gravitational radiation
aa r X i v : . [ g r- q c ] M a r Center of Mass and spin for isolated sources of gravitational radiation
Carlos N. Kozameh † , Gonzalo D. Quiroga † † Instituto de Física Enrique Gaviola, FaMAF, Universidad Nacional de Córdoba, Córdoba, Argentina. (Dated: January 26, 2018)We define the center of mass and spin of an isolated system in General Relativity. The resulting relationshipsbetween these variables and the total linear and angular momentum of the gravitational system are remarkablysimilar to their Newtonian counterparts, though only variables at the null boundary of an asymptotically flatspacetime are used for their definition. We also derive equations of motion linking their time evolution to theemitted gravitational radiation. The results are then compared to other approaches. In particular one obtainsunexpected similarities as well as some differences with results obtained in the Post Newtonian literature .These equations of motion should be useful when describing the radiation emitted by compact sources suchas coalescing binaries capable of producing gravitational kicks, supernovas, or scattering of compact objects.
I. INTRODUCTION
The main goal of this work is to define the notions ofcenter of mass and intrinsic angular momentum for isolatedsystems and obtain their dynamical evolution when gravita-tional radiation is emitted. The evolution of isolated systemsand its gravitational radiation is naturally described using thenotion of asymptotically flat spacetimes. Thus, our approachwill be based on this mathematical framework.Both in Newtonian theory and special relativity one can finda particular trajectory with the property that the mass dipolemoment vanishes at this trajectory. This special trajectory iscalled the center of mass. If one would like to generalize thisconcept to GR, then the goal would be to find a worldline inspacetime with analogous properties to the one described inNewtonian gravity or special relativity. The first step is there-fore to provide an adequate definition of mass dipole momentin GR.One also expects that any suitable definition of center ofmass should be related to other global quantities like theBondi mass M or momentum P i by the relation P i = M V i + radiation terms . However, in contrast to Newton’s theory ofgravity, the Bondi mass or momentum will not be conservedfor an isolated system since gravitational waves carry awaymass and momentum. Therefore, one also expects that thevelocity of the center of mass will change when radiation isemitted.It is also worth mentioning that there is a qualitative differ-ence between the geometrical meaning of the dipole mass mo-ment in Newtonian gravity and in special relativity. Whereasin Newton theory the mass moment is a vector, in special rel-ativity it is a component of the so called, the mass dipolemoment/angular momentum 2-form [1]. Thus, to imple-ment this program one should generalize the mass dipole mo-ment/angular momentum 2 form to GR, and then define thecenter of mass worldline as the special place where the massdipole vanishes. As a bonus one should obtain the intrinsicangular momentum evaluating the non-vanishing part of thisgeneralized 2-form on the center of mass worldline.However, as one can see in the literature, there are manydefinitions of angular momentum/mass dipole moment forisolated systems in general relativity. As a non complete listof authors we could mention Dray and Streubel [2], Bram- son [3], Geroch [4], Helfer [5], Moreschi [6], Penrose [7] andWinicour [8]. Although a recent living review [9] offers acomplete survey of the main results in the field with the mainmotivations and technical aspects of each definition, the factthat there is no agreement among these alternative approachesreflects the difficulty of the subject. However, there is a com-mon link between them that can be used as a starting point:all the approaches agree for quadrupole radiation.This fact has been used in the Adamo-Newman-Kozameh[10] approach. By restricting themselves to quadrupole radi-ation data, it is shown that both the center of mass and an-gular momentum are defined from an asymptotic Weyl scalarwhose l = 1 part of the spherical harmonic decompositiontransforms as a 4-dim two form under the action of the ho-mogeneous Lorentz algebra of the BMS group, the availablekinematic geometry of null infinity [10]. Moreover, the ANKformulation is the only one that gives equations of motion forboth the center of mass and spin of an isolated system. In theANK approach the center of mass and spin are respectivelythe real and imaginary parts of a complex worldline definedin the solution space of the good cut equation. The geometri-cal interpretation of this space is that each solution describes acongruence of asymptotically shear free null geodesics reach-ing null infinity. The novelty of the formulation lies in thedefinition of the spin as an intrinsic property of this complexworldline and thus it can be used to give a classic definition ofa gravitational particle with spin.From our perspective, however, the ANK approach hassome points that deserve further attention1. the angular momentum is only defined for quadrupoleradiation and cannot be extended to generic radiationsince it does not give the expected results when thespace time has a rotational symmetry. Thus, one mustgeneralize this definition to spacetimes with arbitrarygravitational radiation, and include the case when thespacetime is axially symmetric,2. By assumption, the approach is based on null congru-ences with vanishing shear at null infinity. However,at null infinity the shear of the future null cone of anypoint does not vanish. This follows from the opticalequations since a non vanishing Weyl curvature on thenull cone induces non vanishing shear. Thus, the cen-ter of mass worldline defined on the solution space ofasymptotically vanishing shears does not correspond toa worldline of the underlying spacetime.3. The spin is defined as the imaginary part of a complexworldline instead of simply evaluating the angular mo-mentum at the center of mass.In this work we present new definitions of center of massand spin using the available tools on asymptotically flat space-times. In these new definitions we try to answer the above is-sues by constructing one parameter (Newman-Unti) foliationsof null infinity that are related to null cones cuts from pointsof the spacetime and have non vanishing shear at null infin-ity. The spin is simply the angular momentum at the center ofmass and the center of mass is the place where the mass dipolemoment vanishes.Since there are many technical details, some of them in-volved, it is better to outline here the main ideas of our ap-proach. In this way the reader can have a broad picture with-out the technical complications.We first introduce the notion of null cone cuts as the inter-section of the future null cones from points x a of the space-time with null infinity. We then define the regularized nullcone cuts (or RNC cuts) as the Huygens part of the null conecuts. By construction the RNC cuts are smooth 2-surfaces atnull infinity that parametrically depend on the points of thespacetime. If the points x a ( u ) describe a worldline the RNCcuts yield a special Newman-Unti (NU) family of cuts.We then introduce the notion of linkages [8, 11] on thisspecial family of NU cuts to define the dipole mass mo-ment/angular momentum. The main reason for this choiceis that the linkage is a linear generalization of the Komar for-mula which automatically yields the standard Komar defini-tion when the spacetime has a Killing field associated to a ro-tational symmetry. By restricting the linkages to the RNC cutswe fix one of the main problems in the linkage formulation.Instead of having a definition of angular momentum with asupertranslation freedom we restrict the freedom to the RNCfamily, a special 4-dim family of Newman-Unti cuts, wherethe notions of dipole mass moment and angular momentumare introduced (see ref. [9] page 30). Although there are stillinfinite degrees of freedom, one for each worldline, the free-dom is analogous to the choice of origins in the Newtoniandefinition of angular momentum.Finally, by demanding that on one RNC cut the mass dipoleterm vanishes we select a special point associated with this cutthat by definition is called center of mass. Evaluating the an-gular momentum on this special RNC cut yields the intrinsicangular momentum or spin.Note that the notion of a null cone cut as the intersectionof null cones from points of the spacetime with null infinityis purely geometric. Note also that, as pointed out by Gerochand Winicour, the linkages also offer a coordinate free defi-nition. Thus in principle our construction solely depends ona family of NU cuts and it is independent on the coordinatesused for its description.As it was also done in the ANK formulation, this approachyields explicit equations of motion for the center of mass andspin when gravitational radiation is emitted from the source. The equations of motion of both formulations can be com-pared and, as one would expect, they are different. Giventhat there is available in the literature models of binary coa-lescence based on the post Newtonian approximation it is alsoof great interest to compare our equations with these models.It is surprising to find out that the time evolution of the totalmass, linear and angular momentum in our approach agreeswith the PN formulation up to octupole terms in the gravita-tional radiation.It is left for future work to analyse other definitions ofdipole mass moment/angular momentum, that yield the Ko-mar formula for axial symmetry. In this sense one shouldmention that the Gallo-Moreschi definition [12], following acompletely different approach, gives exactly the same formulaas the Linkages on Bondi sections. (The old definition hadsome freedom and the original way to fix it yielded a differentresult [13].)Since the Moreschi approach also defines a preferred fam-ily of Bondi cuts (called nice sections) it is worth making afew remarks about them. The nice sections are found by de-manding that the l ≥ part of the supermomentum at nullinfinity vanishes when restricted to those cuts. The nice sec-tion equation is obtained and the center of mass frame is aspecial solution of the equation. The nice section equation isdifferent from either the null cone cut equation at a local levelor the regularized null cone cut equation at a global level onthe sphere. Whereas the cut equations have (at least) a lineardependence on the Bondi shear, the nice section equation hasa quadratic dependence. In addition, since the RNC cut equa-tion yield monoparametric families of NU cuts whose areasare time dependent and in general are not unit spheres, the nicesections are by construction Bondi surfaces and thus have unitarea. Furthermore, the solutions to the nice sections are spe-cially adapted to get rid of unwanted supermomentum termsand thus define unambiguously the notion of center of massand intrinsic angular momentum at each Bondi time. On theother hand, our formulation is based on a special monopara-metric family of cuts at null infinity together with a coordinatefree approach to find the notion of center of mass and intrinsicangular momentum. In the end however, one uses a Bondi co-ordinate system to obtain explicit description of the approach.Therefore, it is worthwhile to examine in more detail both ap-proaches and find similarities and differences in a future work.The technical material needed for this work is presented inSections 2-4. Section 5 is the main part of this work. We givedefinitions of center of mass and spin, derive the equations ofmotion and compare our results with other approaches. Thework ends with some concluding remarks. II. FOUNDATIONS
In this section, we introduce several of the key ideas andthe basic tools that are needed for our later discussion.
A. Asymptotically flatness and I + We first introduce some mathematical framework. In par-ticular we introduce the notion of an isolated source of grav-itational radiation realized by defining the so called asymp-totically flat spacetimes. Bondi, Sachs and collaborators inthe sixties [14, 15], used a canonical coordinate system weremass, momentum and gravitational radiation could be defined.Later, Penrose gave a geometrical definition using a rescaledmetric together with a null boundary [16]. Both approachescan be found in the review of Newman and Tod [17]. We fol-low Newman and Tod in the following definitions.A spacetime ( M , g ab ) is called asymptotically flat if thecurvature tensor vanishes as infinity is approached alongthe future-directed null geodesics of the spacetime. Thesegeodesics end up at what is referred to as future null infin-ity I + , the future null boundary of the spacetime . Theseideas can be formalized by giving the following, Definition: a future null asymptote is a manifold ˆ M withboundary I + ≡ ∂ ˆ M together with a smooth lorentzian met-ric ˆ g ab , and a smooth function Ω on ˆ M satisfying the follow-ing • ˆ M = M ∪ I + • On M , ˆ g ab = Ω g ab with Ω > • At I + , Ω = 0 , n ∗ a ≡ ∂ a Ω = 0 and ˆ g ab n ∗ a n ∗ b = 0 We assume I + to have topology S × R . A Newman-Unti(N-U) coordinate system [18] is introduced in the neighbor-hood of I + ,as follows. We first give a regular one-parameterfamily of closed 2-dim cuts at null infinity, labelled by theparameter u which meet every generator once. The stere-ographic coordinates ( ζ, ¯ ζ ) label each generator on the cut.We then construct a family of null surfaces whose intersectionwith I are these NU cuts,and use the affine parameter r oneach null surface as our last coordinate.Since I + is a null hypersurface in the rescaled manifold ˆ M the restriction of the rescaled metric on this null boundarytakes the form d ˆ s = 4 dζd ¯ ζP . (1)with P ( u, ζ, ¯ ζ ) a strictly positive function. With the choice of Ω = r − as the conformal factor, the physical metric is thengiven as ds = 4 r dζd ¯ ζP . (2) B. Null Tetrads and Operators on the sphere
Associated with the NU coordinates ( u, r, ζ, ¯ ζ ) , there is anull tetrad system denoted by ( l ∗ a , n ∗ a , m ∗ a , ¯ m ∗ a ). The first nulltetrad covector l ∗ a is defined as [17] l ∗ a = ∇ a u, (3) Thus, l a ∗ is a null vector tangent to the null surface u = const. . The remaining null vectors are then prescribed at I + and then parallel propagated inwards along l a ∗ . The secondtetrad vector n ∗ a is tangent to the null generators of I + andnormalized to l ∗ a n a ∗ l ∗ a = 1 . (4)The null tetrad at I + is finally completed by selecting twocomplex null vectors at the intersection of u = const. and Ω = 0 . The complex vector m a ∗ orthogonal to l a ∗ and n a ∗ isnormalized to m ∗ a ¯ m ∗ a = − . (5)The null tetrad for the spacetime is then constructed fromparallel propagation along l a ∗ . The spacetime metric is givenby g ab = l ∗ a n ∗ b + n ∗ a l ∗ b − m ∗ a ¯ m ∗ b − ¯ m ∗ a m ∗ b . (6)(In this work the letters a, b, c, d take values , , , .) Formore details on the asymptotic form of the metric in NU co-ordinates see ref. [17].Since there is a gauge freedom in the choice of conformalfactor Ω one can freely choose the function P ( u, ζ, ¯ ζ ) . Theparticular choice P = P = (1 + ζ ¯ ζ ) , yields a two-surfacesmetric (2) of unit radius that is Lie derived along the null di-rections of I + . For this particular choice of conformal factora Bondi time u B is introduced as the affine length of the nullgeodesic n a ≡ ˆ g ab ∇ b Ω . The covector l a = ∇ a u B yields aBondi tetrad ( l a , n a , m a , ¯ m a ) following the same procedureas above.Since u B = const. are unit spheres whereas u = const. are not, the description of one cut in terms of the other may bewritten as u B = Z ( u, ζ, ¯ ζ ) , (7) u = T ( u B , ζ, ¯ ζ ) . (8)where Z is a smooth function and T is the inverse of Z . Theysatisfy ˙ T Z ′ = 1 , where "dot" and "prime" denote the deriva-tive with respect to u B and u respectively.We also introduce the concept of spin weight. A quantity η that transforms as η → e isλ η under a rotation m a ∗ → e iλ m a ∗ is said to have a spin weight s . For any function f ( u, ζ, ¯ ζ ) , wedefine the differential operators ð ∗ and ¯ ð ∗ [10] by ð ∗ f = P − s ∂ ( P s f ) ∂ζ , (9) ¯ ð ∗ f = P s ∂ ( P − s f ) ∂ ¯ ζ , (10)where f has a spin weight s and P is the conformal factordefining the metric (2). Likewise, we define ð f = P − s ∂ ( P s f ) ∂ζ , (11) ¯ ð f = P s ∂ ( P − s f ) ∂ ¯ ζ , (12)with P = (1 + ζ ¯ ζ ) . Furthermore, using P = P Z ′ (whichfollows from r B P = rP and r B = Z ′ r [18]) one can relatethese two operators as ð ∗ f = Z ′ ð f + sf ð Z ′ (13) ¯ ð ∗ f = Z ′ ¯ ð f − sf ¯ ð Z ′ . (14)The above equation (which is not a coordinate transformationbetween the NU and Bondi coordinate systems) will be usedbelow to expand regular functions on the sphere in the stan-dard spherical harmonic basis.Now, we are interested in the relationship between the NUand Bondi null tetrads. We start by rewriting eq. (3) in theform l a = ∇ a Z ( u, ζ, ¯ ζ ) and using the orthogonality of thenull vectors to get l ∗ a = 1 Z ′ [ l a − Lr B ¯ m a − ¯ Lr B m a + L ¯ Lr B n a ] , (15) n ∗ a = Z ′ n a , (16) m ∗ a = m a − Lr B n a , (17) ¯ m ∗ a = ¯ m a − ¯ Lr B n a , (18)where L ( u B , ζ, ¯ ζ ) = ð Z ( u, ζ, ¯ ζ ) . C. The Spin Coefficient Formalism
In this subsection we will describe the NP formalism interm of the Bondi coordinates ( u B , r B , ζ, ¯ ζ ) , this means thatall introduced functions depend on these coordinates. First,we introduce the Ricci rotation coefficients γ µνρ [17, 19] γ µνρ = λ aρ λ bν ∇ a λ bµ , (19)the Ricci rotations coefficients satisfy γ µνρ = − γ νµρ . (20)where λ aµ = ( l a , n a , m a , ¯ m a ) , (21)where µ, ν, ρ = 1 , , , are tetrad indexes. The twelve spincoefficients are defined as combinations of the γ µνρ α = 12 ( γ − γ ); λ = − γ ; κ = γ β = 12 ( γ − γ ); µ = − γ ; ρ = γ (22) γ = 12 ( γ − γ ); ν = − γ ; σ = γ ε = 12 ( γ − γ ); π = − γ ; τ = γ The Peeling theorem of Sachs [20] tell us the asymptotic be-havior of the spin coefficients [10]. κ = π = ε = 0; ρ = ¯ ρ ; τ = ¯ α + βρ = − r − B − σ ¯ σ r − B + O ( r − B ) σ = σ r − B + [( σ ) ¯ σ − ψ / r − B + O ( r − B ) α = α r − B + O ( r − B ) β = β r − B + O ( r − B ) (23) γ = γ − ψ (2 r B ) − + O ( r − B ) µ = µ r − B + O ( r − B ) λ = λ r − B + O ( r − B ) ν = ν + O ( r − B ) where the relationships among the r-independent functions α = − ¯ β = − ζ , γ = ν = 0 ,ω = − ¯ ð σ , λ = ˙¯ σ , µ = − , with σ the value of the Bondi shear at null infinity. Thiscomplex scalar is called the Bondi free data (or Bondi news)since ¨ σ yields the gravitational radiation reaching null infin-ity. Since the BOndi shear is a s.w. 2 object it can be writtenas σ = ð ( σ R + iσ I ) . The real functions σ R , σ I are respectively called the elec-tric and magnetic part of the Bondi shear. They are related tothe mass and magnetic nth-poles moments of the gravitationalsource.As the spacetime is assumed to be empty in a neighborhoodof I + the gravitational field is given by the Weyl tensor.Using the available tetrad one defines five complex scalars,whose asymptotic behavior is [20] ψ = C abcd m a l b l c m d ≃ ψ r B , ψ = C abcd l a n b n c ¯ m d ≃ ψ r B .ψ = C abcd n a l b l c m d ≃ ψ r B , ψ = C abcd ¯ m a n b n c ¯ m d ≃ ψ r B .ψ = 12 ( C abcd l a n b m c ¯ m d − C abcd l a n b l c n d ) ≃ ψ r B . Using the peeling theorem the radial part of the Einsteinequations can be integrated leaving only the Bianchi identitiesat I as the unsolved equations. In a Bondi frame the resultingequations look remarkably simple. Some of those equationsrelate the Weyl scalars with the Bondi shear, i.e., [10, 17] ψ + ð ¯ σ + σ ˙¯ σ = ¯ ψ + ¯ ð σ + ¯ σ ˙ σ , (24) ψ = ð ˙¯ σ , (25) ψ = − ¨¯ σ , (26)Here the ð operator is taken at u B = const .In the same way we can define the Weyl scalars in N-U us-ing the fact that the Weyl tensor C abcd is conformally invariant[17]. ψ ∗ = C abcd n a ∗ l b ∗ l c ∗ m ∗ d ≃ ψ ∗ r − ,σ ∗ = m ∗ a m ∗ b ∇ a l ∗ b ≃ σ ∗ r − . From the equations (15-18) we can find transformations fromNU to Bondi for any scalar or spin coefficient [21, 22]. Inparticular we are interested in ψ ∗ Z ′ = [ ψ − Lψ + 3 L ψ − L ψ ] , (27)where ψ ∗ is constructed from the N-U tetrad. Similarly wefind the relation between σ ∗ and σ [21] σ ∗ Z ′ = σ − ð Z. (28)where σ ∗ is the NU shear [18]. D. Evolution equations
Finally, the Bianchi identities (in Bondi coordinates) aregiven by [10, 17] ˙ ψ = − ð ψ + 3 σ ψ , (29) ˙ ψ = − ð ψ + 2 σ ψ , (30) ˙ ψ = − ð ψ + σ ψ . (31)Note that eq. (24) defines a real variable Ψ called the massaspect [14]. Ψ = ψ + ð ¯ σ + σ ˙¯ σ , (32)In term of Ψ is possible to write the Bondi Mass M and Bondilineal momentum P i by M = − c π √ G Z Ψ dS, (33) P i = − c π √ G Z Ψ˜ l i dS, (34)with ˜ l i = 11 + ζ ¯ ζ ( ζ + ¯ ζ, − i ( ζ − ¯ ζ ) , − ζ ¯ ζ ) . (35)with dS = dζ ∧ d ¯ ζP the area element on the unit sphere andwhere i, j, k, l, m = 1 , , are three dimensional Euclidianindices. It is important to note that at I we move upstairsand downstairs indices with the flat metric.It is also quite convenient to give the evolution equation for Ψ . Directly from eq. (31) one obtains ˙Ψ = ˙ σ ˙¯ σ . (36)This equation will be used later. III. REGULARIZED NULL CONE CUTS
Another important construction in this work is a special NUfoliation obtained from the null cone cuts of null infinity orNC cuts for short.Given a point x a on the spacetime and denoting by N x thefuture null cone from x a , we define a null cone cut (NC cut)as N x ∩ I + . The local and global properties of the NC cutshave been extensively analysed [23–25] and some of them aresummarized in the Appendix. In this section we briefly reviewsome results that are needed for this work.In flat spacetime the NC cuts are smooth surfaces that canbe written as a regular functions on the sphere, i.e., Z = x a ℓ a , x a = ( t, x i ) , ℓ a = ( Y , − Y i ) , (37)with x a the apex of the null cone and Y , Y i the ℓ = 0 , spherical harmonics. If the apex describes a timelike world-line x a ( τ ) in Minkowski space the NC cuts describe a oneparameter foliation of Null Infinity.The idea is to generalize this concept for asymptotically flatspacetimes. This is a highly non trivial task since curvatureinduces caustics on the future null cones of points. Thus, theNC cuts have self-intersections and caustics. Nevertheless onecan show that it is always possible to find a neighborhood atnull infinity where a NC cut is a smooth 2-surface. In a Bondicoordinate system, this surface is a graph of a function u B = Z ( x a , ζ, ¯ ζ ) . (38)For the type of systems we are interested in describing, i.e.,gravitational radiation coming from compact sources in theobservation volume of aLIGO, one can always assume the nullcone cut can be described by the above function. Moreover,to recover the point x a from which the radiation is comingone does not need the whole 2-surface, rather a small neigh-borhood of points ( ζ, ¯ ζ ) in the sphere. This follows fromthe dual meaning of Z as the past null cone from ( u B , ζ, ¯ ζ ) .Thus, ( ð Z, ¯ ð Z ) gives you the incoming direction of the nullgeodesic of that past null cone whereas ¯ ðð Z identifies a pointon that null geodesic. Therefore, with a very small array ofobservers one can identify points in the spacetime such thattheir null cone cuts are described by equation (38).One can also show that Z ,a is a null covector, namely, itsatisfies g ab Z ,a Z ,b = 0 . (39)The above equation can also be used to reconstruct the con-formal metric from knowledge of Z . The explicit construc-tion is given on a preferred coordinate system ( u, ω, ¯ ω, R ) =( Z, ð Z, ¯ ð Z, ¯ ðð Z ) , and the metric coefficients are given interms of a function Λ( Z, ð Z, ¯ ð Z, ¯ ðð Z, ζ, ¯ ζ ) , related to Z bythe equation ð Z = Λ . This function plays a central role in the metric reconstructiontechnique. If Λ is given, to obtain a Lorentzian metric from(39), Λ must satisfy a set of PDEs called metricity conditions.This is the core of the Null Surface Formulation of Generalrelativity [26], or NSF for short, and it gives a generalizationof Cartan’s work on third order ODEs and a Lorentzian metricon the solution space [27, 28]. Note that if Λ = 0 we obtain aflat metric and the solution of ð Z = 0 is given by (37). Λ also has a very simple geometric meaning. Using Sachs’theorem one can show that ð Z = σ − σ x , (40)with σ the asymptotic Bondi shear at null infinity and σ x theasymptotic shear of the future null cone from x a evaluated atnull infinity [29]. In general σ x will always be non-vanishingfor a non flat spacetime since the Weyl tensor induces shearon the future null cone from any point x a . It follows fromthe above equation that a vanishing asymptotic shear does notcorrespond to a NC cut. (As a remark we point out that in theANK approach one uses a congruence of null geodesics suchthat the associated asymptotic shear vanishes at null infinity.One thus sets σ x = 0 in eq. (40) to obtain the so called "goodcut equation".)As we are interested in describing a particular worldlinewhose motion will depend on σ ( u B , ζ, ¯ ζ ) we assume ˙ σ isknown. Moreover, the outgoing gravitational radiation we areinterested in is emitted by closed binaries, supernovae or scat-tering of compact sources. For those systems one can alwaysassume they are asymptotically stationary, i.e., ˙ σ vanishesas u B → −∞ . In that limit σ I → , σ is purely electricand by a supertranslation one can get rid of the electric partat that initial time. We thus assume that we work in a definiteBondi system such that σ vanishes as u B → −∞ . This re-stricts the Bondi supertranslation freedom to the translationsof the Poincare group. Following the above results the de-scription of the cuts in any other Bondi system will be givenby ˜ Z ( x a , ζ, ¯ ζ ) = Z ( x a , ζ, ¯ ζ ) + α ( ζ, ¯ ζ ) and the ℓ = 0 , partsof Z and ˜ Z do not depend on the higher harmonics of α ( ζ, ¯ ζ ) .Finally, we want to obtain dynamical equations for Z to ex-hibit the explicit dependence of σ x on σ . It is clear that onecannot hope to obtain Z or Λ in closed form for an arbitraryasymptotically flat spacetime. On the other hand it is not dif-ficult to set up a perturbation procedure off Minkowski spaceand obtain a first order deviation from a flat cut.Writing Z = Z + Z , with Z given by (37) and Λ = ð Z , one can show that Λ satisfies the wave equation inMinkowski space and that it functionally depends on theBondi shear via (see Appendix A) ¯ ð ð Z = ¯ ð σ ( Z , ζ, ¯ ζ ) + ð ¯ σ ( Z , ζ, ¯ ζ ) . (41)The second term in the r.h.s. of eq. (41) gives the relation-ship between σ x and σ . Since the Bondi shear is a smooth s.w. 2 function on I + the above equation admit regular so-lutions on the sphere. Thus, the first order deviation from aflat cut are smooth 2-surfaces (they can be expanded in spher-ical harmonics) at null infinity and are called the (linearized)regularized null cone cuts. If x a ( u ) describes a worldline inMinkowski space, the function Z ( x a ( u ) , ζ, ¯ ζ ) describes a oneparameter family of cuts. To show that this family is NU weperform a Taylor expansion Z ( x a ( u + δu ) , ζ, ¯ ζ ) = Z ( x a ( u ) , ζ, ¯ ζ ) + v a ∂ a Zδu, where v a ≡ ∂ u x a and δu > . If we assume v a is futurepointing with respect to the flat metric, it then follows that v a ∂ a Z > , since Z a is null and future pointing (for the flat metric) forsmall values of σ B . We conclude that this monoparametricfamily never intersects itself and it is a well behaved NU foli-ation.Solving for (41) yields Z = R − R i Y i + σ ijR
12 + √
272 ˙ σ igI R f ǫ gfj ! Y ij , (42)with Y , Y i and Y ij the tensorial spin-s harmonic expansion[30]. Note that Z depends on the real and imaginary part ofthe Bondi shear [22, 29]. This is what one would expect in aperturbation expansion since the imaginary part of the Bondishear is related to the current quadrupole moment whereas thereal part comes from the mass quadrupole moment[31].The first order solution (42) will be used to define center ofmass and spin for isolated sources of gravitational radiation.It will also be used to compare our results with those derivedin the ANK and postNewtonian formulations.Finally, it is a fair question to ask what happens to the aboveconstruction if one goes beyond the linearized approximation.If we assume the spacetime is Ricci flat in a neighborhoodof I + one obtains the field equation for Z [26] (see appendixA). The field equation exhibits the non Huygens nature ofthe NC cuts showing explicitly which term is responsible forcaustics. Thus, a generic NC cut is not a smooth 2-surfaceat null infinity. However, if ˙ σ is small both in the past or inthe future of some small interval of time, one expects that theleading contribution to the solution comes from the Huygenspart of field equation, ¯ ð ð Z = ¯ ð σ ( Z, ζ, ¯ ζ ) + ð ¯ σ ( Z, ζ, ¯ ζ ) , (43)referred to as the Regularized Null Cone cut equation or RNCcut equation for short. Since (43) only contains l > terms ina spherical harmonic decomposition, the kernel of (43) is a 4dim space x a , i.e. a flat cut Z = x a ℓ a .Equation (43) or its linearized version (41) should be com-pared with the good cut equation ð Z C = σ ( Z C , ζ, ¯ ζ ) , (44)Note that the good cut equation yields complex cuts withvanishing shear whereas the NRC cut equation yields NU cutswhose shear depends linearly on the Bondi shear.Thus, from the point of view of available structures at nullinfinity we could start with the RNC cut equation (43). On its4-dim solution space one constructs a Lorentzian metric fol-lowing the NSF procedure[26]. A perturbative solution givesa Minkowski space together with flat cuts (37) at its lowestorder and the linearized RNC cuts (42) at first order. IV. LINKAGES AND THE ANGULARMOMENTUM-CENTER OF MASS TENSOR
For axially symetric spacetimes the Komar integral con-structed from the axial Killing field yields a natural definitionof angular momentum that it is a conserved quantity in vac-uum and has a flux law in Einstein-Maxwell spacetimes [22].This idea can be generalized to asymptotically flat spacetimesby first introducing the notion of asymptotic Killing vectorsand then giving a generalization of the Komar integral, theWinicour-Tamburino linkage [32], which yields the Komarformula when the spacetime has a Killing symmetry. We willuse in future sections these concepts to define the spin, totalangular momentum and center of mass of an asymptoticallyflat spacetime.
A. The Asymptotic Symmetry Group
First we introduce the generators ξ a of asymptotic symme-tries on a neighborhood of I + as smooth solutions of theasymptotic Killing equation [11] ξ a ; b + ξ b ; a = O ( r − n ) (45) ( ξ a ; b + ξ b ; a ) l b ∗ = 0 . (46)Here l b ∗ is a null vector tangent to the generators of eachoutgoing null hypersurface in M and n differs with the choiceof components [18]. The second equation represents theKilling propagation law along the null hypersurface [33]. At I + the collection of all solutions form the BMS algebra L [20]. If ξ a ∝ n a they define the supertranslation subalgebra T and the quotient L / T is isomorphic to the Lorentz group[11]. This subalgebra is realizad by an equivalence class [ ξ a ] where ξ a ∼ ξ ′ a if ξ a − ξ ′ a ∝ n a ∗ . Equations (45) and (46)can be solved by direct integration using the spin-coefficient[34]. The results may be written as ξ a = Al a ∗ + Bn a ∗ + C ¯ m a ∗ + ¯ Cm a ∗ (47)where A = A r + A + A − r − + O ( r − ) B = B C = C r + C + C − r − + O ( r − ) and A = − (1 /Z ′ )( B Z ′ ) ′ ,A = ð ∗ ¯ ð ∗ B + B ð ∗ ¯ ð ∗ ln P,A − = 12 [ B ( ψ ∗ + ¯ ψ ∗ ) + ¯ C ψ ∗ + C ¯ ψ ∗ ] ,C = a ( ζ, ¯ ζ ) /Z ′ , with ð a = 0 ,C = ð ∗ B + ¯ C σ ∗ ,C − = 0 ,B = b ( ζ, ¯ ζ ) /Z ′ − (1 / Z ′ ) Z u Z ′ [ ð (¯ aZ ′− ) + ¯ ð ( aZ ′− )] du. Note that the only freedom is in b ( ζ, ¯ ζ ) , the supertranslationfreedom, and solutions to ð a = 0 , which correspond to thehomogeneous Lorentz transformation. B. Linkages in Asymptotically Flat Spacetimes
Given a u = const. null foliation, which can be either NUor Bondi, introducing an affine parameter r and constructingthe r = const. l ∗ [ a ˆ n ∗ b ] dS , thelinkage integral is defined as [35] L ξ ( I + ) = − π lim r →∞ Z (cid:16) ∇ [ a ξ b ] + ∇ c ξ c l ∗ [ a ˆ n ∗ b ] (cid:17) l ∗ a ˆ n ∗ b dS, (48)Note that ˆ n ∗ b is not one of the associated null vectors of theNU tetrad. Whereas n ∗ b is parallel propagated along l ∗ a , ˆ n ∗ b is orthogonal to the u = const. , r = const. surface. It can berewritten in terms of the NU tetrad via a null rotation around l ∗ b as [34] ˆ n ∗ b = n ∗ b − ¯ ω ∗ m ∗ b − ω ∗ ¯ m ∗ b + ω ¯ ωl ∗ b (49)with ω ∗ = − (¯ ð ∗ σ ∗ ) r − + O ( r − ) (50)This scalar linear functional of ξ b transforms as an adjointrepresentation of the BMS group. If ξ b is a translation eq.(48) yields the Bondi energy momentum vector. Likewise, if ξ b belongs to the Lorentz subgroup, eq. (48) can be used todefine the notion of the mass dipole and angular momentum.Solving the asymptotic Killing equation by making use of theradial dependence of the spin coefficients and tetrad compo-nents, one can show that this linkage integral can be writtenas [34], L = 18 π √ Re Z [ b (cid:18) ψ ∗ + σ ∗ λ ∗ − ð ∗ ¯ σ ∗ Z ′ (cid:19) +¯ a (cid:18) ψ ∗ − σ ∗ ð ∗ ¯ σ ∗ − ð ∗ ( σ ∗ ¯ σ ∗ ) Z ′ (cid:19) ] dS putting b = 0 we obtain L DJ = Re π √ Z ¯ a (cid:20) ψ ∗ − σ ∗ ð ∗ ¯ σ ∗ − ð ∗ ( σ ∗ ¯ σ ∗ ) Z ′ (cid:21) dS (51)where ¯ a = ¯ a i Y − i with ¯ a i three complex constants. The threecomplex, i.e six real, values of (51) are by definition, the com-ponents of the mass dipole - angular momentum tensor (formore details the reader can see ref. [34]). To obtain thosecomponents it is quite convenient to define a complex vector D ∗ i + i c J ∗ i where i symbolize the vectors (1,0,0), (0,1,0) and(0,0,1) as D ∗ i + i c J ∗ i = Z Y − i (cid:20) ψ ∗ − σ ∗ ð ∗ ¯ σ ∗ − ð ∗ ( σ ∗ ¯ σ ∗ )8 π √ Z ′ (cid:21) dS (52)It is worth mentioning that at a linearized level and for sta-tionary spacetimes, the real and imaginary parts of ψ cap-ture the notion of the two form that defines the center of massand angular momentum and transform appropriately under theLorentz transformation. The linkage is a natural generaliza-tion for asymptotically flat spacetimes.It is also worth mentioning that the value of the linkagedepends on the choice of section introduced for its definition[11]. This is analogous to the freedom in special relativitywith the choice of origin for the definition for center of massor angular momentum. The main difference is that whereas inrelativistic mechanics the freedom is a point on the spacetime,in the definition of a linkage the freedom is a whole section, aninfinite set of constants, one for each coefficient in a sphericalharmonic decomposition. Consequently, if one now has a NUfoliation, where each coefficient now depends on the Bonditime, the freedom becomes an infinite set of functions of timea priori without physical meaning.However, in what follows below, we will restrict this infi-nite freedom to four functions that describe a worldline in thesolution space of the RNC cuts. From its geometrical mean-ing there is a one to one correspondence between worldlinesin the solution space and a RNC foliation at null infinity. Fur-thermore, by defining the notion of mass dipole moment andrequiring that for one worldline of the RNC foliation the massdipole moment vanishes, one gets the right number of equa-tions from which a special worldline is found. This specialworldline will be called the center of mass worldline. Fi-nally, restricting the angular momentum to this special RNCcut yields the notion of spin or intrinsic angular momentum. V. MAIN RESULTSA. Definitions of Center of Mass and Angular Momentum
Directly from (52) we define the mass dipole momentand angular momentum associated with a RNC foliation as D ∗ i + i c J ∗ i = − c G − √ (cid:20) ψ ∗ − σ ∗ ð ∗ ¯ σ ∗ − ð ∗ ( σ ∗ ¯ σ ∗ ) Z ′ (cid:21) i . (53) The six functions of the NU time u defined above function-ally depend on the particular worldline x a ( u ) that character-izes each RNC cut.We then impose a condition on a special RNC foliation, i.e.,on a special worldline (42), at each u = const. cut, the massdipole moment D ∗ i vanishes. This condition is given by Re (cid:20) ψ ∗ − σ ∗ ð ∗ ¯ σ ∗ − ð ∗ ( σ ∗ ¯ σ ∗ ) Z ′ (cid:21) i = 0 . (54)By adequately choosing x a ( u ) one has enough freedom tosatisfy the above equation for each value of u . Since the4-velocity of the worldline is normalized to one (using thespacetime metric), we use this norm to fix the timelike com-ponent of the worldline coordinate. Thus, the freedom leftare the spatial components of the worldline x a ( u ) and theabove equation gives three algebraic equations from whichthese components are obtained. This special worldline willbe called the center of mass worldline. The angular momen-tum J i ∗ evaluated at the center of mass will be called intrinsicangular momentum S i , i.e., S i = − c √ G Im (cid:20) ψ ∗ − σ ∗ ð ∗ ¯ σ ∗ − ð ∗ ( σ ∗ ¯ σ ∗ ) Z ′ (cid:21) i . (55)The above equations have been obtained from two surfaceintegrals on a particular RNC cut foliation, namely, the centerof mass foliation. Thus, they have a well defined geometricalmeaning. We now solve eq. (54) explicitly on a Bondi framesince variables like gravitational radiation, mass loss, linearmomentum, are easier to define in Bondi coordinates. To writedown the mass dipole moment and and angular momentum(53) in Bondi coordinates it is convenient to define analogousquantities in a Bondi tetrad, i.e., D i + i c − J i = − c √ G (cid:2) ψ − σ ð ¯ σ − ð ( σ ¯ σ ) (cid:3) i . (56)Using the relations between the NU and the Bondinull vectors given by (15-28) to transform the quantities ( ψ ∗ , σ ∗ , ð ∗ ) → ( ψ , σ , ð ) , one can write (53) as D ∗ i ( u ) = D i ( u B ) + 3 c √ G Re [ ð Z (Ψ − ð ¯ σ ) + F ] i (57) J i ∗ ( u ) = J i ( u B ) + 3 c √ G Im [ ð Z (Ψ − ð ¯ σ ) + F ] i (58)with F = −
12 ( σ ð ¯ ð Z + ð Z ð ¯ σ − ð Z ð ¯ ð Z ) −
16 (¯ σ ð Z + ¯ ð Z ð σ − ¯ ð Z ð Z ) . (59)If we insert the center of mass RNC cut Z in (57), then itsl.h.s. vanishes on a u = const. surface and we obtain analgebraic equation to be solved for R i ( u ) . Equation (58) thengives a relationship between S i and J i , the intrinsic and totalangular momentum respectively. B. Approximations and assumptions
Although the main equations have been presented above, toobtain the explicit form of the worldline in this work we willmake the following assumptions. • σ = 0 for some initial Bondi time, usually taken to be −∞ . • R i is a small deviation form the coordinates origin. • R = u assuming the slow motion approximation. • The Bondi shear only has a quadrupole term.The first assumption fixes the supertranslation freedom andis consistent with our choice of null cone cut, namely, thefreedom in the solution of eq. (41) is only a translation be-tween two Bondi frames. The second assumption is a work-ing simplification. Since we are particularly interested in theacceleration of the center of mass, which is quadratic in thegravitational radiation, we want to ignore terms like R σ . Fi-nally, the third assumption is a physical one. Since in mostastrophysical processes less than of the total mass is lostas gravitational radiation the gamma factor for the center ofmass velocity is about . . Putting it in other words, even iftwo coalescing stars are approaching each other at relativisticspeeds, if the center of mass is initially at rest it will neveracquire a relativistic velocity.In principle all of these assumptions can be relaxed butsince we want to make direct comparisons with other formula-tions, like the ANK approach or the PN equations of motion,they are needed for these purposes.The ANK approach usesthe same Bondi gauge as ours whereas the PN formulationselects an initial time where the system is stationary and themetric is flat.It is possible to extract several important formulae relatingthe dynamical evolution of mass, momentum, etc. by expand-ing the Bianchi identities in a spherical harmonics decompo-sition. Using the tensorial spin-s spherical harmonics [30]; Y , Y i , Y ij , etc., one can expand the relevant scalars at nullinfinity as σ = σ ij ( u B ) Y ij ( ζ, ¯ ζ ) ,ψ = ψ i ( u B ) Y i ( ζ, ¯ ζ ) + ψ ij ( u B ) Y ij ( ζ, ¯ ζ ) , (60) Ψ = − √ Gc M − Gc P i Y i ( ζ, ¯ ζ ) + Ψ ij ( u B ) Y ij ( ζ, ¯ ζ ) , Note that the complex tensor σ ij represents the quadrupolemomentum of the gravitational wave.Now, from eq. (41) if we write x a ( u ) as ( R ( u ) , R i ( u )) ,assuming the Bondi shear only has a quadrupole term, andusing the tensorial spin-s harmonic expansion, this solution isgiven as Z ( u, ζ, ¯ ζ ) = R ( u ) − R i ( u ) Y i + 112 σ ijR ( u ) Y ij . (61)the freedom left in (61) is an arbitrary worldline in a fiducialspacetime. Choosing u as the proper time, we can easily solve for R ( u ) in terms of the spatial components of the 4-velocity.Furthermore, in the slow motion approximation R ( u ) = u + O ( v ) . C. The center of mass and spin
The center of mass worldline R i ( u ) is obtained from (57)by demanding that the l.h.s. vanishes on the u = const. cutwhen u B = Z ( u, ζ, ¯ ζ ) is inserted in the r.h.s. of the equation.Furthermore, since by assumption R i ( u ) and σ ijR ( u ) are small,we write Z = u + δu = u − R i ( u ) Y i + 112 σ ijR ( u ) Y ij , (62)and make a Taylor expansion of the Bondi tetrad variables upto first order in δu . We write (57) as D i ( u + δu ) + 3 c √ G Re [(Ψ − ð ¯ σ ) ð δu + F ] i = D i ( u ) + [ ˙ D ( u ) δu ] i + 3 c √ G Re [(Ψ − ð ¯ σ ) ð δu + F ] i = D i ( u ) + c √ G Re [( ð Ψ − ð ¯ σ ) δu ] i + 3 c √ G Re [(Ψ − ð ¯ σ ) ð δu + F ] i , (63)where we have used eq. (30) to rewrite ˙ D i . Note that in thiscase the second line in the definition of F (59) vanishes since σ only has quadrupole terms. In the remaining terms of (57)we simply replace u B by u as the extra terms are cubic orhigher in the expansion variables. Solving for R i from theabove equation yields M R i = D i + 85 √ c σ ijR P j , (64)where σ ijR and σ ijI are respectively the real and the imaginarypart of σ ij . Note that inserting eq. (64) in (62) yields Z CM = u − M ( D i + 85 √ c σ ijR P j ) Y i + 112 σ ijR Y ij , (65)the special NU foliation that represents the center of massworldline.As it was mentioned previously, replacing eq. (64) in theimaginary part of (53) yields the spin of the system. To dothat we start with the relationship (58) J ∗ i ( u ) = J i ( u + δu ) + 3 c √ G Im [(Ψ − ð ¯ σ ) ð δu + F ] i , perform a Taylor expansion, J ∗ i ( u ) = J i ( u )+[ ˙ J ( u ) δu ] i + 3 c √ G Im [(Ψ − ð ¯ σ ) ð δu + F ] i , J ∗ i = J i ( u ) + c √ G Im [( ð Ψ − ð ¯ σ ) δu ] i (66) + 3 c √ G Im [(Ψ − ð ¯ σ ) ð δu + F ] i . Finally, using eq. (65) gives S i = J i − R j P k ǫ ijk . (67)Note that this equation is exactly the same formula as in New-tonian theory although no post-Newtonian approximation hasbeen assumed. D. Dynamical Evolution
The time evolution of D i and J i follows from the Bianchiidentity for ψ , where we must insert the proper factor of √ to account for the retarded Bondi time, i.e., the retarded time, u ret = √ u B . The use of the retarded time, u ret , is importantin order to obtain the correct numerical factors in the expres-sions for the final physical results [10]. Note that the two lastimportant eqs. (64) and (67) remain unchanged in term of u ret or u B . However, for the rest of the paper, we adopt the symbol“dot” for ∂ u ret .Then, we use the definition (56) and replace the real andimaginary l = 1 component of (30) to obtain ˙ D i = P i , (68) ˙ J i = c G ( σ klR ˙ σ jlR + σ klI ˙ σ jlI ) ǫ ijk . (69)In the same way taking the l = 0 , part of (36) yields the massloss equation and the linear momentum time rate, namely, ˙ M = − c G ( ˙ σ ijR ˙ σ ijR + ˙ σ ijI ˙ σ ijI ) , (70) ˙ P i = 2 c G ˙ σ jlR ˙ σ klI ǫ ijk . (71)Note that in this Bondi gauge σ ijR = h ij + and σ ijI = h ij × strains in the transverse traceless gauge [36]. Now, taking atime derivative of eq. (64), using eq. (68), and writing up toquadratic terms in σ ij , gives M ˙ R i = P i + 85 √ c ˙ σ ijR P j , (72)the relationship between the velocity of the center of mass ˙ R i and the Bondi momentum. It departs from the Newtonianformula by radiation terms.Finally, taking one more Bondi time derivative of (72)yields the equation of motion for the center of mass, M ¨ R i = 2 c G ˙ σ jlR ˙ σ klI ǫ ijk + 85 √ c ¨ σ ijR P j . (73)The r.h.s. of the equation only depends on the gravitacionaldata at null infinity and the initial mass of the system. Similarly, taking a time derivative of (67) together with (69)gives ˙ S i = ˙ J i = c G ( σ klR ˙ σ jlR + σ klI ˙ σ jlI ) ǫ ijk . (74)This equality is also true in Newtonian mechanics for an iso-lated system (with both terms being equal to zero). However,in GR the angular momentum of an isolated system is not con-served since it is being carried away by the gravitational radi-ation. E. Comparison with ANK equations
In this subsection we compare the (ANK) equations ofmotion with the ones obtained in our approach. Before thatwe list the main differences between the approaches,1. We give a definition of angular and mass dipole mo-menta based on TWG linkages, the ANK uses the ℓ = 1 part of ψ for these definitions.2. The ANK approach relies on asymptotically vanish-ing shears, this approach uses non vanishing shears ob-tained from the RNC cut equation.3. The solution space of the good cut equation is complexmanifold, the solution space of the RNC cut equation isreal.4. The ANK approach defines the intrinsic angular mo-mentum as the imaginary part of a complex worldline.We evaluate the angular momentum on the center ofmass to define the spin.Thus, it is interesting to see if the final equations in thesetwo formulations have some similarities. To proceed with thecomparison we identify the flat metric of our construction withthe real flat metric used in the ANK approach to write theequations of motion for the center of mass worldline.It is also important to note that the Bondi mass M andthe linear momentum P i have the same definition in both ap-proaches. First we introduce the mass dipole moment, angularmomentum and spin definitions given in the ANK formalism[10] D i ANK = − c √ G ψ i R , (75) J i ANK = − c √ G ψ i I , (76) S i ANK = cM ξ iI . (77)Now computing the component l = 1 of eq. (56) we can write D i = − c √ G ψ i R + c G σ jlR σ klI ǫ ijk + higher harmonics J i = − c √ G [ ψ − σ ð ¯ σ − ð ( σ ¯ σ )] iI . (78)1The relationship between the mass dipole moment and angu-lar momentum with the asymptotic fields at null infinity aredifferent in both formalisms. These differences are a conse-quence of the definitions used in both formulations. Whereasin our approach we integrate a two form with values on theBMS algebra, in the ANK approach one directly uses ψ i forthe definitions.The angular momenta in the ANK formulation is only de-fined for quadrupole radiation, where most of the definitionsavailable in the literature agree. However, one could forseepotential problems for J i ANK if one considers higher multipolemoments in the radiation data and/or spacetimes with symme-try. The fact that ψ i I is not conserved for axially symmetricspacetimes is a clear indication that the ANK definition mustbe changed when including higher multipole moments [22].It is worth mentioning that only for quadrupole radiation bothformulae agree. We obtain non vanishing extra terms whenoctupole data is included (see appendix B).When comparing the relationship between the center of masworldline, and spin and the geometrical quantities at null in-finity like the Bondi mass, momentum, etc., we will only con-sider quadrupole radiation data.In the ANK approach one has [10] P i = M ˙ ξ iR + 43 c √ σ ijR P j + c G ( σ jlR σ klI ˙) ǫ ijk , (79)where ξ iR is the center of mass worldline. In our formulation,from eq. (72) we get P i = M ˙ R i − √ c ˙ σ ijR P j , (80)The main difference between the above equations is the lastterm in the ANK, which is missing in our equation. Note alsoa different factor with an opposite sign in front of the secondterm. This difference can be traced back in the ANK formu-lation to the use of the relation Ψ ij = − ¯ σ ij in eq. (6.33)[10]. However, this relationship contradicts eq. (36) as onecan see by deriving the relationship with respect to time andgetting ˙Ψ ij = − ˙¯ σ ij . It is clear from eq. (36) that ˙Ψ ij must bequadratic in ˙ σ ij . Thus, some derivations in the ANK formu-lation, and in particular the above relation, are incorrect.The ANK equations for the angular momentum is given by J i ANK = S i ANK + ξ jR P k ǫ ijk + 45 √ P k σ kiI . (81)whereas we obtain J i = S i + R j P k ǫ ijk . (82)Another subtle but important difference is that our definitionof spin is via a linkage formulation whereas in the ANK for-mulation the spin is an intrinsic property of a complex world-line, by definition it is the imaginary part of a complex world-line.Finally in the ANK formalism the equation of motion forthe center of mass is given by M ¨ ξ iR = 2 √ c G ˙ σ jlR ˙ σ klI ǫ ijk − c G ( σ jlR σ klI ¨) ǫ ijk − c √ σ ijR P j , (83) while in our formalism it is given by M ¨ R i = 2 c G ˙ σ jlR ˙ σ klI ǫ ijk + 85 √ c ¨ σ ijR P j . (84)Although both formulations agree for stationary spacetimes,they differ when gravitational radiation is present. F. Comparison with PN equations
In this subsection we partially compare the evolution equa-tions obtained in our approach with those coming from thePN formalism. In principle, a full comparison between theseapproaches can be a formidable task, i.e., the PN start withdefinitions in the near zone with multipoles defined in termsof the source whereas the asymptotic formulation defines ra-diative multipole moments. The asymptotic formulation hasexact equations of motion for mass, momentum and angu-lar momentum whereas in the PN approach one builds up theloss due to gravitational radiation valid up to the level of ap-proximation considered since apriori one does not have avail-able an exact formula. Nevertheless it is very useful to try tobuild a bridge between these approaches and see whether ornot they yield equivalent equations of motion for a compactsource emitting gravitational radiation.We compare below the evolution equations for the to-tal mass, momentum and angular momentum of a compactsource of gravitational radiation. In both formalisms, a dotderivative means derivation with respect with the retardedtime, as one can see following ref.[37] page 6 and 27.In the PN equations the radiative energy loss, the linear andangular momentum loss are given by (in units of G = c = 1 )[38, 39] ˙ E PN = −
15 ˙ U ij ˙ U ij − V ij ˙ V ij − U ijk ˙ U ijk −
184 ˙ V ijk ˙ V ijk (85) ˙ P i PN = (cid:18) U kl ˙ V jl + 1126 ˙ U klm ˙ V jlm (cid:19) ǫ ijk −
263 ( ˙ U jk ˙ U ijk + 2 ˙ V jk ˙ V ijk ) (86) ˙ J i PN = − (cid:18) U kl ˙ U jl + 3245 V kl ˙ V jl (cid:19) ǫ ijk − (cid:18) U klm ˙ U jlm + 128 V klm ˙ V jlm (cid:19) ǫ ijk (87)where in the above equations the quadrupole as well as oc-tupole terms have been included.To compare both approaches, we must include in our for-malism the octupole contribution to the equations for themass, angular and linear momentum (see Appendix B). In this2way we can write our equations (in term of G = c = 1 ) as ˙ M = −
110 ( ˙ σ ijR ˙ σ ijR + ˙ σ ijI ˙ σ ijI ) −
37 ( ˙ σ ijkR ˙ σ ijkR + ˙ σ ijkI ˙ σ ijkI ) , ˙ P i = −
215 ˙ σ klR ˙ σ jlI ǫ ijk − √
27 ( ˙ σ jkR ˙ σ ijkR + ˙ σ jkI ˙ σ ijkI ) −
37 ˙ σ klmR ˙ σ jlmI ǫ ijk . (88) ˙ J i = 15 ( σ klR ˙ σ jlR + σ klI ˙ σ jlI ) ǫ ijk + 97 ( σ klmR ˙ σ jlmR + σ klmI ˙ σ jlmI ) ǫ ijk . (89)Since the r.h.s of the above equations are quadratic in theradiation terms we only need a linear relationship betweenthe radiation data and the PN multipole expansion. Using thelinearized Einstein’s equation in the TT gauge and following[37], one finds that σ ijR = −√ U ij σ ijI = 83 √ V ij σ ijkR = − U ijk σ ijkI = 16 V ijk Thus, both have identical r.h.s. to this order. This is a re-markable result since the evolution equations come from com-pletely different approaches. On the other hand one must becareful with the final equations of motion for the center ofmass, energy and spin of the system since their relationshipto kinematical variables are different in both formulations. Inseveral PN papers, the recoil velocity of the center of massis defined as ∆ P i M which is the integral of eq. (71) dividedby the total mass. However, it follows from eq. (72) that inour formulation one obtains a different result. This a straight-forward consequence that in this formalism the gravitationalradiation is part of the total linear momentum. In some sensethis is analogous to the definition of momentum in electrody-namics where the kinematical definition Σ i m i −→ v i as well asthe electromagnetic radiation enter in the definition of P i . Amore careful look into these differences will be addressed inthe future. VI. FINAL COMMENTS AND CONCLUSIONS
We summarize our results and make some final remarks. • We have defined the notion of center of mass andspin for asymptotically flat spacetimes, i.e., spacetimeswhere there is a precise notion of an isolated gravita-tional system. • The main tools used in our approach are the linkages to-gether with a canonical NU foliation constructed fromsolutions to the Regularized Null Cone cut equation.The RNC cut foliation is given in the so called Newman Penrose gauge with a vanishing shear in the asymptoticpast. Physically this corresponds to an isolated gravita-tional system which is asymptotically stationary in thepast and it is specially useful to describe the emission ofsources like of closed binary coalescence, supernovas orscattering of compact objects. • The RNC cut equation is an important ingredient in thisconstruction. Its 4-dim solution space together with thelorentzian metric constructed from the solutions of theRNC cut equation provide the background to define thecenter of mass worldline. In this work we have used aperturbative approach to the RNC cut equation to intro-duce a flat metric at the zeroth order and a first ordersolution of the RNC cut equation to obtain NU folia-tions, one for each timelike worldline (with respect tothe flat metric) on the solution space. • We have obtained the center of mass worldline by re-quiring that the mass dipole moment vanishes at thespecial NU foliation associated with this worldline. Wehave derived equations of motion for the center of massand spin linking their time evolution to the emitted grav-itational radiation. They are given by a very simpleset of equations that resemble their Newtonian coun-terparts and thus should be useful in generalizing manywell known results in astrophysics when very energeticprocesses are considered. • In astrophysics very often one assumes conservationlaws for isolated systems. However, our equations showthat for highly energetic processes were a fair percent-age of energy is emitted as gravitational radiation, thisis far from being true. We have shown here that thisradiation affects the motion of the center of mass andthe spin of the system and the solutions to the aboveequations yield their dynamical evolution. • We have compared our approach with the ANK formu-lation to check for differences and similarities. Thiscomparison suggest that our definitions of mass dipolemoment and angular momentum are better suited to al-low for higher multipole radiation or spacetimes withrotational symmetry. • We have also compared our equations with those de-rived from the PN formalism. Although we have mainlydone so for a very simple set of global variables, the re-sults are very encouraging since, to second order ap-proximation, the r.h.s. of the evolution equation forthese variables are identical in both formulations. How-ever, the relationship between total linear momentumand the velocity of the center of mass is different inboth approaches. This difference might disappear af-ter taking a careful look at other variables like radiativevs local shears, etc. A lot more work is needed to find abridge between these formulations that start at oppositeends, one at null infinity, the other from local definitionsbased on the sources.3 • It is believed that in late 2017 aLIGO will be operationalto detect radiation from coalescing neutron stars and/orblack holes. As we are all aware, numerical waveformswill never be able to fill out the parameter space neededfor a coincidence check. Therefore, several ODE mod-els like the PN approach or the EOB have been pursuedwith that goal in mind. Our approach should be usefulto the ODE models for the reasons outlined below.The PN approach has several tentative definitions ofcenter of mass with vanishing acceleration while emit-ting gravitational radiation. Since the motion of the cen-ter of mass is crucial in analyzing the motion of the co-alescing sources, evolution of the mass and current mo-menta, and finally in the plot of the waveform in timedomain, it is important to know whether or not the cen-ter of mass has an acceleration during this process. Inthis work we have shown that the center of mass hasan acceleration which is partially given by the radiationreaction of M , P i , and J i and partially given by the re-lationship between the center of mass velocity, gravita-tional radiation and the global quantities M , P i , and J i .Following our results, the equations of motion for thecoalescing sources should be revised if it can be shownthat the waveform changes when the center of mass hasacceleration. This will be addressed in the future. • Finally we want to address an important conceptual is-sue, the meaning of the observational space, i.e. thesolution space of the RNC cut equation.In this work we have used worldlines on a 4-dimMinkowski space constructed on the solution space ofthe RNC cut equation. This flat metric can be re-garded as the zeroth order approximation on a perturba-tion procedure on NSF to construct Lorentzian metrics.Note also that, both the in ANK and PN approaches, aflat background metric is used to introduce worldlinesand propagate the gravitational radiation. Thus, in thethree formulations that provide equations of motion forthe gravitational sources one uses the same Minkowskibackground to compare results.However, the RNC cut equation provides a method toconstruct a 4-dim observational space with a regularmetric constructed from fields at null infinity. In whichsense is the solution space points x a and regular met-ric associated with the RNC cut equation related to the"real spacetime" from which the gravitational radiationis obtained at null infinity?If the gravitational source is composed of ordinary mat-ter. Then the NSF equations provides in principle amethod to construct null cone cuts for "real" points ofthe spacetime. The equation has three different terms,a Huygens part made of gravitational radiation, a grav-itational tail and a source term that includes integralsalong spacetime lines and is responsible for caustics andsingularities. Therefore, if one is able to detect gravita-tional waves one can then safely assume that is not on acaustic region. Moreover, we do not have the technol-ogy to detect gravitational tails and we conclude that the dominant part of the NSF equation for the situationassumed above is the Huygens one.The RNC cut equation is the smoothed version of theNSF equation, obtained by neglecting the other contri-butions and extending de validity of the Huygens part tothe whole sphere. Thus, one could define a "norm" formetrics constructed from the NSF and RNC cut equa-tions using energy methods to see how far apart are thesolutions. One should mention that this comparison isa highly non trivial task that is worth addressing in thefuture.Even if the space time contains black holes our ap-proach can also assign a center of mass worldline. Fromthe gravitational radiation reaching null infinity oneconstructs the observation space with a regular metricand in that space one defines the center of mass world-line associated with this radiation. In this case the reg-ular metric of the solution space has no relationship tothe spacetime with black holes. Nevertheless, from thegravitational radiation reaching null infinity one com-putes the equation of motion including the back reactioneffects. If a black hole is formed after the coalescence,one can also compute its final position and velocity al-though one knows that a black hole evolution is not aworldline in the real spacetime. We find this a desir-able feature of this formalism since it gives a methodof defining particle worldlines without the infinities thatappear when one introduces delta functions in stress en-ergy tensors. It has been pointed out that this secondmethod yields ill defined quantities [40]. Acknowledgements:
We are grateful to two anonymousreferees for their questions/comments that help to improve thequality of this paper. This research has been supported bygrants from CONICET and the Agencia Nacional de Cienciay Tecnología.
Appendix A: Null Cone Cuts
We present here some properties of the NC cuts comingfrom a worldline on the spacetime. • Globally the NC cuts are projections from smooth 2-dim Legendre submanifolds of the projective cotangentbundle of I + [23–25]. It follows from this property thata generic NC cut has a finite number of singularitiesand those singularities can be classified as either cuspsor swallowtails. Thus, locally the NC cuts are smooth2-surfaces at null infinity. • For the gravitational systems we would like to de-scribe, compact sources such in the observational vol-ume space of aLIGO, it is always possible to give a lo-cal description of the cuts in a given Bondi coordinatesystem as u B = Z ( x a , ζ, ¯ ζ ) . (A1)4 • The above equation has also a second meaning, namely,for fixed values of ( u B , ζ, ¯ ζ ) the points x a that satisfythe above equation form the past null cone from thepoint ( u B , ζ, ¯ ζ ) at null infinity. Thus, Z satisfies g ab ( x ) ∂ a Z∂ b Z = 0 , (A2) • Under a Bondi supertranslation ˜ u B = u B + α ( ζ, ¯ ζ ) , Z transform as ˜ Z = Z + α ,. However, neither the confor-mal metric nor the field equations for Z change under asupertranslation as they all depend on spacetime deriva-tives of Z . • The explicit algebraic construction of the conformalmetric is done by first selecting a (( ζ, ¯ ζ ) family of)null coordinate system u = Z ( x a , ζ, ¯ ζ ) , ω = ð Z, ¯ ω =¯ ð Z, R = ¯ ðð Z and then extracting the metric compo-nents from (A2 by successive ð and ¯ ð derivatives of(A2). • It can be shown that all the non trivial components of theconformal metric are obtained in terms of spacetimesderivatives of a function Λ( x a , ζ, ¯ ζ ) defined as ð Z = Λ( x a , ζ, ¯ ζ ) . (A3)This function Λ plays a major role in the field equationsfor the NSF. Note that from its definition it follows that ¯ ð Λ = ð ¯Λ . (A4)This condition is called the reality condition and it willbe used below to restrict the free data in the field equa-tions. • A perfectly valid question is whether a conformal met-ric can be constructed from any arbitrary function Z ( x a , ζ, ¯ ζ ) . In general the answer is no since for a fixedvalue of x a equation (A2) is an algebraic equation fornine constants whereas ( ζ, ¯ ζ ) can take any value. Thus,conditions must be imposed on Z for a metric to exist.It can be shown that the so called metricity conditionsare given by ð (cid:0) g ab ( x ) ∂ a Z∂ b Z (cid:1) = 0 , (A5)and they must be satisfied by Z before one looks for aconformal metric. These conditions generalize work byCartan[27], and Chern[28] originally derived for thirdorder ODEs although coming from a completely differ-ent approach. In three dimensions, one derives exactlythe same condition from either the NSF[41] or the Car-tan approach. • Further insight into the geometrical meaning of Λ canbe gained by Using Sachs theorem. One can show thatthe Λ satisfies Λ = σ ( Z, ζ, ¯ ζ ) − σ Z ( x a , ζ, ¯ ζ ) , (A6)with σ the Bondi shear at null infinity and σ Z theasymptotic shear of the future null cone from x a evalu-ated at null infinity [29]. • Under a Bondi supertranslation the above equation re-mains valid in form as σ ′ = σ + ð α and σ Z remainsthe same. • As a particular case of equation (A6), one can obtainthe null cone cuts in Minkowski space. Since σ Z = 0 (null cones are shear free) and the imaginary part of theshear vanishes we have, ð Z M = σ ( ζ, ¯ ζ ) = ð σ R , with σ R a real function on the sphere. Using a super-translation one eliminates σ R and obtains a canonicalequation ð Z = 0 , whose solution will be given and used in this work.To obtain the dynamical equations for Z one uses the alge-braic relation between the conformal metric of the spacetimeand Z directly from (A2). One then constructs the Ricci andWeyl tensor and imposes the Einstein equations. Since the re-sulting equations are technically involved we present first thelinearized version of the field equations for Λ and Z .Keeping only terms of order Λ in the metric componentsand writing down the Ricci flat equation to linear order in Λ one gets ✷ Λ = 0 , with ✷ the D’Alembertian in flat space. Thus, Λ satisfies Huy-gens principle and its solution only depends on the data givenon the flat null cone cut. If in addition one imposes the metric-ity conditions and the reality condition for Λ one gets ∂∂u (cid:2) ¯ ð Λ − ¯ ð σ ( u, ζ, ¯ ζ ) − ð ¯ σ ( u, ζ, ¯ ζ ) (cid:3) = 0 , (A7)and as expected, the above equation is supertranslation invari-ant.To go from the above equation to the RNC cut equation onereplaces Λ by ð Z , u by Z , etc. obtaining ¯ ð ð ( Z − Z i ) = ¯ ð ( σ − σ i ) + ð (¯ σ − ¯ σ i ) (A8)with Z i some initial cut, and σ i = σ ( Z i , ζ, ¯ ζ ) . Thus, Z − Z i is supertranslation invariant and only depends on σ − σ i .Defining [ Z ] = Z − Z i , [ σ ] = σ − σ i , one writes theformal linearized solution as [ Z ]( x a , ζ, ¯ ζ ) = x a ℓ a + I K ( ζ, ζ ′ )(¯ ð ′ [ σ ′ ] + ð ′ [¯ σ ′ ]) dS ′ , (A9)with x a ℓ a and K ( ζ, ζ ′ ) the kernel and the Green function ofthe ¯ ð ð operator on the sphere. In the above equation thefour constants x a = ( R , R i ) are interpreted as points in thespacetime whereas ℓ a = ( Y , − Y i ) are the l = 0 . spherical harmonics. Also σ ′ = σ ( x a ℓ ′ a , ζ ′ , ¯ ζ ′ ) .5We now seek for a one parameter family of solutions thatrepresents worldlines on the spacetime, i.e., x a = x a ( τ ) . Forthis family we set Z i = Z ( x a ( τ i ) , ζ, ¯ ζ ) . Instead of finding themost general form of the solution to the above equation wewant to concentrate in the compact systems we are interestedin describing. Essentially we would like to describe sourcesin the volume of space that can be observed by aLIGO such asclosed binaries, supernovae or gravitational kicks. For thosesituations it is fair to assume that the system is asymptoticallystationary both in the past and in the future. We thus assumethat as τ i → −∞ the imaginary part of the Bondi vanishes andthe cut Z i is shear free. In that limit we get σ i = ð σ R ( ζ, ¯ ζ ) , Z i = x a ( τ i ) ℓ a + σ R ( ζ, ¯ ζ ) . Selecting a Bondi frame withvanishing σ R ( ζ, ¯ ζ ) we obtain ¯ ð ð Z = ¯ ð σ ( Z, ζ, ¯ ζ ) + ð ¯ σ ( Z, ζ, ¯ ζ ) . (A10)This equation is used in this work and is referred to as theregularized null cone cut equation or RNC cut equation forshort. Its linearized version was independently derived by L.Mason [42] and by Fritelli and collaborators [43].Although the RNC cut does not corresponds to any space-time point, the full NSF equation[29] can be used to checkhow far apart is the RNC cut from a "real" cut coming froma spacetime point. Assuming the propagation is mostly alongthe characteristics and there are no caustics in the propagation(this is the type of situation aLIGO will be operating) then themain contribution to the NSF equation is the Huygens part.Work of Luc Blanchet show that the contribution of gravita-tional tails on binary coalescence are 5 to 7 orders of magni-tude smaller than the leading part of the radiation. Thus, thereal null cone cut is locally smooth and close to, in a preciseway given by the non-Huygens terms of the NSF equation, aRNC cut.This seems to be the case for the gravitational radiation thatcan be detected by aLIGO. If aLIGO can only detect radia-tion for null directions where the intensity is higher it is safeto assume that for such isolated system the RNC cut will ad-equately describe the null cone from the center of mass sinceone only detects the Huygens part of the gravitational wave.We thus claim that the solution space of the RNC cut equa-tions is useful to describe the dynamical behaviour of globalvariables such as the center of mass and intrinsic angular mo-mentum defined in this work. Appendix B: Octupole Contribution
To include the octupole contribution in our equation of ˙ M , ˙ P i and ˙ J i we write the expansion of the Bondi shear of the set of eqs. (60) in the form σ = σ ij ( u B ) Y ij ( ζ, ¯ ζ ) + σ ijk ( u B ) Y ijk ( ζ, ¯ ζ ) . (B1)The energy and the linear momentum loss are the l = 0 , component of the eq. (36). Introducing the above equation in(36) and using the tensorial harmonics products table of refs.[22] and [30] we get ˙ M = − c G ( ˙ σ ijR ˙ σ ijR + ˙ σ ijI ˙ σ ijI ) − c G ( ˙ σ ijkR ˙ σ ijkR + ˙ σ ijkI ˙ σ ijkI ) , ˙ P i = 2 c G ˙ σ jlR ˙ σ klI ǫ ijk − √ c G ( ˙ σ jkR ˙ σ ijkR + ˙ σ jkI ˙ σ ijkI )+ 3 c G ˙ σ jlmR ˙ σ klmI ǫ ijk . To obtain the quadrupole and octupole contribution to theangular momentum loss we compute the imaginary l = 1 component of the definition (56). For this we use (30) andthe tensorial harmonics products table to get ˙ J i = c G ( σ klR ˙ σ jlR + σ klI ˙ σ jlI ) ǫ ijk + 9 c G ( σ klmR ˙ σ jlmR + σ klmI ˙ σ jlmI ) ǫ ijk In a similar way one can write the n-pole contribution to theevolution equation for M , P i and J i . Appendix C: The Tensor Spin-s Harmonics
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