Multipole Expansion of Gravitational Waves:from Harmonic to Bondi coordinates

(or “Monsieur de Donder meets Sir Bondi”)

Luc Blanchet,a1 Geoffrey Compère,b2

Guillaume Faye,a3 Roberto Oliveri,c4 Ali Seraj b5

a GRεCO, Institut d’Astrophysique de Paris, UMR 7095,CNRS & Sorbonne Université, 98bis boulevard Arago, 75014 Paris, France

b Université Libre de Bruxelles, Centre for Gravitational Waves,International Solvay Institutes, CP 231, B-1050 Brussels, Belgium

c CEICO, Institute of Physics of the Czech Academy of Sciences,Na Slovance 2, 182 21 Praha 8, Czech Republic

Abstract

We transform the metric of an isolated matter source in the multipolar post-Minkowskianapproximation from harmonic (de Donder) coordinates to radiative Newman-Unti (NU)coordinates. To linearized order, we obtain the NU metric as a functional of the massand current multipole moments of the source, valid all-over the exterior region of thesource. Imposing appropriate boundary conditions we recover the generalized Bondi-van der Burg-Metzner-Sachs residual symmetry group. To quadratic order, in the caseof the mass-quadrupole interaction, we determine the contributions of gravitational-wave tails in the NU metric, and prove that the expansion of the metric in terms of theradius is regular to all orders. The mass and angular momentum aspects, as well as theBondi shear, are read off from the metric. They are given by the radiative quadrupolemoment including the tail terms.

1luc.blanchet@iap.fr2gcompere@ulb.ac.be3faye@iap.fr4roliveri@fzu.cz5aseraj@ulb.ac.be

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Contents

1 Introduction 21.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Notation and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 From harmonic gauge to Newman-Unti gauge 52.1 Linear metric in harmonic coordinates . . . . . . . . . . . . . . . . . . . . . 52.2 Algorithm to transform harmonic to NU metrics . . . . . . . . . . . . . . . . 6

3 Newman-Unti metric to linear order 73.1 Solving the NU gauge conditions . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Boundary conditions and the BMS group . . . . . . . . . . . . . . . . . . . . 93.3 Bondi data to linear order . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Newman-Unti metric to quadratic order 134.1 Tails and the mass-quadrupole interaction . . . . . . . . . . . . . . . . . . . 144.2 Mass and angular momentum losses . . . . . . . . . . . . . . . . . . . . . . . 19

5 Conclusion and perspectives 21

A Map between Bondi and Newman-Unti gauges 22

B Equations for any PM order 23

1 Introduction

1.1 Motivations

Gravitational waves (GWs), whose physical existence was controversial for years, were es-tablished rigorously in the seminal works of Bondi, van der Burg, Metzner and Sachs [1, 2].The Bondi-Sachs formalism describes the asymptotic structure near future null infinity ofthe field generated by isolated self-gravitating sources. This asymptotic structure was fur-ther elucidated thanks to the tools of the Newman-Penrose formalism [3] and conformalcompactifications [4] leading to the concept of asymptotically simple spacetimes in the senseof Penrose [5]. Asymptotically simple spacetimes are now proven to follow from large setsof initial data which are stationary at spatial infinity, see e.g. the review [6].

Bondi coordinates or Bondi tetrad frames are defined from an outgoing light cone con-gruence with radial sections parametrized by the luminosity (areal) distance. A variant ofthese coordinates are the Newman-Unti (NU) coordinates whose radial coordinate is insteadan affine parameter [7]. Bondi gauge and NU gauge share all essential features and can easilybe mapped to each other [8, 9]. Under the assumption of asymptotic simplicity, Einstein’sequations admit a consistent asymptotic solution [10,11]. Such an asymptotic series is how-ever limited to the vicinity of null infinity and it does not, in particular, resolve the sourcethat generates the radiation.

2

Recent interest in Bondi gauge arose from the fact that it is preserved under an infiniteset of residual symmetries, dubbed the generalized BMS group, that is generated by super-translations and arbitrary diffeomorphisms on the two-sphere [11–16], which gives rise to twoinfinite sets of flux-balance laws [8, 11, 14, 17–32]. Thanks to junction conditions at spatialinfinity [18], the generalized BMS group is a symmetry of the quantum gravity S-matrix,which gives rise to Ward identities that are identical to Weinberg’ soft graviton theorem [33]and to the subleading soft graviton theorem [15,34].

For GW generation and applications to the data analysis of the GW events one needs theconnection between the asymptotic structure of the field and explicit matter sources. Thisis achieved by the multipolar post-Minkowskian (MPM) expansion [35–38] which combinesthe multipole expansion for the field in the exterior region of the source with a nonlinearityexpansion in powers of the gravitational constant G. The MPM formalism is defined inharmonic coordinates, also known as de Donder coordinates. At linear order the MPMexpansion reduces to the linear metric as written by Thorne [39] and is characterized interms of two infinite sets of canonical multipole moments, namely the mass and currentmultipole moments. A class of radiative coordinate systems exists such that the MPMexpansion leads to asymptotically simple spacetimes for sources that are stationary beforesome given time in the remote past [36]. In such radiative coordinates, two infinite sets ofradiative multipoles can be defined in terms of the canonical multipoles. They parametrizethe asymptotic transverse-traceless waveform or, equivalently, the two polarizations of theBondi shear.

In addition, the MPM formalism has to be matched to the post-Newtonian (PN) fieldin the near-zone and the interior of the source, which allows us to express the canonicalmultipoles in terms of the actual source multipoles and, furthermore, yields the radiation-reaction forces caused by the radiation onto their sources [40–42]. The MPM-PN formalismwas applied to compact binary systems and permitted to compute the GW phase evolutionof inspiralling compact binaries to high PN order, see notably [43–46].

The main objective of this paper is to make explicit the relationship between Bondiexpansions and the MPM formalism. The Bondi and NU gauges belong to the general class ofradiative gauges in the sense of [36,47]. Here we will describe the construction of the explicitdiffeomorphism transforming the metric in the MPM expansion from harmonic coordinates toNU coordinates. The diffeomorphism is perturbative in powers of G and, for each PM order,it is valid everywhere outside the source. After imposing standard boundary conditions, wefind this diffeomorphism to be unique up to generalized BMS transformations [11–16, 24,48–50], as we will cross-check in details. This allows us to transpose known results on theexterior MPM metric in harmonic gauge for a particular multipolar mode coupling and agiven post-Minkowskian order to a metric in NU gauge, written as an exact expression toall orders in the radial expansion. As an illustration, we will explicitly derive the Bondimetric of the second-order post-Minkowskian (2PM or G2) perturbation corresponding tomass-quadrupole interactions [38, 51]. In particular this entails the description of GW tailswithin the Bondi asymptotic framework.

The rest of the paper is organized as follows. Section 1.2 is devoted to our notation andconventions. Section 2.1 recalls the harmonic-coordinates description of the metric in termsof canonical moments at linearized order. In Sec. 2.2 we present an algorithm implementingthe transformation from harmonic coordinates to NU coordinates. In Sec. 3.1 we derive

3

the NU metric as a function of the mass and current multipoles to linearized order. InSec. 3.2 we impose standard boundary conditions for the asymptotic metric and naturallyrecover from our algorithm the gauge freedom associated with the BMS group. Notably, inSec. 3.3, we obtain the Bondi mass aspect, the angular momentum aspect and the Bondishear as multipole expansions parametrized by the canonical moments. In Sec. 4 we applythe algorithm to the quadratic metric (i.e. to 2PM order in the MPM formalism), focusingon the quadratic interaction between the mass monopole and the mass quadrupole. Explicitresults on GW tails obtained in harmonic coordinates, are then conveyed into the NU metricin Sec. 4.1, to any order in the radial expansion. Finally we discuss in Sec. 4.2 the mass andangular momentum GW losses in the Bondi-NU framework at the level of the quadrupole-quadrupole interaction. The paper ends with a short conclusion and perspectives in Sec. 5.Two appendices gather technical details on the map between Bondi and NU gauges (A), andthe all-order PM formulæ for the coordinate change equations (B).

1.2 Notation and conventions

We adopt units with the speed of light c = 1. The Newton gravitational constant G iskept explicit to bookmark post-Minkowskian (PM) orders. We will refer to lower case Latinindices from a to h as indices on the two-dimensional sphere, while lower case Latin indicesfrom i to z will refer to three-dimensional Cartesian indices. The Minkowski metric isηµν = diag(−1,+1,+1,+1).

We denote Cartesian coordinates as xµ = (t,x) and spherical ones as (t, r, θa). Here,the radial coordinate r is defined as r = |x| and θa = (θ, ϕ) with a, b, · · · = {1, 2}. Theunit directional vector is denoted as ni = ni(θa) = xi/r. Euclidean spatial indices i, j, · · · ={1, 2, 3} are raised and lowered with the Kronecker metric δij. Furthermore, we definethe Minkowskian outgoing vector kµ∂µ = ∂t + n

i∂i with ∂i = ∂/∂xi, or, in components,

kµ = (1, ni) and kµ = (−1, ni). In retarded spherical coordinates (u, r, θa) with u = t− r, wehave kµ∂µ = ∂r

∣∣u. We employ the natural basis on the unit 2-sphere ea =

∂∂θa

embedded in

R3 with components eia = ∂ni/∂θa. Given the unit metric on the sphere γab = diag(1, sin2 θ)we have: nieia = 0, ∂iθ

a = r−1γabeib, γab = δijeiaejb and γ

abeiaejb =⊥ij, where ⊥ij= δij − ninj

is the projector onto the sphere. We also use the notation ei〈aejb〉 = e

i(ae

jb) −

12γab⊥ij for the

trace-free product of basis vectors. Introducing the covariant derivative Da compatible withthe sphere metric, Daγbc = 0, we have Dae

ib = Dbe

ia = DaDbn

i = −γabni.Given a general manifold, harmonic/de Donder coordinates are specified by using a tilde:

x̃µ = (t̃, x̃) or (t̃, r̃, θ̃a). The metric tensor is g̃µν(x̃). Asymptotically flat spacetimes admitas a background structure the Minkowskian outgoing vector k̃µ = (1, ñi), the basis on thesphere ẽia = ∂ñ

i/∂θ̃a, etc. We define the retarded time ũ in harmonic coordinates as ũ = t̃−r̃,such that k̃µ = −∂̃µũ.

Newman-Unti (NU) coordinates are denoted xµ = (u, r, θa) with θa = (θ, ϕ). The metrictensor in NU coordinates is denoted as gµν(x), with all other notation, such as the naturalbasis on the sphere eia and the metric γab, as previously.

We denote by L = i1i2 . . . i` a multi-index made of ` spatial indices. We use short-handsfor: the multi-derivative operator ∂L = ∂i1 . . . ∂i` , the product of vectors nL = ni1 . . . ni`and xL = xi1 . . . xi` = r

`nL. The multipole moments ML and SL are symmetric and trace-free (STF). The transverse-trace-free (TT) projection operator is denoted ⊥ijklTT =⊥k(i⊥j)l

4

−12⊥ij⊥kl. Time derivatives are indicated by superscripts (q) or by dots.

2 From harmonic gauge to Newman-Unti gauge

2.1 Linear metric in harmonic coordinates

We work with the gothic metric deviation defined as hµν =√|g̃|g̃µν − ηµν and satisfying

the de Donder (or harmonic) gauge condition ∂̃µhµν = 0. The Einstein field equations in

harmonic coordinates read as

�̃hµν =16πG

c4|g̃|T µν + Λµν(h, ∂h, ∂2h) , (1)

where �̃ = �̃η is the flat d’Alembertian operator, and the right-hand side contains the matterstress-energy tensor T µν as well as the back-reaction from the metric itself, in the form of aninfinite sum Λµν of quadratic or higher powers of h and its space-time derivatives. We shallconsider the metric generated by an isolated matter system, in the form of a non-linearityor post-Minkowskian (PM) expansion, labeled by G,

hµν =+∞∑n=1

Gnhµνn . (2)

Furthermore we consider the metric in the vacuum region outside the isolated matter system,and assume that each PM coefficient hµνn in Eq. (2) is in the form of a multipole expansion,parametrized by so-called canonical multipole moments. We call this the multipolar-post-Minkowskian approximation [35]. In the linearized approximation the vacuum Einstein fieldequations in harmonic coordinates read �̃hµν1 = ∂̃νh

µν1 = 0, whose most general retarded

solution, modulo an infinitesimal harmonic gauge transformation, is [39]

h001 = −4+∞∑`=0

(−)`

`!∂̃L

(ML(ũ)

r̃

), (3a)

h0j1 = 4+∞∑`=1

(−)`

`!

[∂̃L−1

(M

(1)jL−1(ũ)

r̃

)+

`

`+ 1∂̃pL−1

(εjpqSqL−1(ũ)

r̃

)], (3b)

hjk1 = −4+∞∑`=2

(−)`

`!

[∂̃L−2

(M

(2)jkL−2(ũ)

r̃

)+

2`

`+ 1∂̃pL−2

(εpq(jS

(1)k)qL−2(ũ)

r̃

)], (3c)

given in terms of symmetric-trace-free (STF) canonical mass and current multipole momentsML and SL depending on the harmonic coordinate retarded time ũ = t̃ − r̃. Among thesemoments, the mass monopole M is the constant (ADM) total mass of the system, P i = M

(1)i

is the constant linear momentum and Si is the constant angular momentum. We can expandthe linear metric in powers of 1/r̃ using the formula (valid for arbitrary STF tensors ML)

∂̃L

(ML(ũ)

r̃

)= (−)` ñL

∑̀k=0

ak`M

(`−k)L (ũ)

r̃k+1, (4a)

5

with ak` =(`+ k)!

2kk!(`− k)!. (4b)

A method has been proposed in [35] to compute each of the PM coefficients up to anyorder n, starting from the linear metric (3). Each of the PM approximation is then obtainedas a functional of the canonical multipole moments ML and SL. The construction representsthe most general solution of the Einstein field equations outside a matter source withoutany incoming flux from past null infinity. This is the so-called MPM formalism. The rela-tion between the canonical moments and the source moments depending on actual sourceparameters is known [40–42].

In this paper we assume that the metric is stationary in the past in the sense that all themultipole moments are constant before some finite instant in the past, say ML(ũ) = constand SL(ũ) = const when ũ 6 −T . Under this assumption all non-local integrals we shallmeet will be convergent at their bound in the infinite past.1

2.2 Algorithm to transform harmonic to NU metrics

Consistently with the PM expansion (2), we assume that the NU coordinates are related tothe harmonic coordinates by the following class of transformations

u = ũ++∞∑n=1

GnUn(ũ, r̃, θ̃a) , (5a)

r = r̃ ++∞∑n=1

GnRn(ũ, r̃, θ̃a) , (5b)

θa = θ̃a ++∞∑n=1

GnΘan(ũ, r̃, θ̃b) , (5c)

where the PM coefficients Un, Rn and Θan are functions of the harmonic coordinates (ũ, r̃, θ̃

a)to be determined, with ũ = t̃− r̃.

The NU gauge2 is defined by the following conditions:

gur = −1 , grr = 0 , gra = 0 . (6)

For computational reasons, it is more convenient to work with the inverse metric components,for which the NU gauge reads as

guu = 0 , gur = −1 , gua = 0 , (7)

The gauge is constructed such that (i) the outgoing vector kµ = −∂µu is null, (ii) the angularcoordinates are constant along null rays kµ∂µθ

a = 0, and (iii) the radial coordinate is an

1This assumption may be weakened to the situation where the source is initially made of free particlesmoving on unbound hyperbolic like orbits (initial scattering). In this case we would have ML(ũ) ∼ (−ũ)`and SL(ũ) ∼ (−ũ)` when ũ → −∞, and the tail integrals in the radiative moment, Eq. (47) below, wouldstill be convergent for such initial state [52].

2The NU and Bondi gauges differ by a choice of the radial coordinate. See more details in [8] and in theAppendix A below.

6

affine parameter on outgoing null curves, i.e. kµ∂µr = 1. The strategy to construct theperturbative diffeomorphism is the following. From the NU gauge conditions (7), one findsthe following constraints on the transformation laws (5), namely

g̃µν(x̃)∂u

∂x̃µ∂u

∂x̃ν= 0 , (8a)

g̃µν(x̃)∂u

∂x̃µ∂r

∂x̃ν= −1 , (8b)

g̃µν(x̃)∂u

∂x̃µ∂θa

∂x̃ν= 0 . (8c)

Inserting the linear metric (3) this permits to solve for the linear corrections U1, R1 and Θa1,

modulo an arbitrariness related in fine to BMS transformations. Then one uses

grr(x) = g̃µν(x̃)∂r

∂x̃µ∂r

∂x̃ν, (9a)

gra(x) = g̃µν(x̃)∂r

∂x̃µ∂θa

∂x̃ν, (9b)

gab(x) = g̃µν(x̃)∂θa

∂x̃µ∂θb

∂x̃ν, (9c)

to deduce guu = −grr + gragua, gua = grbgab, gab = (gab)−1 to linear order. We can thenread off, respectively, the Bondi mass aspect, the Bondi angular momentum aspect andthe Bondi shear. To quadratic order one inserts the metric hµν2 (x̃) in harmonic coordinatessolving Eq. (1) to order G2, and obtain U2, R2, Θ

a2 and the NU metric to order G

2. In theend we have to re-express the metric in terms of NU coordinates using the inverse of Eq. (5).This algorithm can be iterated in principle at any arbitrary order in powers of G.

3 Newman-Unti metric to linear order

3.1 Solving the NU gauge conditions

At linear order in G, the constraints (8) are equivalent to the following equations for thelinear coefficients U1, R1 and Θ

a1, involving the directional derivative along the direction

k̃µ = (1, ñi) of the Minkowski null cone:

k̃µ∂̃µU1 =1

2k̃µk̃νh

µν1 , (10a)

k̃µ∂̃µR1 = −1

2ñiñjh

ij1 +

1

2hii1 − U̇1 , (10b)

k̃µ∂̃µΘa1 =

ẽair̃

(∂̃iU1 − k̃µhµi1

). (10c)

where the overdot denotes the derivative with respect to ũ. Notice that hii1 = 0 for themetric (3). Using the explicit form of the linearized metric (3)–(4) one readily obtains themost general solutions of those equations as

U1 = −2(M − ñiPi

)ln(r̃/P) + 4

+∞∑`=1

1

`!

∑̀k=1

(2k − 1)ak`(`+ k − 1)(`+ k)

ñLM(`−k)L

r̃k− ξu1 , (11a)

7

R1 = M + [3− 2 ln(r̃/P)] ñiPi + 2+∞∑`=2

1

`!

`−1∑k=1

(`− k)(`+ 3k + 1)ak`(`+ k)(`+ k − 1)(k + 1)

ñLM(`−k)L

r̃k− ξr1 ,

(11b)

Θa1 =ẽair̃

[2Pi[1− ln(r̃/P)

](11c)

− 4+∞∑`=1

1

`!

∑̀k=1

akl(`+ k)(k + 1)

ñL−1r̃k

(2k2 − ``+ k − 1

M(`−k)iL−1 +

2k`

`+ 1εijkñjS

(`−k)kL−1

)]− ξa1 ,

where P is an irrelevant constant. We recognize the standard logarithmic deviation u =ũ− 2GM ln(r̃/P) +O(r̃−1) from harmonic to radiative coordinates; see e.g. [36].

Furthermore we have added the most general homogeneous solution of the differentialequations (10) denoted by ξµ. These are indeed the residual gauge transformations preservingthe NU gauge (8), at linearized order, i.e. xµ → xµ + ξµ with ξµ = O(G1). The linear gaugetransformation, ξ = Gξ1 takes the form

ξu1 = f , ξr1 = −r̃ḟ +Q , ξa1 = Y a −

1

r̃D̃af , (12)

where f , Q and Y a are arbitrary functions of ũ = t̃− r̃ and the angles θ̃a. Note that for laterconvenience, we made explicit into the expression of R1 given by Eq. (11b) some constantmonopolar and dipolar (` = 0, 1) contributions corresponding to a redefinition of the radialcoordinate as r̃ → r̃ +G(M + 3ñiPi), thanks to the arbitrary function Q in Eq. (12).

The metric in NU gauge is immediately obtained at linear order in G from the linearmetric hµν1 (and its trace h1 = ηµνh

µν1 = −h001 ) as given by Eq. (3) together with the linear

coefficients U1, R1 and Θa1 as

guu = −1 + 2G(Ṙ1 + U̇1 +

1

4h1

)+O(G2) , (13a)

gua = Gr2[−Θ̇1a + r−1eiah0i1 + r−2Da

(R1 + U1

)]+O(G2) , (13b)

gab = r2γab −Gr2

(2D(aΘ1b) + 2r

−1γabR1 + eiaejbh

ij1 −

1

2γab h1

)+O(G2) . (13c)

Note that the final result for the metric has been written in terms of the NU coordinates xµ.As a result the spatial metric gab is given by a covariant tensorial expression on the sphere,involving the Lie derivative LΘ1γab = 2D(aΘ1b). To this end, we have written the leadingcontribution in the spatial metric gab in terms of NU coordinates to linear order in G as

r̃2γ̃ab = r2[γab − 2G

(r−1R1γab + Θ

c1Γ

ec(aγb)e

)]+O(G2) , (14)

where Γabc denotes the Christoffel symbol on the sphere. At linear order in G, we canequivalently replace the harmonic coordinates by the NU ones, as the correction will be atO(G2). Plugging the results (11) into the metric (13), we find

guu = −1 + 2G+∞∑`=0

(`+ 1)(`+ 2)

`!

∑̀k=0

ak`(k + 1)(k + 2)

nLM(`−k)L

rk+1+ δξguu +O(G2) , (15a)

8

gua = Geia

{−

+∞∑`=2

`+ 2

`!nL−1

[M

(`)iL−1 −

2`

`+ 1εipqnpS

(`)qL−1

]+ 2

+∞∑`=1

`+ 2

`!nL−1

∑̀k=1

ak`k + 2

1

rk

[M

(`−k)iL−1 +

2`

`+ 1εipqnpS

(`−k)qL−1

]}+ δξgua +O(G2) , (15b)

gab = r2

[γab + 4Ge

i〈ae

jb〉

+∞∑`=2

1

`!

nL−2r

{M

(`)ijL−2 −

2`

`+ 1εipqnpS

(`)jqL−2

+∑̀k=2

k − 1k + 1

ak`rk

[M

(`−k)ijL−2 +

2`

`+ 1εipqnpS

(`−k)jqL−2

]}]+ δξgab +O(G2) , (15c)

where we have posed ei〈aejb〉 = e

i(ae

jb) −

12γab ⊥ij. The last terms correspond to the freedom

left in the metric, which is associated with the gauge vector (12), and are given by

δξguu = 2G[−Q̇− ḟ + rf̈

]+O(G2) , (16a)

δξgua = G[−Da

(Q+ f

)+ r2Ẏa

]+O(G2) , (16b)

δξgab = 2G[r2(−γabḟ +D(aYb)

)+ r(−DaDbf + γabQ

)]+O(G2) . (16c)

3.2 Boundary conditions and the BMS group

An asymptotic frame is defined from boundary Dirichlet gauge fixing conditions, which picka specific foliation by constant u surfaces and a specific measure on the codimension 2boundary. The boundary gauge fixing conditions when r →∞ are

guu = O(r0) , (17a)gua = O(r0) , (17b)

det gab = r4 sin2θ +O(r2) , (17c)

where the first term in Eq. (17c) is the determinant of the metric on the unit sphere metric.Notice that Eq. (17c) not only fixes the measure on the sphere, but also requires thatthe shear which appears at order O(r3) is trace-free, see the discussion around Eq. (2.5)of [8]. The boundary condition (17c) only determines the leading order determinant, whichis compatible with Newmann-Unti gauge.3 The metric (15)–(16) does not yet respect theboundary conditions (17). Thus one has to implement an infinitesimal transformation inorder to achieve the gauge with appropriate asymptotic behavior.

The first condition (17a) implies that f̈ = 0, hence ḟ must only be a function of theangles θa. The second condition (17b) implies that Ẏ a = 0, i.e. that Y a also is only afunction of the angles. To impose the last condition (17c), we note that the leading metricon the sphere γab already satisfies the leading behavior of (17c), i.e., its measure is that ofa unit metric on the sphere. Therefore the leading term in Eq. (16) must be trace-free, thusḟ = 1

2DaY

a, which is consistent with f̈ = 0. Similarly the next-to-leading term O(r) in gab3In contrast, the Bondi gauge fixing condition ∂r(detgab/r

4) = 0 fixes the determinant at any r, exceptat leading order.

9

must also be trace-free, hence Q = 12∆f where ∆ = DaDa is the Laplacian on the sphere.

Summarizing all these, we have

Q =1

2∆f , f(u, θa) = T (θa) +

u

2DaY

a , Y a = Y a(θb) . (18)

The simplest choice that brings the metric (15) into the form (17) is of course obtained bysetting f = Q = T = Y a = 0. This choice is generally assumed in the perturbative approachto gravitational waves in harmonic coordinates, see e.g. [53]. However, after fulfilling all theconditions, i.e. the gauge conditions (6) and the asymptotic boundary conditions (17), weare still left with the infinitesimal coordinate transformations generated by the gauge vectorfield ξµBMS ≡ ξµ, with components

ξuBMS = T +u

2DaY

a , (19a)

ξrBMS = −r

2DaY

a +1

2∆(T +

u

2DaY

a), (19b)

ξaBMS = Ya − 1

rDa(T +

u

2DbY

b). (19c)

The coordinate transformation generated by the above vector fields form the symmetriesof the space of solutions which are parametrized by a time-independent function T (θa)generating super-translations, and a time-independent vector Y a(θb) on the sphere gen-erating super-Lorentz transformations. These form the celebrated generalized BMS alge-bra [15, 16, 24, 48, 49] (i.e., the smooth version of [11–14]). The modification of the metricunder the BMS group reads4

δBMS guu = −G(∆ + 2

)ḟ +O(G2) = −G

2Da

(Y a + ∆Y a

)+O(G2) , (20a)

δBMS gua = −G

2Da(∆ + 2

)f +O(G2)

= −GDa[T +

1

2∆T +

u

4Db

(Y b + ∆Y b

)]+O(G2), (20b)

δBMS gab = G[2r2D〈aYb〉 − 2r D〈aDb〉f

]+O(G2) , (20c)

where we recall that D〈aYb〉 = D(aYb) − 12γabDcYc and D〈aDb〉f = DaDbf − 12γab∆f . The

transformation law of the asymptotic metric on the sphere qab defined from gab = r2qab+O(r1)

agrees with Eq. (2.20) of [24]. We note that the leading uu component of the metric is givenby Eq. (20a) where the divergence DaY

a only involves the determinant of the metric on theunit sphere. This is consistent with Eqs. (2.5) and (2.25) or Eqs. (3.11) and (3.21) of [24].

It is worth pointing out that the kernel of the operator ∆ + 2 appearing in the BMStransformation of the uu component of the metric in Eq. (20a) is the ` = 1 harmonics, i.e.(∆ + 2)f = 0 if and only if f is made of the ` = 1 harmonics. Similarly the kernel ofthe operator D〈aDb〉 appearing in the BMS transformation of the ab component (20c) [see

4We have ∆DaYa = Da(∆Y

a − Y a). Note that the Ricci tensor Rab = γab on the unit sphere.

10

also the shear (28)] is the ` = 0 and ` = 1 harmonics. In order to make this explicit, wedecompose the function f into STF spherical harmonics

f = T +u

2DaY

a =+∞∑`=0

nLfL(u) , (21)

where the STF coefficients fL are linear functions of u, and find

(∆ + 2

)f = −

+∞∑`=0

(`+ 2)(`− 1)nLfL , (22a)

D〈aDb〉f = ei〈ae

jb〉

+∞∑`=0

`(`− 1)nL−2fijL−2 . (22b)

For completeness, we can now detail the boundary conditions at spatial infinity thatcould be imposed in order to completely fix the asymptotic frame, even though we willnot enforce these conditions in the following sections since they remove the generalized BMSasymptotic symmetry group at spatial infinity [18,50,54,55]. First, upon fixing the boundarymetric to be the unit sphere metric, qab = γab, all proper super-Lorentz transformations arediscarded and the generalized BMS algebra reduces to the original BMS algebra. Second,upon imposing stationarity in the asymptotic past u→ −∞, one sets the momenta to zero,Pi = 0 and the boost are discarded. Since the Bondi news Nab is zero or decays in theasymptotic past, the electric part of the Bondi shear defined as C+ in the decompositionCab = −2GD〈aDb〉C+ + 2G�c(aDb)DcC− satisfies limu→−∞C+ = C(θ, φ) with C → C +T under a supertranslation. One can then discard all supertranslations but the Poincarétranslations by fixing all harmonics ` > 1 of C. On the other hand, the rotations not alignedwith the total angular momentum can be discarded by setting the Bondi angular momentumNa(u = −∞) to canonical form, Na = −3J sin2 θ∂aφ. Finally, the spatial translations canbe discarded by setting the mass dipole to zero, Mi = 0, which is equivalent to choosingthe center-of-mass frame. The BMS symmetry group is then gauge-fixed to R× SO(2), thesymmetry group of asymptotically stationary solutions consisting of time translations androtations around the axis of the total angular momentum. In conclusion, one can reduce thefour-dimensional diffeomorphism group to R×SO(2) after imposing Newman-Unti gauge (6),boundary gauge fixing conditions (17) and additional boundary conditions at spatial infinityas just described.

3.3 Bondi data to linear order

Finally, we shall write the metric (15) including the bulk terms in the form

guu = −1−G(∆ + 2

)ḟ +

2G

r

[m+

+∞∑k=1

1

rkKk

]+O(G2) , (23a)

gua = G

(1

2DbC

ba +

1

r

[2

3Na + e

ia

+∞∑k=1

1

rkP ik

])+O(G2) , (23b)

11

gab = r2

[γab + 2GD〈aYb〉 +

G

r

(Cab + e

i〈ae

jb〉

+∞∑k=1

1

rkQijk

)]+O(G2) . (23c)

The sub-dominant contributions in 1/r in the metric (23) read as

Kk =1

(k + 1)(k + 2)

+∞∑`=k

(`+ 1)(`+ 2)

`!ak` nLM

(`−k)L +O(G) , (24a)

P ik =2

k + 3

+∞∑`=k+1

`+ 2

`!ak+1` nL−1

[M

(`−k−1)iL−1 +

2`

`+ 1εipqnpS

(`−k−1)qL−1

]+O(G) , (24b)

Qijk = 4k − 1k + 1

+∞∑`=k

1

`!ak` nL−2

[M

(`−k)ijL−2 +

2`

`+ 1εipqnpS

(`−k)jqL−2

]+O(G) . (24c)

Note that Qij1 = 0 for k = 1. Therefore, at linear order in G the next order correction termin 1/r in the metric gab beyond the shear Cab is absent. This is just a feature of the linearmetric, since at quadratic order O(G2) there is a well-known term quadratic in the shear.

To leading order when r → ∞ the metric (23) is defined by the so-called Bondi massaspect m, angular momentum aspect Na and shear Cab (see e.g. [10, 27, 32]). These arefunctions of time u and the angles θa. The mass and angular momentum aspects are givenin terms of the multipole moments to linear order in G by

m =+∞∑`=0

(`+ 1)(`+ 2)

2`!nLM

(`)L +O(G) , (25a)

Na = eia

+∞∑`=1

(`+ 1)(`+ 2)

2(`− 1)!nL−1

[M

(`−1)iL−1 +

2`

`+ 1εipqnpS

(`−1)qL−1

]+O(G) . (25b)

In the next section we will work out the mass loss and angular momentum loss formulasfor ṁ and Ṅa to quadratic order in G. But we already note that

Ṅa = Dam+ eia

+∞∑`=1

`(`+ 2)

(`− 1)!εipqnpL−1S

(`)qL−1 +O(G) , (26)

in agreement with the Einstein equation for the angular momentum aspect.To define the shear we introduce the usual asymptotic waveform in transverse-trace-free

(TT) gauge, given in terms of the multipole moments by (see e.g. [56])

H ijTT = 4⊥ijklTT

+∞∑`=2

nL−2`!

[M

(`)klL−2 −

2`

`+ 1εkpqnpS

(`)lqL−2

]+O(G) , (27)

where ⊥ijklTT is the TT projection operator. Then the shear is given by

Cab = ei〈ae

jb〉H

ijTT − 2D〈aDb〉f . (28)

12

The first term is directly related to the usual two polarization waveforms at infinity. PosingH+ = limr→∞(rh+) and H× = limr→∞(rh×) we have

5

ei〈aejb〉H

ijTT =

(H+ H× sin θ

H× sin θ −H+ sin2 θ

). (29)

The second term in Eq. (28) comes from the BMS transformation as

δBMS Cab = −2D〈aDb〉f . (30)

In the stationary limit, the Bondi mass and angular momentum aspects reduce to theconserved ADM mass M and angular momentum Na = 3e

ia�ipqnpSq, and the shear Cab

vanishes up to the supertranslation shift (30) with f = T . In the metric (23), the canonicalmultipole moments ML, SL appear in guu, r

−1gua, r−2gab exactly at order r

−`+1 and match(up to a normalisation) with the standard Geroch-Hansen multipole moments [39, 57–59].In the zero supertranslation frame (i.e. ∆(∆ + 2)T = 0) and in a Lorentz frame (i.e.D〈aYb〉 = 0), the stationary limit of Eq. (23) is, modulo O(G2),

gstatuu = −1 + 2G+∞∑`=0

(−)`

`!ML∂L

(1

r

), (31a)

gstatua = −2Geia+∞∑`=1

(−)`

`!(2`− 1)

[MiL−1 +

2`

`+ 1εipqnpSqL−1

]∂L−1

(1

r

), (31b)

gstatab = r2γab + 4Ge

i〈ae

jb〉

+∞∑`=2

(−)`

`!

(`− 1)(2`− 1)(2`− 3)`+ 1

×

×[MijL−2 +

2`

`+ 1εipqnpSjqL−2

]∂L−2

(1

r

). (31c)

4 Newman-Unti metric to quadratic order

At second order in G the perturbation (2) reads as6√|g̃|g̃µν = ηµν + hµν = ηµν +Ghµν1 +G2h

µν2 +O(G3). (32)

In the following we will denote h2 ≡ ηµνhµν2 and the indices are lowered by the backgroundMinkowski metric ηµν . At second order in G, the NU gauge conditions (8) imply the followingequations for the functions U2, R2, Θ

a2, respectively,

5We adopt for the polarization vectors εiθ = eiθ and ε

iϕ = e

iϕ/ sin θ such that ε

iθεjθ + ε

iϕεjϕ =⊥ij= γabeiae

jb.

6It implies√|g̃| = 1 + G2 h1 +G

2(

12h2 +

18h

21 − 14h

ρσ1 h1ρσ

)+O(G3) and

g̃µν = ηµν+G(−h1µν + 12h1ηµν

)+G2

[−h2µν − 12h1h1µν +

(12h2 +

18h

21 − 14h

ρσ1 h1ρσ

)ηµν + h1µρh

ρ1 ν

]+O(G3) ,

g̃µν = ηµν +G

(hµν1 −

1

2h1η

µν

)+G2

[hµν2 −

1

2h2η

µν − 12h1h

µν1 +

(1

8h21 +

1

4hαβ1 h1αβ

)ηµν]

+O(G3).

13

k̃µ∂̃µU2 =1

2k̃µk̃νh

µν2 +

(1

2∂̃µU1 − k̃νhµν1

)∂̃µU1 , (33a)

k̃µ∂̃µR2 =1

8h21 −

1

4hµν1 h1µν +

1

2h2 + ñi

[∂̃iU2 − k̃µhµi2 + (∂̃µU1)h

µi1

]+(∂̃µU1 − k̃νhµν1

)∂̃µR1 , (33b)

k̃µ∂̃µΘa2 =

ẽair̃

[∂̃iU2 − k̃µhµi2 + (∂̃µU1)h

µi1

]+(∂̃µU1 − k̃νhµν1

)∂̃µΘ

a1 . (33c)

See Appendix B for a formal generalization of these equations to any PM order.In the following, we will show how the explicit solution for the quadratic metric in

harmonic coordinates, i.e., solving the Einstein field equations (1) to order G2 for somegiven multipole interactions, can be used as an input in our algorithm in order to generatethe corresponding Bondi-NU metric.

The main features of the quadratic metric in harmonic coordinates are [38, 51, 60]: (i)the presence of gravitational-wave tails, corresponding to quadratic interactions between theconstant mass M and varying multipole moments ML and SL (for ` > 2); (ii) the massand angular momentum losses describing the corrections of the constant ADM quantitiesintroduced in the linear metric (M and Si) due to the GW emission;

7 (iii) the presence ofthe non-linear memory effect. We investigate the effects (i) and (ii) in the subsections belowbut postpone (iii) to future work.

4.1 Tails and the mass-quadrupole interaction

In this subsection we construct the NU metric corresponding to the monopole-quadrupoleinteraction M ×Mij, starting from the explicit solution in harmonic coordinates given by(see Appendix B of [38], or Eq. (2.8) of [51])

h002 = Mñpq r̃−4 {−21Mpq − 21r̃M (1)pq + 7r̃2M (2)pq + 10r̃3M (3)pq }

+ 8Mñpq

∫ +∞1

dxQ2(x)M(4)pq (t̃− r̃x) , (34a)

h0i2 = Mñipq r̃−3{−M (1)pq − r̃M (2)pq −

1

3r̃2M (3)pq

}+Mñp r̃

−3{−5M (1)pi − 5r̃M

(2)pi +

19

3r̃2M

(3)pi

}+ 8Mñp

∫ +∞1

dxQ1(x)M(4)pi (t̃− r̃x) , (34b)

hij2 = Mñijpq r̃−4{−15

2Mpq −

15

2r̃M (1)pq − 3r̃2M (2)pq −

1

2r̃3M (3)pq

}+Mδijñpq r̃

−4{−1

2Mpq −

1

2r̃M (1)pq − 2r̃2M (2)pq −

11

6r̃3M (3)pq

}+Mñp(i r̃

−4{

6Mj)p + 6r̃M(1)j)p + 6r̃

2M(2)j)p + 4r̃

3M(3)j)p

}7Similarly there are corrections associated with the losses of linear momentum (or recoil) and the position

of the center of mass, see e.g. [32, 53].

14

+M r̃−4{−Mij − r̃M (1)ij − 4r̃2M

(2)ij −

11

3r̃3M

(3)ij

}+ 8M

∫ +∞1

dxQ0(x)M(4)ij (t̃− r̃x) . (34c)

The metric is composed of two types of terms: the so-called “instantaneous” ones dependingon the quadrupole moment Mij and its derivatives at time ũ = t̃−r̃, and the “hereditary” tailterms depending on all times from −∞ in the past until ũ. The tail integrals are expressedin Eq. (34) by means of the Legendre function of the second kind Q` (with branch cut from−∞ to 1), given by the explicit formula in terms of the Legendre polynomial P`:

Q`(x) =1

2P`(x) ln

(x+ 1

x− 1

)−∑̀j=1

1

jP`−j(x)Pj−1(x) . (35)

We recall that the Legendre function Q` behaves like 1/x`+1 when x → +∞, and that its

leading expansion when y ≡ x − 1 → 0+ reads (with H` =∑`

j=11j

being the `th harmonic

number)

Q`(1 + y) = −1

2ln(y

2

)−H` +O (y ln y) . (36)

With the known harmonic metric (34) [or see below Eq. (46)], we apply our algorithmto generate the Bondi-NU metric. We focus on the case of the mass-quadrupole interactionM×Mij, keeping track of all instantaneous and tail terms. Plugging hµν2 given by Eq. (34) aswell as hµν1 and U1 given in the previous section in the right-side of Eq. (33a), and retainingonly the mass-quadrupole interaction we get

k̃µ∂̃µU2 = Mñpq

[−6r̃−4Mpq − 3r̃−3M (1)pq + 6r̃−2M (2)pq

]+ 4Mñpq

∫ +∞1

dx

[Q2 − 2Q1 +Q0 −

1

2

]M (4)pq (t̃− r̃x) . (37)

We remark that, as an intermediate step to obtain (37), an instantaneous term of the form

−2r̃−1MñpqM (3)pq has been equivalently written as the last term in the second line. In thisform, it is explicit that the integrand of Eq. (37) does not diverge in the limit x → 1+,despite the logarithmic pole, thanks to the factor (x− 1) in the sum of Legendre functions,

Q2(x)− 2Q1(x) +Q0(x)−1

2=

1

4(x− 1)

[(3x− 1) ln

(x+ 1

x− 1

)− 6]. (38)

This permits to immediately integrate Eq. (37) over r̃ (while keeping ũ fixed) with result8

U2 = Mñpq

[2r̃−3Mpq +

3

2r̃−2M (1)pq −

∫ +∞1

dx (3x− 1) ln(x+ 1

x− 1

)M (3)pq (t̃− r̃x)

]. (39)

In principle this is valid up to an homogeneous solution corresponding to a linear gaugetransformation starting to order G2. It will be of the form −ξu2 = −f2 where f2 is a function

8Note that t̃− r̃x = ũ− r̃(x− 1) and ∂r̃M (3)[ũ− r̃(x− 1)]∣∣ũ=const

= −(x− 1)M (4)[ũ− r̃(x− 1)].

15

of ũ = t̃− r̃ and θ̃a. It thus takes the same form as the linear gauge transformation alreadyintroduced to order G in Eq. (11a). Hence, we can absorb f2 into the redefinition of fthrough the replacement f → f + Gf2, and the solution (39) is the most general in oursetup. Following the same procedure outlined above to compute U2, we obtain

R2 = Mñpq

[r̃−2M (1)pq +

9

2r̃−1M (2)pq − 3

∫ +∞1

dx ln

(x+ 1

x− 1

)M (3)pq (t̃− r̃x)

], (40a)

Θa2 =Mñpẽ

aq

r̃

[r̃−3Mpq +

2

3r̃−2M (1)pq + 2r̃

−1M (2)pq + 2

∫ +∞1

dx (x− 1) ln(x+ 1

x− 1

)M (3)pq (t̃− r̃x)

].

(40b)

Having determined U2, R2 and Θa2 we continue our algorithm and successively obtain

the contravariant components grr, gra and gab of the NU metric, and then its covariantcomponents guu, gua and gab, see Sec. 2.2. We consistently keep only the terms correspondingto the mass-quadrupole M×Mij interaction. In the end we recall that we have to express themetric components in terms of the NU coordinates xµ = (u, r, θa) by applying (the inverseof the) coordinate transformation (5). In order to present the result in the best way weintroduce the following tail-modified quadrupole moment as defined by [56]9

M radij (u) = Mij(u) + 2GM

∫ +∞0

dz

[ln( z

2P

)+

11

12

]M

(2)ij (u− z) +O

(G2). (41)

Such definition agrees with the expression of the radiative quadrupole moment parametrizingthe leading r−1 piece of the metric at future null infinity. Restoring the powers of c−1 we seethat the tail provides a 1.5PN correction ∼ c−3 to the quadrupole. Generally the radiativequadrupole moment is rather defined as the second-time derivative of M radij , see Eq. (76a)of [56]. But here, as we not only control the leading term r−1 but also all the subleadingterms r−2, r−3, etc. in the expansion of the metric at infinity, it will turn out to be betterto define the radiative moment simply as M radij .

We find that the NU metric guu to quadratic orderG2 for the mass-quadrupole interaction,

including all terms in the expansion at infinity, reads

guu = −1 +G[2Mr−1 + 6r−1nij

(2)

Mradij + 6r

−2nij(1)

Mradij + 3r

−3nijM radij

](42)

+3

2G2Mr−3nij

[(1)

Mradij + r

−2∫ +∞

0

dzM radij (u− z)(

1 + z2r

)2 ]+O (G3) .We recover Eq. (15a) for the linear part, and we see that to quadratic order the tails nicelyenter the metric only through the replacement of the canonical moment Mij by the radiativemoment M radij defined by Eq. (41). In fact, with this approximation (neglecting G

3 terms),we can use either Mij or M

radij in the second line of Eq. (42).

9We have changed the integration variable to z = r(x − 1). In previous formulæ, it is convenient todecompose ln(x+1x−1 ) = −ln(

z2P ) + ln(1 +

z2r ) + ln(

rP ), where P is the constant introduced in Eq. (11). The

first term gives the tail in the quadrupole (41), the second term gives the tail in the metric (42)–(44) andthe third term is cancelled after reexpressing the metric in NU coordinates.

16

Note that the last term of Eq. (42), involving a time integral over the radiative moment,is “exact” all over the exterior region of the source. The integral is convergent under ourassumption of stationarity in the past. Furthermore, this term is of order O(r−4) at nullinfinity where it admits an expansion involving only powers of r−1. We have the regularexpansion when r → +∞ for u = const:∫ +∞

0

dzM radij (u− z)

(1 + z2r

)2=

+∞∑p=0

(−)p(p+ 1)(2r)p

∫ u+T0

dz zpM radij (u− z) +4r2M radij (−T )2r + u+ T

(43)

=+∞∑p=0

(−)p(p+ 1)(2r)p

∫ +∞0

dz zp[M radij (u− z)−M radij (−T )

]+ 2rM radij (−T ) ,

where −T is the finite instant in the remote past before which the multipoles are constant.Further processing we obtain the other components of the NU metric as

gua = Geian

j

{−2

(2)

Mradij + 2r

−1(εijkSk + 2

(1)

Mradij

)+ 3r−2M radij (44a)

+1

2GM

[3r−2

(1)

Mradij + r

−4∫ +∞

0

dz5 + 3z

2r(1 + z

2r

)3 M radij (u− z)]}+O (G3) ,gab = r

2

[γab + 2Ge

i〈ae

jb〉

(r−1

(2)

Mradij + r

−3M radij

)(44b)

+G2Mei〈aejb〉

(r−3

(1)

Mradij +

1

4r−5∫ +∞

0

dz18 + 8z

r+ z

2

r2(1 + z

2r

)4 M radij (u− z))]

+O(G3).

Again we find some remaining tail integrals, but which rapidly fall off when r → ∞ andadmit an expansion in simple powers of r−1. Finally we conclude that the expansion ofthe NU metric at infinity is regular, without the powers of ln r which plague the expansionof the metric in harmonic coordinates. In intermediate steps of the computation, however,logarithmic divergences occur in the quadratic term, but they are cancelled by the expansionof the linear term taking into account ũ = u+ 2GM ln(r/P) +O(G2).

The fact that the NU metric admits a regular (smooth) expansion when r → +∞ toall orders, without logarithms, is nicely consistent with the earlier work [36] which provedthe property of asymptotic simplicity in the sense of Geroch and Horowitz [61], i.e., witha smooth conformal boundary at null infinity, for the large class of radiative coordinatesystems, containing the Bondi and NU coordinates. Indeed, a crucial assumption in theproof of [36] as well as in our work, see Eq. (43), is that the metric is stationary in the past(for u 6 −T ).

To second order in G, as already commented, we could still add to the construction somearbitrary homogeneous solutions of the equations for U2, R2 and Θ

a2, but the corresponding

terms in the metric will have exactly the same form as those found to linear order in G, seeEqs. (16), and shown to describe with appropriate boundary conditions the modification ofthe metric under the BMS group.

From the results (42)–(44), one can easily deduce the mass and angular momentumaspects m and Na, and the Bondi shear Cab, for the case of the mass-quadrupole interaction

17

to order G2. As expected the Bondi data are entirely determined by the radiative quadrupolemoment (41). Recalling the expression of the metric in the NU gauge as given by Eqs. (62)and (65) in Appendix A, where Na is defined according to the convention of [20], we get

m = M + 3nij(2)

Mradij +O

(G2), (45a)

Na = 3eian

j(εijkSk + 2

(1)

Mradij

)+O

(G2), (45b)

Cab = 2ei〈ae

jb〉

(2)

Mradij +O

(G2). (45c)

We have added in the angular momentum aspect the linear contribution due to the totalconstant (ADM) angular momentum or spin Si, as read off from Eq. (25b).

Notice that the difference between the Newman-Bondi and Bondi radii is a term quadraticin Cab, see Eq. (61). This term is thus quadratic in the source moment Mij, and so, for themass-quadrupole interactionM×Mij considered in this section, there is no difference betweenthe NU and Bondi gauges.

In the stationary limit, the Bondi mass and angular momentum aspects as well as theshear (45) reduce to their linear expressions. Moreover, the radiative quadrupole M radijas defined in Eq. (41) reduces to the canonical one Mij. More generally, it follows fromdimensional analysis that no perturbative non-linear correction exists to the Bondi data orto the multipole moments in the stationary case. Indeed, suppose a non-linear correctionto the moment ML, built from n moments ML1 , · · · , MLn . In the stationary case thiscorrection must be of the type ∼ Gn−1

c2n−2ML1 · · ·MLn with ` = n− 1 +

∑`i in order to match

the dimension. Furthermore, we must also have∑`i = ` + 2k for the correspondence

of indices, where k is the number of contractions among the indices L1 · · ·Ln. The twoconditions are clearly incompatible. This entails that the canonical multipoles ML, SL agreewith the Geroch-Hansen multipoles [57,58] at the non-linear level.

We can in principle generalize the latter results to multipole interactions M ×ML andM ×SL (with any ` > 2), starting from the known expressions of tail terms in the metric inharmonic coordinates:10

h002 = 16MñL`!

∫ +∞1

dxQ`(x)M(`+2)L (t̃− r̃x) + · · · , (46a)

h0i2 = 16MñL−1`!

∫ +∞1

dx

[Q`−1(x)M

(`+2)iL−1 −

`

`+ 1Q`(x) εipq ñp S

(`+2)qL−1

]+ · · · , (46b)

hij2 = 16MñL−2`!

∫ +∞1

dx

[Q`−2(x)M

(`+2)ijL−2 −

2`

`+ 1Q`−1(x) ñp εpq(iS

(`+2)j)qL−2

]+ · · · . (46c)

Here the ellipsis refer to many non-tail contributions, in the form of instantaneous (i.e.,local-in-time) terms depending on the multipole moments only at time ũ. Considering theprevious results we can conjecture that the mass and angular momentum aspects will takethe same form as in Eqs. (25) but with the canonical moments ML and SL replaced by theradiative moments M radL and S

radL [62]

M radL (u) = ML(u) + 2GM

∫ +∞0

dz[ln( z

2P

)+ κ`

]M

(2)L (u− z) +O

(G2), (47a)

10This is a straightforward generalization of the mass quadrupole tail terms in Eq. (34).

18

SradL (u) = SL(u) + 2GM

∫ +∞0

dz[ln( z

2P

)+ π`

]S

(2)L (u− z) +O

(G2), (47b)

where the constants are given by (with H` =∑`

j=11j)

κ` =2`2 + 5`+ 4

`(`+ 1)(`+ 2)+H`−2 , π` =

`− 1`(`+ 1)

+H`−1 . (48)

More work would be needed to generalize our algorithm in order to include any multipoleinteractions M ×ML and M × SL (especially instantaneous ones).

4.2 Mass and angular momentum losses

Taking the angular average of the mass aspect m we obtain the Bondi mass MB ≡∫

dΩ4πm.

At this stage, we find from Eqs. (45a) or (25a) that the Bondi mass just equals the ADM massMADM ≡M . This is because we have not yet included the mass loss by GW emission whicharises in this formalism from the quadratic interaction between two quadrupole moments, sayMij ×Mkl, as well as higher multipole moment interactions. The losses of mass and angularmomentum are straightforward to include in the formalism, starting from the known resultsin harmonic coordinates.

The terms responsible for mass and angular momentum losses (at the lowest quadrupole-quadrupole interaction level) in the harmonic-coordinate metric are (see e.g. Eq. (4.12)in [60]):

h002 =4

5r̃

∫ ũ−∞

dvM (3)pq M(3)pq (v) + · · · , (49a)

h0j2 =4

5εjpq∂̃p

(1

r̃εqrs

∫ ũ−∞

dvM(2)rt M

(3)st (v)

)+ · · · , (49b)

hjk2 = · · · , (49c)

where again, the ellipsis denote many instantaneous (local-in-time) terms, in contrast withthe non-local time anti-derivative integrals over the multipole moments in Eq. (49). Impor-tantly, the ellipsis in Eq. (49) also contain another type of non-local terms that are associ-ated with the non-linear memory effect, but which we shall not discuss here. The completequadrupole-quadrupole interaction Mij ×Mkl has been computed in harmonic coordinatesin [60], including the description of the various GW losses and the non-linear memory effect.

We thus apply our algorithm to generate the corresponding mass and angular momentumlosses in the NU metric. In this calculation we only keep track of the non-local-in-time (or“hereditary”) integrals, and neglect all the instantaneous terms. Furthermore, as we saidwe do not consider the memory effect, which is disconnected from GW losses (see e.g. [60]).Finally we are restricted to the quadrupole-quadrupole interaction, as in Eq. (49).

Looking at the second-order equations (33) we see that we are just required to solve

k̃µ∂̃µU2 =1

2k̃µk̃νh

µν2 + · · · , (50a)

k̃µ∂̃µR2 =1

2h2 + ñi

[∂̃iU2 − k̃µhµi2

]+ · · · , (50b)

19

k̃µ∂̃µΘa2 =

ẽair̃

[∂̃iU2 − k̃µhµi2

]+ · · · . (50c)

We obtain successively (changing consistently harmonic to NU coordinates)

U2 =2

5ln(r/P)

∫ u−∞

dvM (3)pq M(3)pq (v) + · · · , (51a)

R2 = · · · , (51b)

Θa2 = −2

5

eai nj

r2εijp εpqr

∫ u−∞

dvM (3)qs M(2)rs (v) + · · · . (51c)

We find no such hereditary terms in R2. The logarithmic term in U2 corrects the light conedeviation at linear order as given by Eq. (11a). The corresponding contributions in the NUmetric follow as

guu = −1−2G

5r−1∫ u−∞

dvM (3)pq M(3)pq (v) + · · · , (52a)

gua = −4G

5

eai nj

rεijp εpqr

∫ u−∞

dvM (2)qs M(3)rs (v) + · · · , (52b)

gab = r2γab

[1− 2G

5r−1∫ u−∞

dvM (3)pq M(3)pq (v)

]+ · · · . (52c)

Combining this with previous results (45a) or (25a) we obtain the mass aspect which is nowaccurate enough to include the physical GW mass loss

m = M + 3nij(2)

Mradij −

G

5

∫ u−∞

dvM (3)pq M(3)pq (v) + · · · . (53)

Hence the Bondi mass MB =∫

dΩ4πm reads (where M is the constant ADM mass)

MB = M −G

5

∫ u−∞

dvM (3)pq M(3)pq (v) + · · · . (54)

The mass loss in the right-side is characterized by the hereditary (or “semi-hereditary”)11

non-local integral, in contrast with the instantaneous contributions indicated by dots. Suchinstantaneous terms will be in the form of total time derivatives in the corresponding fluxbalance equation, and may be neglected in average over a typical orbital period for quasi-periodic systems. Thus the averaged balance equation reduces to

〈dMBdt〉 = −G

5M (3)pq M

(3)pq , (55)

which is of course nothing but (with this approximation) the balance equation correspondingto the standard Einstein quadrupole formula.

11We distinguish [38] semi-hereditary integrals that are just time anti-derivatives of products of multipolemoments as in Eq. (54), from truly hereditary integrals extending over the past, like the tail terms in Eq. (46).

20

In a similar way we obtain the angular momentum aspect and Bondi shear as

Na = 6eian

j

[1

2εijpSp+

(1)

Mradij −

G

5εijpεpqr

∫ u−∞

dvM (2)qs M(3)rs (v) + · · ·

], (56a)

Cab = 2ei〈ae

jb〉

(2)

Mradij −

2G

5γab

∫ u−∞

dvM (3)pq M(3)pq (v) + · · · . (56b)

The Bondi angular momentum is defined from the angular momentum aspect by

SBi ≡1

2εipq

∫dΩ

4πepa n

q(Na −

α

4GCabDcC

bc). (57)

As shown in [32], this quantity requires a prescription for α which is fixed to α = 1 in [14,20, 24, 28], α = 0 in [22, 63] or α = 3 in [27]. Since the α-term gives instantaneous terms aswell as higher order terms, we can simply ignore it for this computation. Hence we have

SBi = Si −2G

5εipq

∫ u−∞

dvM (2)ps M(3)qs (v) + · · · . (58)

Upon averaging this leads to the usual quadrupole balance equation for angular momentum12

〈dSBi

dt〉 = −2G

5εipqM

(2)ps M

(3)qs . (59)

Note that the discussion of the GW losses in the linear momentum (or recoil) and thecenter-of-mass position would require the coupling between the mass quadrupole and themass octupole moments, which is outside the scope of the present calculation.

5 Conclusion and perspectives

In this paper we have shown how to implement practically the transformation of the metricof an isolated matter source in the MPM (multipolar post-Minkowskian) approach fromharmonic (de Donder) coordinates to Bondi-like NU (Newman-Unti) coordinates. This is ofinterest because the asymptotic properties of radiative space-times are generally discussedwithin the Bondi-Sachs-Penrose formalism, while the connection to the source’s propertiesis done by a matching procedure to the source using the MPM expansion.

In particular we obtain explicit expressions for the NU metric valid at any order in theradial distance to the source (while staying outside the domain of the source), expressedin terms of the canonical mass and current multipole moments. Under the assumptionof stationarity in the remote past, we prove that the NU metric (for particular multipolemoment couplings) admits a regular expansion at future null infinity. This is consistent withthe fact that the MPM expansion satisfies the property of asymptotic simplicity [36].

12The angular momentum aspect itself satisfies, see also Eq. (26),

dNadt

= Dam+ 3eia εipqnp

dSBqdt

+ · · · .

21

On the other hand the canonical moments are known in terms of the source’s parametersto high PN (post-Newtonian) order. Our approach permits to rewrite explicit results de-rived in harmonic coordinates using the MPM approximation into the Bondi-Sachs-Penroseformalism for the asymptotic structure, including the notions of Bondi shear, and mass andangular momentum aspects. In particular, we recover from our construction the generalizedBMS (Bondi-van der Burg-Metzner-Sachs) residual symmetry group leaving invariant theNU metric under appropriate boundary conditions at future null infinity.13

To non-linear order our construction is in principle valid for any coupling between thecanonical moments. In this paper we have worked out the coupling between the mass and thequadrupole, including the contributions due to non-local (hereditary) tail effects but also alllocal (instantaneous) terms. Including the non-local (semi-hereditary) terms arising from thecoupling between two quadrupoles, we obtain the mass and angular momentum losses dueto the GW emission through the expressions of the mass and angular momentum aspects.However we ignored all the instantaneous terms in the quadrupole-quadrupole metric, aswell as the contributions from the non-linear memory effect. In future work we intendto thoroughly investigate the quadrupole-quadrupole interaction in our framework, and inparticular discuss the occurrence of the non-linear memory effect, thereby contrasting theperspective from approximation methods in harmonic coordinates with that from asymptoticstudies in Bondi-like coordinates confined close to future null infinity.

Acknowledgments R.O. and A.S. are grateful to Bernard Whiting for enlightening dis-cussions on related topics. G.C. acknowledges Y. Herfray and A. Puhm for interestingdiscussions. G.F., R.O. and A.S. would like to thank the Munich Institute for Astro-and Particle Physics (MIAPP), which is funded by the Deutsche Forschung-sgemeinschaft(DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2094 –390783311, for giving them the opportunity of preliminary discussions that triggered thecurrent project. R.O. and A.S. thank the Institut d’Astrophysique de Paris for the hos-pitality when this work was initiated and the COST Action GWverse CA16104 for par-tial financial support. R.O. is funded by the European Structural and Investment Funds(ESIF) and the Czech Ministry of Education, Youth and Sports (MSMT), Project CoGraDS- CZ.02.1.01/0.0/0.0/15003/0000437. A.S. receives funding from the European Union’s Hori-zon 2020 research and innovation program under the Marie Sklodowska-Curie grant agree-ment No 801505. G.C. is Senior Research Associate from the Fonds de la Recherche Sci-entifique F.R.S.-FNRS (Belgium) and he acknowledges support from the FNRS researchcredit J.0036.20F, bilateral Czech convention PINT-Bilat-M/PGY R.M005.19 and the IISNconvention 4.4503.15.

A Map between Bondi and Newman-Unti gauges

Bondi gauge and Newman-Unti gauge differ by a choice of radial coordinate [8]. They bothadmit identical asymptotic symmetry groups, phase spaces and physical quantities [8]. We

13By contrast, harmonic coordinates are preserved by a distinct residual symmetry group which includesthe Poincaré group as well as multipole symmetries whose associated Noether charges are the canonicalmultipole moments [64].

22

denote in both coordinate systems the angular coordinates as θa and the coordinate labellingthe foliation of null hypersurfaces as u. Let us refer to rB as the Bondi radius and rNU as theNewman-Unti radius. The Newman-Unti radius rNU is the affine parameter along the outgo-ing null rays, while the Bondi radius is the luminosity distance such that ∂rB [det(gab)/r

4B] = 0.

There are certain advantages of NU coordinates over the Bondi coordinates, in particularthe bulk extension of NU is larger than Bondi [65]. The relationship between the radii isgiven by [8]

rNU = rB +

∫ ∞rB

dr′(grBu + 1

), rB =

(det gabdet γab

)1/4. (60)

For large radii, we have

rNU = rB +1

16rBCabC

ab +O(r−2B ) , (61a)

rB = rNU −1

16rNUCabC

ab +O(r−2NU) . (61b)

The deviation only starts from order 1/rB or 1/rNU. We deduce that Cab and m can be readoff from the metric in Newman-Unti gauge as

gNUuu = −1 +2mNUrNU

+O(r−2NU) , (62a)

gNUab = r2NUγab + rNUCab +O(r0NU) , (62b)

with mNU = m+116∂u(CabC

ab). Instead,

gNUua = gBua +

1

16rDa(CbcC

bc) +O(r−2) , (63)

where r is either rB or rNU. In the convention of [20], the angular momentum aspect Na isread in Bondi gauge from

gBua =1

2DbCab +

1

r

[2

3Na −

1

16Da(CbcC

bc)]

+O(r−2) . (64)

We deduce from Eq. (63) that it is read in Newman-Unti gauge from

gNUua =1

2DbCab +

2

3rNa +O(r−2) . (65)

B Equations for any PM order

At any given PM order p ∈ N, the NU gauge conditions (8) imply the following equationsfor Up, Rp and Θ

ap, respectively,

k̃µ∂̃µUp =1

2k̃µk̃νh

µνp +

∑m,n>1m+n=p

(12∂̃µUm − k̃νhµνm

)∂̃µUn +

1

2

∑m,n,q>1m+n+q=p

(∂̃νUm)(∂̃µUn)hµνq , (66a)

23

k̃µ∂̃µRp =∑m>1

m+n=p

(12

m

)[∑n≥1

|g̃|n]m

+ ñi

[∂̃iUp − k̃µhµip +

∑m,n>1m+n=p

(∂̃µUn)hµim

]+

+∑m,n>1m+n=p

(∂̃µUm − k̃νhµνm

)∂̃µRn +

∑m,n,q>1m+n+q=p

(∂̃νUm)(∂̃µRn)hµνq , (66b)

k̃µ∂̃µΘap =

ẽair̃

[∂̃iUp − k̃µhµip +

∑m,n>1m+n=p

(∂̃µUn)hµim

]+∑m,n>1m+n=p

(∂̃µUm − k̃νhµνm

)∂̃µΘ

an+

+∑

m,n,q>1m+n+q=p

(∂̃νUm)(∂̃µΘan)h

µνq . (66c)

To derive the equation for Rp, one formally writes

|g̃| = 1 +∑n>1

Gn|g̃|n −→√|g̃| =

∑m>0

(12

m

)[∑n>1

Gn(|g̃|)n]m

. (67)

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http://arXiv.org/abs/1310.1528http://arXiv.org/abs/gr-qc/9710037http://arXiv.org/abs/gr-qc/9501030http://arXiv.org/abs/1502.06120http://arXiv.org/abs/1711.08806http://arXiv.org/abs/1609.01731

1 Introduction1.1 Motivations1.2 Notation and conventions

2 From harmonic gauge to Newman-Unti gauge2.1 Linear metric in harmonic coordinates2.2 Algorithm to transform harmonic to NU metrics

3 Newman-Unti metric to linear order3.1 Solving the NU gauge conditions3.2 Boundary conditions and the BMS group3.3 Bondi data to linear order

4 Newman-Unti metric to quadratic order4.1 Tails and the mass-quadrupole interaction4.2 Mass and angular momentum losses

5 Conclusion and perspectivesA Map between Bondi and Newman-Unti gaugesB Equations for any PM order