-
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.
[email protected]@[email protected]@[email protected]
1
arX
iv:2
011.
1000
0v2
[gr
-qc]
4 F
eb 2
021
-
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)
References
[1] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner,
Gravitational Waves inGeneral Relativity. VII. Waves from
Axi-Symmetric Isolated Systems, Proceedings ofthe Royal Society of
London Series A 269 (Aug., 1962) 21–52.
[2] R. K. Sachs, Gravitational Waves in General Relativity.
VIII. Waves inAsymptotically Flat Space-Time, Proceedings of the
Royal Society of London Series A270 (Oct., 1962) 103–126.
[3] E. Newman and R. Penrose, An Approach to gravitational
radiation by a method ofspin coefficients, J. Math. Phys. 3 (1962)
566–578.
[4] R. Penrose, Asymptotic properties of fields and space-times,
Phys. Rev. Lett. 10(1963) 66.
[5] R. Penrose, Zero rest-mass fields including gravitation -
asymptotic behaviour, Proc.R. Soc. London, Ser. A 284 (1965)
159.
[6] H. Friedrich, Peeling or not peeling—is that the question?,
Class. Quant. Grav. 35(2018), no. 8 083001 [1709.07709].
[7] E. T. Newman and T. Unti, A class of null flat-space
coordinate systems, Journal ofMathematical Physics 4 (1963), no. 12
1467–1469.
[8] G. Barnich and P.-H. Lambert, A Note on the Newman-Unti
group and the BMScharge algebra in terms of Newman-Penrose
coefficients, J. Phys. Conf. Ser. 410(2013) 012142 [1102.0589].
24
http://arXiv.org/abs/1709.07709http://arXiv.org/abs/1102.0589
-
[9] G. Barnich and P.-H. Lambert, Asymptotic symmetries at null
infinity and localconformal properties of spin coefficients, TSPU
Bulletin 2012 (2012), no. 13 28–31[1301.5754].
[10] L. A. Tamburino and J. H. Winicour, Gravitational Fields in
Finite and ConformalBondi Frames, Phys. Rev. 150 (1966)
1039–1053.
[11] G. Barnich and C. Troessaert, Aspects of the BMS/CFT
correspondence, JHEP 05(2010) 062 [1001.1541].
[12] J. de Boer and S. N. Solodukhin, A Holographic reduction of
Minkowski space-time,Nucl. Phys. B665 (2003) 545–593
[hep-th/0303006].
[13] G. Barnich and C. Troessaert, Symmetries of asymptotically
flat 4 dimensionalspacetimes at null infinity revisited, Phys. Rev.
Lett. 105 (2010) 111103 [0909.2617].
[14] G. Barnich and C. Troessaert, BMS charge algebra, JHEP 12
(2011) 105 [1106.0213].
[15] M. Campiglia and A. Laddha, Asymptotic symmetries and
subleading soft gravitontheorem, Phys. Rev. D90 (2014), no. 12
124028 [1408.2228].
[16] M. Campiglia and A. Laddha, New symmetries for the
Gravitational S-matrix, JHEP04 (2015) 076 [1502.02318].
[17] G. Barnich and C. Troessaert, Comments on holographic
current algebras andasymptotically flat four dimensional spacetimes
at null infinity, JHEP 11 (2013) 003[1309.0794].
[18] A. Strominger, On BMS Invariance of Gravitational
Scattering, JHEP 07 (2014) 152[1312.2229].
[19] A. Strominger and A. Zhiboedov, Gravitational Memory, BMS
Supertranslations andSoft Theorems, JHEP 01 (2016) 086
[1411.5745].
[20] E. E. Flanagan and D. A. Nichols, Conserved charges of the
extendedBondi-Metzner-Sachs algebra, Phys. Rev. D95 (2017), no. 4
044002 [1510.03386].
[21] G. Compère and J. Long, Classical static final state of
collapse with supertranslationmemory, Class. Quant. Grav. 33
(2016), no. 19 195001 [1602.05197].
[22] S. W. Hawking, M. J. Perry and A. Strominger, Superrotation
Charge andSupertranslation Hair on Black Holes, JHEP 05 (2017) 161
[1611.09175].
[23] G. Barnich and C. Troessaert, Finite BMS transformations,
JHEP 03 (2016) 167[1601.04090].
[24] G. Compère, A. Fiorucci and R. Ruzziconi, Superboost
transitions, refraction memoryand super-Lorentz charge algebra,
JHEP 11 (2018) 200 [1810.00377]. [Erratum:JHEP 04, 172 (2020)].
25
http://arXiv.org/abs/1301.5754http://arXiv.org/abs/1001.1541http://arXiv.org/abs/hep-th/0303006http://arXiv.org/abs/0909.2617http://arXiv.org/abs/1106.0213http://arXiv.org/abs/1408.2228http://arXiv.org/abs/1502.02318http://arXiv.org/abs/1309.0794http://arXiv.org/abs/1312.2229http://arXiv.org/abs/1411.5745http://arXiv.org/abs/1510.03386http://arXiv.org/abs/1602.05197http://arXiv.org/abs/1611.09175http://arXiv.org/abs/1601.04090http://arXiv.org/abs/1810.00377
-
[25] D. A. Nichols, Spin memory effect for compact binaries in
the post-Newtonianapproximation, Phys. Rev. D95 (2017), no. 8
084048 [1702.03300].
[26] D. A. Nichols, Center-of-mass angular momentum and memory
effect inasymptotically flat spacetimes, Phys. Rev. D98 (2018), no.
6 064032 [1807.08767].
[27] B. Bonga and E. Poisson, Coulombic contribution to angular
momentum flux ingeneral relativity, Phys. Rev. D99 (2019), no. 6
064024 [1808.01288].
[28] J. Distler, R. Flauger and B. Horn, Double-soft graviton
amplitudes and the extendedBMS charge algebra, JHEP 08 (2019) 021
[1808.09965].
[29] G. Barnich, P. Mao and R. Ruzziconi, BMS current algebra in
the context of theNewman–Penrose formalism, Class. Quant. Grav. 37
(2020), no. 9 095010[1910.14588].
[30] A. Ashtekar, T. De Lorenzo and N. Khera, Compact binary
coalescences: Constraintson waveforms, Gen. Rel. Grav. 52 (2020),
no. 11 107 [1906.00913].
[31] A. Ashtekar, T. De Lorenzo and N. Khera, Compact binary
coalescences: The subtleissue of angular momentum, Phys. Rev. D 101
(2020), no. 4 044005 [1910.02907].
[32] G. Compère, R. Oliveri and A. Seraj, The Poincaré and BMS
flux-balance laws withapplication to binary systems, JHEP 10 (2020)
116 [1912.03164].
[33] T. He, V. Lysov, P. Mitra and A. Strominger, BMS
supertranslations and Weinberg’ssoft graviton theorem, JHEP 05
(2015) 151 [1401.7026].
[34] D. Kapec, V. Lysov, S. Pasterski and A. Strominger,
Semiclassical Virasoro symmetryof the quantum gravity S-matrix,
JHEP 08 (2014) 058 [1406.3312].
[35] L. Blanchet and T. Damour, Radiative gravitational fields
in general relativity I.general structure of the field outside the
source, Phil. Trans. Roy. Soc. Lond. A320(1986) 379–430.
[36] L. Blanchet, Radiative gravitational fields in general
relativity. 2. Asymptoticbehaviour at future null infinity, Proc.
Roy. Soc. Lond. A409 (1987) 383–399.
[37] L. Blanchet and T. Damour, Tail Transported Temporal
Correlations in the Dynamicsof a Gravitating System, Phys. Rev. D37
(1988) 1410.
[38] L. Blanchet and T. Damour, Hereditary effects in
gravitational radiation, Phys. Rev.D46 (1992) 4304–4319.
[39] K. S. Thorne, Multipole Expansions of Gravitational
Radiation, Rev. Mod. Phys. 52(1980) 299–339.
[40] L. Blanchet, On the multipole expansion of the
gravitational field, Class. Quant. Grav.15 (1998) 1971–1999
[gr-qc/9801101].
26
http://arXiv.org/abs/1702.03300http://arXiv.org/abs/1807.08767http://arXiv.org/abs/1808.01288http://arXiv.org/abs/1808.09965http://arXiv.org/abs/1910.14588http://arXiv.org/abs/1906.00913http://arXiv.org/abs/1910.02907http://arXiv.org/abs/1912.03164http://arXiv.org/abs/1401.7026http://arXiv.org/abs/1406.3312http://arXiv.org/abs/gr-qc/9801101
-
[41] O. Poujade and L. Blanchet, Post-newtonian approximation
for isolated systemscalculated by matched asymptotic expansions,
Phys. Rev. D 65 (2002) 124020[gr-qc/0112057].
[42] L. Blanchet, G. Faye and S. Nissanke, Structure of the
post-newtonian expansion ingeneral relativity, Phys. Rev. D 72
(2005) 044024.
[43] L. Blanchet, G. Faye, B. R. Iyer and B. Joguet,
Gravitational-wave inspiral of compactbinary systems to 7/2
post-newtonian order, Phys. Rev. D 65 (2002)
061501(R)[gr-qc/0105099]. Erratum Phys. Rev. D, 71:129902(E),
2005.
[44] L. Blanchet, T. Damour, G. Esposito-Farèse and B. R. Iyer,
Gravitational radiationfrom inspiralling compact binaries completed
at the third post-newtonian order, Phys.Rev. Lett. 93 (2004) 091101
[gr-qc/0406012].
[45] L. Blanchet, G. Faye, B. R. Iyer and S. Sinha, The Third
post-Newtonian gravitationalwave polarisations and associated
spherical harmonic modes for inspiralling compactbinaries in
quasi-circular orbits, Class. Quant. Grav. 25 (2008) 165003
[0802.1249].[Erratum: Class.Quant.Grav. 29, 239501 (2012)].
[46] G. Faye, L. Blanchet and B. R. Iyer, Non-linear multipole
interactions andgravitational-wave octupole modes for inspiralling
compact binaries to third-and-a-halfpost-Newtonian order, Class.
Quant. Grav. 32 (2015), no. 4 045016 [1409.3546].
[47] J. Madore, Gravitational radiation from a bounded source.
i, Ann. Inst. Henri Poincaré12 (1970) 285.
[48] M. Campiglia and A. Laddha, Subleading soft photons and
large gaugetransformations, JHEP 11 (2016) 012 [1605.09677].
[49] M. Campiglia and J. Peraza, Generalized BMS charge algebra,
Phys. Rev. D 101(2020), no. 10 104039 [2002.06691].
[50] G. Compère, A. Fiorucci and R. Ruzziconi, The Λ-BMS4
charge algebra, JHEP 10(2020) 205 [2004.10769].
[51] L. Blanchet, Gravitational wave tails of tails, Class.
Quant. Grav. 15 (1998) 113–141[gr-qc/9710038]. [Erratum: Class.
Quant. Grav.22,3381(2005)].
[52] L. Blanchet and G. Schäfer, Gravitational wave tails and
binary star systems, Class.Quant. Grav. 10 (1993) 2699–2721.
[53] L. Blanchet and G. Faye, Flux-balance equations for linear
momentum andcenter-of-mass position of self-gravitating
post-Newtonian systems, Class. Quant.Grav. 36 (2019), no. 8 085003
[1811.08966].
[54] C. Troessaert, The BMS4 algebra at spatial infinity,
1704.06223.
[55] M. Henneaux and C. Troessaert, BMS Group at Spatial
Infinity: the Hamiltonian(ADM) approach, JHEP 03 (2018) 147
[1801.03718].
27
http://arXiv.org/abs/gr-qc/0112057http://arXiv.org/abs/gr-qc/0105099http://arXiv.org/abs/gr-qc/0406012http://arXiv.org/abs/0802.1249http://arXiv.org/abs/1409.3546http://arXiv.org/abs/1605.09677http://arXiv.org/abs/2002.06691http://arXiv.org/abs/2004.10769http://arXiv.org/abs/gr-qc/9710038http://arXiv.org/abs/1811.08966http://arXiv.org/abs/1704.06223http://arXiv.org/abs/1801.03718
-
[56] L. Blanchet, Gravitational Radiation from Post-Newtonian
Sources and InspirallingCompact Binaries, Living Rev. Rel. 17
(2014) 2 [1310.1528].
[57] R. P. Geroch, Multipole moments. II. Curved space, J. Math.
Phys. 11 (1970)2580–2588.
[58] R. O. Hansen, Multipole moments of stationary space-times,
J. Math. Phys. 15 (1974)46–52.
[59] Y. Gürsel, Multipole moments for stationary systems: The
equivalence of theGeroch-Hansen formulation and the Thorne
formulation, General Relativity andGravitation 15 (Aug., 1983)
737–754.
[60] L. Blanchet, Quadrupole-quadrupole gravitational waves,
Class. Quant. Grav. 15(1998) 89–111 [gr-qc/9710037].
[61] R. Geroch and G. Horowitz, Asymptotically simple does not
imply asymptoticallyminkowskian, Phys. Rev. Lett. 40 (1978)
203–206.
[62] L. Blanchet, Second postNewtonian generation of
gravitational radiation, Phys. Rev.D51 (1995) 2559–2583
[gr-qc/9501030].
[63] S. Pasterski, A. Strominger and A. Zhiboedov, New
Gravitational Memories, JHEP 12(2016) 053 [1502.06120].
[64] G. Compère, R. Oliveri and A. Seraj, Gravitational
multipole moments from Noethercharges, JHEP 05 (2018) 054
[1711.08806].
[65] T. Mädler and J. Winicour, Bondi-Sachs Formalism,
Scholarpedia 11 (2016) 33528[1609.01731].
28
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