3 Asymptotic Gravitational Waveform
3.1 The radiative multipole moments
The leading-order term of the metric in radiative coordinates
as given in Theorem 4,
neglecting
, yields the operational definition of two sets of STF radiative multipole moments,
mass-type
and current-type
. As we have seen, radiative coordinates are such that the
retarded time
becomes asymptotically a null coordinate at future null infinity. The radiative
moments are defined from the spatial components
of the metric in a transverse-traceless (TT) radiative
coordinate system. By definition, we have [403*]
We have formally re-summed the whole post-Minkowskian series in Eq. (56*) from up to
.
As before we denote for instance
and so on, where
and
.
The TT algebraic projection operator
has already been defined at the occasion of the
quadrupole-moment formalism in Eq. (2*); and obviously the multipole decomposition (66) represents the
generalization of the quadrupole formalism. Notice that the meaning of Eq. (66) is for the
moment rather empty, because we do not yet know how to relate the radiative moments to
the actual source parameters. Only at the Newtonian level do we know this relation, which is



Next we introduce two unit polarization vectors and
, orthogonal and transverse to the
direction of propagation
(hence
). Our convention for the choice of
and
will be clarified in Section 9.4. Then the two “plus” and “cross” polarization states of the
asymptotic waveform are defined by
Although the multipole decomposition (66) is completely general, it will also be important, having
in view the comparison between the post-Newtonian and numerical results (see for instance
Refs. [107*, 34, 237, 97, 98*]), to consider separately the various modes of the asymptotic
waveform as defined with respect to a basis of spin-weighted spherical harmonics of weight
. Those
harmonics are function of the spherical angles
defining the direction of propagation
, and given
by
where and
. We thus decompose
and
onto the basis of such spin-weighted spherical harmonics, which means (see e.g., [107, 272*])






Here denotes the STF tensor connecting together the usual basis of spherical harmonics
to the
set of STF tensors
(where the brackets indicate the STF projection). Indeed both
and
are basis of an irreducible representation of weight
of the rotation group; the two basis are related
by22
In Section 9.5 we shall present all the modes of gravitational waves from inspiralling compact
binaries up to 3PN order, and even 3.5PN order for the dominant mode
.
3.2 Gravitational-wave tails and tails-of-tails
We learned from Theorem 4 the general method which permits the computation of the radiative multipole
moments ,
in terms of the source moments
, or in terms of the intermediate
canonical moments
,
discussed in Section 2.4. We shall now show that the relation
between
,
and
,
(say) includes tail effects starting at the relative 1.5PN
order.
Tails are due to the back-scattering of multipolar waves off the Schwarzschild curvature generated by
the total mass monopole of the source. They correspond to the non-linear interaction
between
and the multipole moments
and
, and are given by some non-local
integrals, extending over the past history of the source. At the 1.5PN order we find [59*, 44*]
where is the length scale introduced in Eq. (42*), and the constants
and
are given by
Recall from the gauge vector found in Eq. (58*) that the retarded time
in
radiative coordinates is related to the retarded time
in harmonic coordinates by

















The tail integrals in Eqs. (76) involve all the instants from in the past up to the
current retarded time
. However, strictly speaking, they do not extend up to infinite past,
since we have assumed in Eq. (29*) that the metric is stationary before the date
. The
range of integration of the tails is therefore limited a priori to the time interval
.
But now, once we have derived the tail integrals, thanks to the latter technical assumption of
stationarity in the past, we can argue that the results are in fact valid in more general situations for
which the field has never been stationary. We have in mind the case of two bodies moving
initially on some unbound (hyperbolic-like) orbit, and which capture each other, because of the
loss of energy by gravitational radiation, to form a gravitationally bound system around time
.
In this situation let us check, using a simple Newtonian model for the behaviour of the
multipole moment when
, that the tail integrals, when assumed to
extend over the whole time interval
, remain perfectly well-defined (i.e., convergent) at
the integration bound
. Indeed it can be shown [180] that the motion of initially
free particles interacting gravitationally is given by
,
where
,
and
denote constant vectors, and
when
. From
that physical assumption we find that the multipole moments behave when
like






To obtain the results (76), we must implement in details the post-Minkowskian algorithm presented in
Section 2.3. Let us flash here some results obtained with such algorithm. Consider first the case of the
interaction between the constant mass monopole moment (or ADM mass) and the time-varying
quadrupole moment
. This coupling will represent the dominant non-static multipole interaction in
the waveform. For these moments we can write the linearized metric using Eq. (35*) in which by definition
of the “canonical” construction we insert the canonical moments
in place of
(notice that
). We must plug this linearized metric into the quadratic-order part
of the gravitational
source term (24) – (25*) and explicitly given by Eq. (26). This yields many terms; to integrate
these following the algorithm [cf. Eq. (45*)], we need some explicit formulas for the retarded
integral of an extended (non-compact-support) source having some definite multipolarity
. A
thorough account of the technical formulas necessary for handling the quadratic and cubic
interactions is given in the Appendices of Refs. [50*] and [48*]. For the present computation the most
crucial formula, needed to control the tails, corresponds to a source term behaving like
:



The metric is composed of two types of terms: “instantaneous” ones depending on the values of the
quadrupole moment at the retarded time , and “hereditary” tail integrals, depending on all
previous instants
.
Let us investigate now the cubic interaction between two mass monopoles with the mass
quadrupole
. Obviously, the source term corresponding to this interaction will involve
[see Eq. (40b)] cubic products of three linear metrics, say
, and quadratic
products between one linear metric and one quadratic, say
and
.
The latter case is the most tricky because the tails present in
, which are given
explicitly by Eqs. (84), will produce in turn some tails of tails in the cubic metric
.
The computation is rather involved [48*] but can now be performed by an algebraic computer
programme [74*, 197*]. Let us just mention the most difficult of the needed integration formulas for this
calculation:25
where is the time anti-derivative of
. With this formula and others given in Ref. [48*] we are
able to obtain the closed algebraic form of the cubic metric for the multipole interaction
,
at the leading order when the distance to the source
with
. The result
is26
where all the moments are evaluated at the instant
. Notice that the
logarithms in Eqs. (86) contain either the ratio
or
. We shall discuss in Eqs. (93*) – (94*) below
the interesting fate of the arbitrary constant
.
From Theorem 4, the presence of logarithms of in Eqs. (86) is an artifact of the harmonic
coordinates
, and it is convenient to gauge them away by introducing radiative coordinates
at
future null infinity. For controling the leading
term at infinity, it is sufficient to take into account the
linearized logarithmic deviation of the light cones in harmonic coordinates:
,
where
is the gauge vector defined by Eq. (58*) [see also Eq. (78*)]. With this coordinate change one
removes the logarithms of
in Eqs. (86) and we obtain the radiative (or Bondi-type [93])
logarithmic-free expansion
where the moments are evaluated at time . It is trivial to compute the contribution
of the radiative moments corresponding to that metric. We find the “tail of tail” term which will be
reported in Eq. (91) below.
3.3 Radiative versus source moments
We first give the result for the radiative quadrupole moment expressed as a functional of the
intermediate canonical moments
,
up to 3.5PN order included. The long calculation follows from
implementing the explicit MPM algorithm of Section 2.3 and yields various types of terms:
- The instantaneous (i.e., non-hereditary) piece
up to 3.5PN order reads
The Newtonian term in this expression contains the Newtonian quadrupole moment
and recovers the standard quadrupole formalism [see Eq. (67*)];
- The hereditary tail integral
is made of the dominant tail term at 1.5PN order in agreement with Eq. (76a) above:
The length scale
is the one that enters our definition of the finite-part operation
[see Eq. (42*)] and it enters also the relation between the radiative and harmonic retarded times given by Eq. (78*);
- The hereditary tail-of-tail term appears dominantly at 3PN order [48*] and is issued from the radiative metric computed in Eqs. (87):
- Finally the memory-type hereditary piece
contributes at orders 2.5PN and 3.5PN and is given by
The 2.5PN non-linear memory integral – the first term inside the coefficient of – has been obtained
using both post-Newtonian methods [42, 427*, 406, 60*, 50] and rigorous studies of the field at future null
infinity [128]. The expression (92) is in agreement with the more recent computation of the non-linear
memory up to any post-Newtonian order in Refs. [189*, 192].
Be careful to note that the latter post-Newtonian orders correspond to “relative” orders when counted in the local radiation-reaction force, present in the equations of motion: For instance, the 1.5PN tail integral in Eq. (90) is due to a 4PN radiative effect in the equations of motion [58*]; similarly, the 3PN tail-of-tail integral is expected to be associated with some radiation-reaction terms occurring at the 5.5PN order.
Note that , when expressed in terms of the intermediate moments
and
, shows a
dependence on the (arbitrary) length scale
; cf. the tail and tail-of-tail contributions (90) – (91). Most
of this dependence comes from our definition of a radiative coordinate system as given by (78*). Exactly as
we have done for the 1.5PN tail term in Eq. (79*), we can remove most of the
’s by inserting
back into (89) – (92), and expanding the result when
, keeping the
necessary terms consistently. In doing so one finds that there remains a
-dependent term at the 3PN
order, namely











The previous formulas for the 3.5PN radiative quadrupole moment permit to compute the dominant
mode of the waveform up to order 3.5PN [197*]; however, to control the full waveform one has also
to take into account the contributions of higher-order radiative moments. Here we list the most accurate
results we have for all the moments that permit the derivation of the waveform up to order 3PN
[74*]:28
For all the other multipole moments in the 3PN waveform, it is sufficient to assume the agreement between the radiative and canonical moments, namely
In a second stage of the general formalism, we must express the canonical moments in
terms of the six types of source moments
. For the control of the
mode in the waveform up to 3.5PN order, we need to relate the canonical quadrupole
moment
to the corresponding source quadrupole moment
up to that accuracy. We
obtain [197*]
Here, for instance, denotes the monopole moment associated with the moment
, and
is
the dipole moment corresponding to
. Notice that the difference between the canonical and source
moments starts at the relatively high 2.5PN order. For the control of the full waveform up to 3PN order we
need also the moments
and
, which admit similarly some correction terms starting at the 2.5PN
order:
The remainders in Eqs. (98) are consistent with the 3PN approximation for the full waveform. Besides
the mass quadrupole moment (97), and mass octopole and current quadrupole moments (98), we can state
that, with the required 3PN precision, all the other moments ,
agree with their source
counterparts
,
:
With those formulas we have related the radiative moments parametrizing the
asymptotic waveform (66) to the six types of source multipole moments
.
What is missing is the explicit dependence of the source moments as functions of the actual
parameters of some matter source. We come to grips with this important question in the next
section.