6 Non-linear Multipole Interactions
We shall now show that the radiative mass-type quadrupole moment
includes a quadratic tail at
the relative 1.5PN order (or
), corresponding to the interaction of the mass
of the source
and its quadrupole moment
. This is due to the back-scattering of quadrupolar waves off
the Schwarzschild curvature generated by
. Next,
includes a so-called non-linear
memory integral at the 2.5PN order, due to the quadrupolar radiation of the stress-energy
distribution of linear quadrupole waves themselves, i.e. of multipole interactions
.
Finally, we have also a cubic tail, or “tail of tail”, arising at the 3PN order, and associated with
the multipole interaction
. The result for
is better expressed in terms of the
intermediate quadrupole moment
already discussed in Section 4.2. This moment reads [7
]
where
means
as given by Eq. (84) in the case
(of course, in Eq. (90) we need only the
Newtonian value of
). The difference between the two moments
and
is a small 2.5PN
quantity. Henceforth, we shall express many of the results in terms of the mass moments
and the
corresponding current ones
. The complete formula for the radiative quadrupole, valid through the 3PN
order, reads [12
, 10
]
The retarded time in radiative coordinates is denoted
. The constant
is the one that
enters our definition of the finite-part operation
(see Eq. (36)). The “Newtonian” term in Eq. (91)
contains the Newtonian quadrupole moment
(see Eq. (89)). The dominant radiation tail at the 1.5PN
order was computed within the present formalism in Ref. [17]. The 2.5PN non-linear memory integral -
the first term inside the coefficient of
- has been obtained using both post-Newtonian
methods [4, 154, 145] and rigorous studies of the field at future null infinity [44]. The other multipole
interactions at the 2.5PN order can be found in Ref. [12
]. Finally the “tail of tail” integral appearing at the
3PN order has been derived in this formalism in Ref. [10
]. 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. (91) is
due to a 4PN radiative effect in the equations of motion [15]; similarly, the 3PN tail-of-tail
integral is (presumably) associated with some radiation-reaction terms occuring at the 5.5PN
order.
Notice that all the radiative multipole moments, for any
, get some tail-induced contributions. They
are computed at the 1.5PN level in Appendix C of Ref. [6]. We find
where the constants
and
are given by
Recall that the retarded time
in radiative coordinates is given by
where
are harmonic coordinates; recall the gauge vector
in Eq. (51). Inserting
as given by
Eq. (94) into Eqs. (92) we obtain the radiative moments expressed in terms of source-rooted coordinates
, e.g.,
This expression no longer depends on the constant
(i.e. the
gets replaced by
).
If we now change the harmonic coordinates
to some new ones, such as, for instance, some
“Schwarzschild-like” coordinates
such that
and
, we get
where
. Therefore the constant
(and
as well) depends on the choice of
source-rooted coordinates
: For instance, we have
in harmonic coordinates (see
Eq. (91)), but
in Schwarzschild coordinates [31
].
The tail integrals in Eqs. (91, 92) involve all the instants from
in the past up to the current time
. However, strictly speaking, the integrals must not extend up to minus infinity in the past,
because we have assumed from the start that the metric is stationary before the date
; see
Eq. (19). 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 in part to the 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 bound system at our current
epoch. In this situation we can check, using a simple Newtonian model for the behaviour of the
quadrupole 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
. We regard this fact as a solid a posteriori justification
(though not a proof) of our a priori too restrictive assumption of stationarity in the past. This
assumption does not seem to yield any physical restriction on the applicability of the final
formulas.
To obtain the result (91), we must implement in details the post-Minkows-kian algorithm presented in
Section 4.1. Let us outline here this computation, limiting ourselves to the interaction between one or two
masses
and the time-varying quadrupole moment
(that is related to the source
quadrupole
by Eq. (90)). For these moments the linearized metric (26, 27, 28) reads
where the monopole part is nothing but the linearized piece of the Schwarzschild metric in harmonic
coordinates,
and the quadrupole part is
(We pose
until the end of this section.) Consider next the quadratically non-linear metric
generated by these moments. Evidently it involves a term proportional to
, the mixed
term corresponding to the interaction
, and the self-interaction term of
. Say,
The first term represents the quadratic piece of the Schwarzschild metric,
The second term in Eq. (100) represents the dominant non-static multipole
interaction, that is between the mass and the quadrupole moment, and that we now
compute.
We apply the equations (39, 40, 41, 42, 43) in Section 4. First we obtain the source for this term, viz.
where
denotes the quadratic-order part of the gravitational source, as defined by Eq. (16). To
integrate this term 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. [12]
and [10
]. For the present computation the crucial formula corresponds to a source term behaving like
:
where
is the Legendre function of the second
kind.
With the help of this and other formulas we obtain the object
given by Eq. (39). Next we compute
the divergence
, and obtain the supplementary term
by applying Eqs. (42). Actually,
we find for this particular interaction
and thus also
. Following Eq. (43), the result is
the sum of
and
, and we get
The metric is composed of two types of terms: “instantaneous” ones depending on the values of the
quadrupole moment at the retarded time
, and “non-local” or tail integrals, depending on all
previous instants
.
Let us investigate now the cubic interaction between two mass monopoles
with the quadrupole
. Obviously, the source term corresponding to this interaction reads
(see Eq. (33)). Notably, the
-terms in Eq. (105) involve the interaction between a linearized metric,
or
, and a quadratic one,
or
. So, included into these terms are the tails
present in the quadratic metric
computed previously with the result (104). These tails will
produce in turn some “tails of tails” in the cubic metric
. The rather involved computation will
not be detailed here (see Ref. [10
]). Let us just mention the most difficult of the needed integration
formulas:
where
is the time anti-derivative of
. With this formula and others given in Ref. [10
] we are
able to obtain the closed algebraic form of the metric
, at the leading order in the distance to the
source. The net result is
where all the moments
are evaluated at the instant
(recall that
). Notice that
some of the logarithms in Eqs. (107) contain the ratio
while others involve
. The indicated
remainders
contain some logarithms of
; in fact they should be more accurately written as
for some
.
The presence of logarithms of
in Eqs. (107) is an artifact of the harmonic coordinates
, and we
need to gauge them away by introducing the radiative coordinates
at future null infinity (see
Theorem 4). As it turns out, it is sufficient for the present calculation to take into account the “linearized”
logarithmic deviation of the light cones in harmonic coordinates so that
, where
is the gauge vector defined by Eq. (51) (see also Eq. (94)). With this coordinate change
one removes all the logarithms of
in Eqs. (107). Hence, we obtain the radiative metric
where the moments are evaluated at time
. It is trivial to compute the contribution of
the radiative moments
and
corresponding to that metric. We find the “tail of tail” term
reported in Eq. (91).