## 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 [16]
where means as given by Equation (87) in the case (of course, in Equation (96) 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 [2119]
The retarded time in radiative coordinates is denoted . The constant is the one that enters our definition of the finite-part operation (see Equation (36)). The “Newtonian” term in Equation (97) contains the Newtonian quadrupole moment (see Equation (92)). The dominant radiation tail at the 1.5PN order was computed within the present formalism in Ref. [29]. The 2.5PN non-linear memory integral - the first term inside the coefficient of - has been obtained using both post-Newtonian methods [132222132921] and rigorous studies of the field at future null infinity [71]. The other multipole interactions at the 2.5PN order can be found in Ref. [21]. Finally the “tail of tail” integral appearing at the 3PN order has been derived in this formalism in Ref. [19]. 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 Equation (97) is due to a 4PN radiative effect in the equations of motion [27]; 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. [15]. 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 Equation (51). Inserting as given by Equation (100) into Equations (98) 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 Equation (97)), but in Schwarzschild coordinates [50].

The tail integrals in Equations (97, 98) 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 Equation (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 (97), 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 Equation (96)). 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 Equation (106) represents the dominant non-static multipole interaction, that is between the mass and the quadrupole moment, and that we now compute. We apply 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 Equation (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. [21] and [19]. 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 Equation (39). Next we compute the divergence , and obtain the supplementary term by applying Equations (42). Actually, we find for this particular interaction and thus also . Following Equation (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 Equation (33)). Notably, the -terms in Equation (111) 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 (110). 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. [19]). 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. [19] 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 Equations (113) 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 Equations (113) 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 Equation (51) (see also Equation (100)). With this coordinate change one removes all the logarithms of in Equations (113). 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 Equation (97).