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 )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 computeLet 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 formulasThe 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).

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