## 6 Non-linear Multipole Interactions

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 [1210]
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 [4154145] 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,