4.1 The post-Minkowskian solution

We insert the ansatz (29) into the vacuum Einstein field equations (12, 13), i.e. with , and we equate term by term the factors of the successive powers of our book-keeping parameter . We get an infinite set of equations for each of the ’s: ,
The right-hand side of the wave equation (30) is obtained from inserting the previous iterations, up to the order , into the gravitational source term. In more details, the series of equations (30) reads
The quadratic, cubic and quartic pieces of are defined by Eq. (16).

Let us now proceed by induction. Some being given, we assume that we succeeded in constructing, from the linearized coefficient , the sequence of post-Minkowskian coefficients , , , , and from this we want to infer the next coefficient . The right-hand side of Eq. (30), , is known by induction hypothesis. Thus the problem is that of solving a wave equation whose source is given. The point is that this wave equation, instead of being valid everywhere in , is correct only outside the matter (), and it makes no sense to solve it by means of the usual retarded integral. Technically speaking, the right-hand side of Eq. (30) is composed of the product of many multipole expansions, which are singular at the origin of the spatial coordinates , and which make the retarded integral divergent at that point. This does not mean that there are no solutions to the wave equation, but simply that the retarded integral does not constitute the appropriate solution in that context.

What we need is a solution which takes the same structure as the source term , i.e. is expanded into multipole contributions, with a singularity at , and satisfies the d’Alembertian equation as soon as . Such a particular solution can be obtained, following the suggestion in Ref. [14], by means of a mathematical trick in which one first “regularizes” the source term by multiplying it by the factor , where . Let us assume, for definiteness, that is composed of multipolar pieces with maximal multipolarity . This means that we start the iteration from the linearized metric (26, 27, 28) in which the multipolar sums are actually finite. The divergences when of the source term are typically power-like, say (there are also powers of the logarithm of ), and with the previous assumption there will exist a maximal order of divergency, say . Thus, when the real part of is large enough, i.e. , the “regularized” source term is regular enough when so that one can perfectly apply the retarded integral operator. This defines the -dependent retarded integral

where the symbol stands for the retarded integral (21). It is convenient to introduce inside the regularizing factor some arbitrary constant length scale in order to make it dimensionless. Everywhere in this article we pose
The fate of the constant in a detailed calculation will be interesting to follow, as we shall see, because it provides some check that the calculation is going well. Now the point for our purpose is that the function on the complex plane, which was originally defined only when , admits a unique analytic continuation to all values of except at some integer values. Furthermore, the analytic continuation of can be expanded, when (namely the limit of interest to us) into a Laurent expansion involving in general some multiple poles. The key idea, as we shall prove, is that the finite part, or the coefficient of the zeroth power of in that expansion, represents the particular solution we are looking for. We write the Laurent expansion of , when , in the form
where , and the various coefficients are functions of the field point . When there are poles; , which depends on , refers to the maximal order of the poles. By applying the box operator onto both sides of (37), and equating the different powers of , we arrive at
As we see, the case shows that the finite-part coefficient in Eq. (37), namely , is a particular solution of the requested equation: . Furthermore, we can prove that this term, by its very construction, owns the same structure made of a multipolar expansion singular at . Let us forget about the intermediate name , and denote, from now on, the latter solution by , or, in more explicit terms,
where the finite-part symbol means the previously detailed operations of considering the analytic continuation, taking the Laurent expansion, and picking up the finite-part coefficient when . The story is not complete, however, because does not fulfill the constraint of harmonic coordinates (31); its divergence, say , is different from zero in general. From the fact that the source term is divergence-free in vacuum, (see Eq. (18)), we find instead
The factor comes from the differentiation of the regularization factor . So, is zero only in the special case where the Laurent expansion of the retarded integral in Eq. (40) does not develop any simple pole when . Fortunately, when it does, the structure of the pole is quite easy to control. We find that it necessarily consists of a solution of the source-free d’Alembertian equation, and, what is more (from its stationarity in the past), the solution is a retarded one. Hence, taking into account the index structure of , there must exist four STF-tensorial functions of the retarded time , say , , and , such that
From that expression we are able to find a new object, say , which takes the same structure as (a retarded solution of the source-free wave equation) and, furthermore, whose divergence is exactly the opposite of the divergence of , i.e. . Such a is not unique, but we shall see that it is simply necessary to make a choice for (the simplest one) in order to obtain the general solution. The formulas that we adopt are
Notice the presence of anti-derivatives, denoted, e.g., by ; there is no problem with the limit since all the corresponding functions are zero when . The choice made in Eqs. (42) is dictated by the fact that the component involves only some monopolar and dipolar terms, and that the spatial trace is monopolar: . Finally, if we pose
we see that we solve at once the d’Alembertian equation (30) and the coordinate condition (31). That is, we have succeeded in finding a solution of the field equations at the th post-Minkowskian order. By induction the same method applies to any order , and, therefore, we have constructed a complete post-Minkowskian series (29) based on the linearized approximation given by (26, 27, 28). The previous procedure constitutes an algorithm, which could be implemented by an algebraic computer programme.