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 Equation (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 Equation (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. [26],
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^{9}.
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

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 Equation (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 Equation (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 Equations (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 Equations (26, 27, 28). The previous procedure constitutes an algorithm, which could be implemented by an algebraic computer programme.http://www.livingreviews.org/lrr-2006-4 |
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