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. , 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 finite9. 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 integralanalytic 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
Let us forget about the intermediate name , and denote, from now on, the latter solution by , or, in more explicit terms,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 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.
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