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4.1 The post-Minkowskian solution

We insert the ansatz (29View Equation) into the vacuum Einstein field equations (12View Equation, 13View Equation), i.e. with tab = c4/(16pG)/\ab, and we equate term by term the factors of the successive powers of our book-keeping parameter G. We get an infinite set of equations for each of the ab hn’s: A n > 2,
[]habn = /\anb[h1,h2,...,hn -1], (30) @ ham = 0. (31) m n
The right-hand side of the wave equation (30View Equation) is obtained from inserting the previous iterations, up to the order n - 1, into the gravitational source term. In more details, the series of equations (30View Equation) reads
[]hab = N ab[h ,h ], (32) 2ab 1 1 []h 3 = M ab[h1,h1,h1] + N ab[h1,h2] + N ab[h2,h1], (33) []hab = Lab[h ,h ,h ,h ] 4 a1b 1 1 1 ab ab + M [h1,h1,h2] + M [h1,h2,h1] + M [h2,h1,h1] + N ab[h2,h2] + N ab[h1,h3] + N ab[h3, h1] . .. (34)
The quadratic, cubic and quartic pieces of ab /\ are defined by Equation (16View Equation).

Let us now proceed by induction. Some n being given, we assume that we succeeded in constructing, from the linearized coefficient h1, the sequence of post-Minkowskian coefficients h2,h3,...,hn -1, and from this we want to infer the next coefficient hn. The right-hand side of Equation (30View Equation), /\ab n, 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 R3, is correct only outside the matter (r > a), and it makes no sense to solve it by means of the usual retarded integral. Technically speaking, the right-hand side of Equation (30View Equation) is composed of the product of many multipole expansions, which are singular at the origin of the spatial coordinates r = 0, 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 ab /\n, i.e. is expanded into multipole contributions, with a singularity at r = 0, and satisfies the d’Alembertian equation as soon as r > 0. Such a particular solution can be obtained, following the suggestion in Ref. [26Jump To The Next Citation Point], by means of a mathematical trick in which one first “regularizes” the source term /\ab n by multiplying it by the factor B r, where B (- C. Let us assume, for definiteness, that ab /\n is composed of multipolar pieces with maximal multipolarity lmax. This means that we start the iteration from the linearized metric (26View Equation, 27View Equation, 28View Equation) in which the multipolar sums are actually finite9. The divergences when r-- > 0 of the source term are typically power-like, say 1/rk (there are also powers of the logarithm of r), and with the previous assumption there will exist a maximal order of divergency, say kmax. Thus, when the real part of B is large enough, i.e. R (B) > kmax - 3, the “regularized” source term B ab r /\n is regular enough when r --> 0 so that one can perfectly apply the retarded integral operator. This defines the B-dependent retarded integral

ab -1[ B ab ] I (B) =_ [] ret r /\n , (35)
where the symbol [] -r1et stands for the retarded integral (21View Equation). It is convenient to introduce inside the regularizing factor some arbitrary constant length scale r 0 in order to make it dimensionless. Everywhere in this article we pose
r r =_ --. (36) r0
The fate of the constant r0 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 Iab(B) on the complex plane, which was originally defined only when R (B) > k - 3 max, admits a unique analytic continuation to all values of B (- C except at some integer values. Furthermore, the analytic continuation of ab I (B) can be expanded, when B --> 0 (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 B in that expansion, represents the particular solution we are looking for. We write the Laurent expansion of Iab(B), when B --> 0, in the form
+ sum oo Iab(B) = iapb Bp, (37) p=p0
where p (- Z, and the various coefficients ab ip are functions of the field point (x,t). When p0 < - 1 there are poles; - p0, which depends on n, refers to the maximal order of the poles. By applying the box operator onto both sides of Equation (37View Equation), and equating the different powers of B, we arrive at
p0 < p < - 1 ===> []iapb = 0, p p > 0 ===> []iab = (ln-r) /\ab. (38) p p! n
As we see, the case p = 0 shows that the finite-part coefficient in Equation (37View Equation), namely ab i0, is a particular solution of the requested equation: ab []i 0 = /\anb. Furthermore, we can prove that this term, by its very construction, owns the same structure made of a multipolar expansion singular at r = 0.

Let us forget about the intermediate name iab 0, and denote, from now on, the latter solution by ab ab u n =_ i0, or, in more explicit terms,

ab -1[ B ab ] un = FPB=0 [] ret r /\n , (39)
where the finite-part symbol F P B=0 means the previously detailed operations of considering the analytic continuation, taking the Laurent expansion, and picking up the finite-part coefficient when B --> 0. The story is not complete, however, because ab un does not fulfill the constraint of harmonic coordinates (31View Equation); its divergence, say wan = @muamn, is different from zero in general. From the fact that the source term is divergence-free in vacuum, @m/\am = 0 n (see Equation (18View Equation)), we find instead
[ ] a -1 B ni ai w n = F PB=0 [] ret B r r /\ n . (40)
The factor B comes from the differentiation of the regularization factor B r. So, a w n is zero only in the special case where the Laurent expansion of the retarded integral in Equation (40View Equation) does not develop any simple pole when B --> 0. 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 wan, there must exist four STF-tensorial functions of the retarded time u = t - r/c, say NL(u), PL(u), QL(u) and RL(u), such that
+ oo 0 sum [ - 1 ] w n = @L r NL(u) , l=0 (41) + sum oo [ ] sum + oo { [ ] [ ]} win = @iL r-1PL(u) + @L -1 r- 1QiL -1(u) + eiab@aL-1 r-1RbL- 1(u) . l=0 l=1
From that expression we are able to find a new object, say ab vn, which takes the same structure as a wn (a retarded solution of the source-free wave equation) and, furthermore, whose divergence is exactly the opposite of the divergence of uanb, i.e. @mvanm = - wan. Such a vanb is not unique, but we shall see that it is simply necessary to make a choice for vab n (the simplest one) in order to obtain the general solution. The formulas that we adopt are
[ ( )] v00= -r- 1N (- 1) + @a r- 1 - N (- 1) + C(- 2) - 3Pa , n a a 0i -1( (-1) (1)) [ - 1 (-1)] + sum oo [ -1 ] vn = r -Q i + 3P i - eiab@a r Rb - @L- 1 r NiL -1 , l=2 sum + oo { (42) vij= -d r-1P + 2d @ [r-1P ]- 6@ [r -1P ] n ij ij L-1 L- 1 L-2(i j)L-2 l=2 } [ -1 (1) (2) ] [ -1 ] + @L-2 r (NijL-2 + 3PijL-2- QijL-2) - 2@aL-2 r eab(iRj)bL -2 .
Notice the presence of anti-derivatives, denoted, e.g., by integral u N (- 1)(u) = - oo dvN (v); there is no problem with the limit v --> - oo since all the corresponding functions are zero when t < -T. The choice made in Equations (42View Equation) is dictated by the fact that the 00 component involves only some monopolar and dipolar terms, and that the spatial trace ii is monopolar: ii -1 v n = - 3r P. Finally, if we pose
ab ab ab hn = un + vn , (43)
we see that we solve at once the d’Alembertian equation (30View Equation) and the coordinate condition (31View Equation). That is, we have succeeded in finding a solution of the field equations at the nth post-Minkowskian order. By induction the same method applies to any order n, and, therefore, we have constructed a complete post-Minkowskian series (29View Equation) based on the linearized approximation ha1b given by Equations (26View Equation, 27View Equation, 28View Equation). The previous procedure constitutes an algorithm, which could be implemented by an algebraic computer programme.
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