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5.3 Equivalence with the Will-Wiseman formalism

Recently, Will and Wiseman [152Jump To The Next Citation Point] (see also Refs. [151112]), extending previous work of Epstein and Wagoner [70] and Thorne [142], have obtained a different-looking multipole decomposition, with different definitions for the multipole moments of a post-Newtonian source. They find, instead of our multipole decomposition given by Eq. (67View Equation),
+ sum oo l { } M(hab) = [] -1[M(/\ab)] - 4G- (-)-@L 1Wab (t- r/c) . (76) ret |R c4 l=0 l! r L
There is no F P operation in the first term, but instead the retarded integral is truncated, as indicated by the subscript R, to extend only in the “far zone”: i.e. |x'|> R in the notation of Eq. (21View Equation), where R is a constant radius enclosing the source (R > a). The near-zone part of the retarded integral is thereby removed, and there is no problem with the singularity of the multipole expansion ab M(/\ ) at the origin. The multipole moments WL are then given, in contrast with our result (68View Equation), by an integral extending over the “near zone” only:
integral Wab (u) = d3x x tab(x, u). (77) L |x|<R L
Since the integrand is compact-supported there is no problem with the bound at infinity and the integral is well-defined (no need of a F P).

Let us show that the two different formalisms are equivalent. We compute the difference between our moment HL, defined by Eq. (68View Equation), and the Will-Wiseman moment WL, given by Eq. (77View Equation). For the comparison we split HL into far-zone and near-zone integrals corresponding to the radius R. Since the finite part F P present in HL deals only with the bound at infinity, it can be removed from the near-zone integral, which is then seen to be exactly equal to WL. So the difference between the two moments is simply given by the far-zone integral:

integral ab ab 3 -ab H L (u) - W L (u) = F P d xxL t (x,u). (78) |x|>R
Next, we transform this expression. Successively we write tab = M(t -ab) because we are outside the source, and -------- M(t -ab) = M(t ab) from the matching equation (65View Equation). At this stage, we recall from our reasoning right after Eq. (74View Equation) that the finite part of an integral over the whole space 3 R of a quantity having the same structure as -----ab- M(t ) is identically zero by analytic continuation. The main trick of the proof is made possible by this fact, as it allows us to transform the far-zone integration |x| > R in Eq. (78View Equation) into a near-zone one |x|< R, at the price of changing the overall sign in front of the integral. So,
integral -------- HabL (u)- WabL (u) = - F P d3x xLM(t ab)(x,u). (79) |x|<R
Finally, it is straightforward to check that the right-hand side of this equation, when summed up over all multipolarities l, accounts exactly for the near-zone part that was removed from the retarded integral of M(/\ab) (first term in Eq. (76View Equation)), so that the “complete” retarded integral as given by the first term in our own definition (67View Equation) is exactly reconstituted. In conclusion, the formalism of Ref. [152] is equivalent to the one of Refs. [6Jump To The Next Citation Point11Jump To The Next Citation Point].
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