The above definition of the post-Newtonian hierarchy gives us an asymptotic series in which each term in the series is manifestly finite. This is based on the dependence of the domain of dependence of the field point . The region is finite with finite values of , and the diameter of the region increases like as . Without this linkage of the region with the expansion parameter , the post-Newtonian approximation leads to divergences in the higher orders. This is closely related to the retarded expansion. Namely, it is assumed that the slow motion assumption enables one to Taylor expand the retarded integrals in retarded time such as3. This sort of divergence may be eliminated if we remember that the upper bound of the integral does depend on as . Thus we would get something like instead of in the usual approach. This shows that the asymptotic Newtonian sequence is not differentiable in at , but there is no divergence in the expansion and it has still an asymptotic approximation in that involves logarithms.
Other than the initial value formulation method [79, 82] mentioned above, various methods have been proposed to solve this problem of the divergent integrals. It is known that a higher order post-Newtonian metric does not respect the asymptotically flat condition. This does not mean that the post-Newtonian approximation is useless at such a high order. The problem is related to the fact that a simple post-Newtonian iteration is meaningful only in the near zone – about one wavelength distance away from the material source – and is not useful outside of the near zone, called far zone, where the wave effect (retardation effect) is manifest. So roughly speaking, if a far zone metric satisfying proper boundary conditions at infinity is solved so that we have a boundary condition to the field equations for a post-Newtonian metric in the buffer zone, we can find a post-Newtonian metric which is meaningful in the sense that it respects the correct behaviour at the near zone boundary.
Blanchet and Damour have developed a systematic approach of a matched asymptotic expansion. They solved the far zone metric using a multipolar post-Minkowskinan expansion. The far zone metric satisfies a stationarity condition and is parametrized by radiative multipole moments. On the other hand, they solve a post-Newtonian near zone metric up to a homogeneous solution. They then establish an association between those radiative multipole moments and the source multipole moments that characterize a post-Newtonian near zone metric to fix the homogeneous solution, and find a post-Newtonian metric which satisfies the correct behaviour in the buffer zone.
Will and Wiseman have developed the DIRE method [129, 162, 165] where they split the integral region in the retarded integral into two – one being the near zone and the other being the far zone. The near zone metric is solved by a post-Newtonian expansion. The retarded integral over the far zone is directly evaluated with the assumption of sufficient stationarity of the system in the infinite past.
In fact, both the Blanchet–Damour method and the Will–Wiseman method are proved to give a physically equivalent result . In this paper, for our computation of the 3 PN equations of motion, we will use the Will and Wiseman method to solve the problem of the breakdown of the post-Newtonian approximation in the near zone.
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