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2.2 Post-Newtonian hierarchy

We shall now define the Newtonian, post-Newtonian, and higher approximations of various quantities as the appropriate higher tangents of the corresponding quantities to the above integral curve at ε = 0. For example the hierarchy of approximations for the spacetime metrics can be expressed as follows:
i i i gμν(ε,τ,x ) = gμν(0,τ,x ) + ε(ℒV gμν)(0,τ,x ) 1-2 2 i εn n i + 2ε (ℒV gμν)(0, τ,x ) + ⋅⋅⋅ + n!(ℒ Vgμν)(0,τ,x ) + Rn+1, (11 )
where ℒV is the Lie derivative with respect to the tangent vector of the curves defined above, and the remainder term μν R n+1 is
μν εn+1 ∫ 1 n+1 n+1 i Rn+1 = -------- d ℓ(1 − ℓ) (ℒV gμν)(ℓε,τ,x ). (12 ) (n + 1)! 0
Taylor’s theorem guarantees that the series is an asymptotic expansion about ε = 0 under certain assumptions mentioned above. It may be useful to point out that the above definition of the approximation scheme may be formulated purely geometrically in terms of a jet bundle.

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 ε dependence of the domain of dependence of the field point k (τ, x ). The region is finite with finite values of ε, and the diameter of the region increases like ε−1 as ε → 0. Without this linkage of the region with the expansion parameter ε, 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 as

∫ ∫ ∫ dr f(τ − εr,...) = dr f(τ,...) − ε drrf (τ,...),τ + ..., (13 )
and assign the second term to a higher order because of its explicit ε in front. This is incorrect because r → ε−1 as ε → 0 and thus εr is not uniformly small in the Newtonian limit. Only if the integrand falls off sufficiently fast, the retardation can be ignored. This happens in the lower order PN terms. But at some higher order there appear many terms which do not fall off sufficiently fast because of the nonlinearity of the Einstein equations. This is the reason that the formal PN approximation produces the divergent integrals. It turns out that such a divergence appears at 4 PN order, indicating a breakdown of the PN approximation in harmonic coordinates [20Jump To The Next Citation Point]3. This sort of divergence may be eliminated if we remember that the upper bound of the integral does depend on ε as ε−1. Thus we would get something like εn ln ε instead of εnln ∞ in the usual approach. This shows that the asymptotic Newtonian sequence is not differentiable in ε at ε = 0, but there is no divergence in the expansion and it has still an asymptotic approximation in ε that involves logarithms.

Other than the initial value formulation method [79Jump To The Next Citation Point82Jump To The Next Citation Point] 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 [129Jump To The Next Citation Point162Jump To The Next Citation Point165Jump To The Next Citation Point] 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 [19Jump To The Next Citation Point]. 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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