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5.1 The matching equation

We shall provide the answer in the case of a post-Newtonian source for which the post-Newtonian parameter 1/c defined by Equation (1View Equation) is small. The fundamental fact that permits the connection of the exterior field to the inner field of the source is the existence of a “matching” region, in which both the multipole and the post-Newtonian expansions are valid. This region is nothing but the exterior near zone, such that r > a (exterior) and r « c (near zone). It always exists around post-Newtonian sources.

Let us denote by M(h) the multipole expansion of h (for simplicity, we suppress the space-time indices). By M(h) we really mean the multipolar-post-Minkowskian exterior metric that we have constructed in Sections 3 and 4:

+ sum oo M(h) =_ hext = Gnhn[IL, ...,ZL]. (62) n=1
Of course, h agrees with its own multipole expansion in the exterior of the source,
r > a ===> M(h) = h. (63)
By contrast, inside the source, h and M(h) disagree with each other because h is a fully-fledged solution of the field equations with matter source, while M(h) is a vacuum solution becoming singular at r = 0. Now let us denote by -- h the post-Newtonian expansion of h. We have already anticipated the general structure of this expansion as given in Equation (47View Equation). In the matching region, where both the multipolar and post-Newtonian expansions are valid, we write the numerical equality
-- a < r « c ===> M(h) = h. (64)
This “numerical” equality is viewed here in a sense of formal expansions, as we do not control the convergence of the series. In fact, we should be aware that such an equality, though quite natural and even physically obvious, is probably not really justified within the approximation scheme (mathematically speaking), and we take it as part of our fundamental assumptions.

We now transform Equation (64View Equation) into a matching equation, by replacing in the left-hand side M(h) by its near-zone re-expansion ------ M(h), and in the right-hand side -- h by its multipole expansion -- M( h). The structure of the near-zone expansion (r --> 0) of the exterior multipolar field has been found in Equation (46View Equation). We denote the corresponding infinite series M(h)-- with the same overbar as for the post-Newtonian expansion because it is really an expansion when r/c --> 0, equivalent to an expansion when c-- > oo. Concerning the multipole expansion of the post-Newtonian metric, -- M( h), we simply postulate its existence. Therefore, the matching equation is the statement that

------ -- M(h) = M( h), (65)
by which we really mean an infinite set of functional identities, valid A (x, t) (- R3*× R, between the coefficients of the series in both sides of the equation. Note that such a meaning is somewhat different from that of a numerical equality like Equation (64View Equation), which is valid only when x belongs to some limited spatial domain. The matching equation (65View Equation) tells us that the formal near-zone expansion of the multipole decomposition is identical, term by term, to the multipole expansion of the post-Newtonian solution. However, the former expansion is nothing but the formal far-zone expansion, when r --> oo, of each of the post-Newtonian coefficients. Most importantly, it is possible to write down, within the present formalism, the general structure of these identical expansions as a consequence of Theorem 3, Equation (46View Equation):
------ sum -- M(h) = n^Lrm(ln r)pFL,m,p(t) = M( h), (66)
where the functions sum FL,m,p = n>1 GnFL,m,p,n. The latter expansion can be interpreted either as the singular re-expansion of the multipole decomposition when r --> 0 (first equality in Equation (66View Equation)), or the singular re-expansion of the post-Newtonian series when r --> + oo (second equality). We recognize the beauty of singular perturbation theory, where two asymptotic expansions, taken formally outside their respective domains of validity, are matched together. Of course, the method works because there exists, physically, an overlapping region in which the two approximation series are expected to be numerically close to the exact solution.
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