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5.4 The source multipole moments

In principle the bridge between the exterior gravitational field generated by the post-Newtonian source and its inner field is provided by Theorem 5; however, we still have to make the connection with the explicit construction of the general multipolar and post-Minkowskian metric in Sections 3 and 4. Namely, we must find the expressions of the six STF source multipole moments IL, JL, ..., ZL parametrizing the linearized metric (26View Equation, 27View Equation, 28View Equation) at the basis of that construction10. To do this we impose the harmonic-gauge conditions (12View Equation) onto the multipole decomposition as given by (67View Equation), and we decompose the multipole functions HabL (u) into STF irreducible pieces (refer to [11] for the details). Theorem 6 The STF multipole moments IL and JL of a post-Newtonian source are given, formally up to any post-Newtonian order, by (l > 2)
integral integral 3 1 { ---4(2l-+-1)---- (1) IL(u) = F P d x dz dl^xLS - c2(l + 1)(2l + 3)dl+1x^iLS i -1 2(2l + 1) } + -4-------------------- dl+2^xijLS(2ij) (x, u + z| x |/c), (80) c (l + 1)(l + 2)(2l + 5) integral integral 1 { } J (u) = F P d3x dz e d ^x S - -----2l +-1-----d ^x S(1) (x,u + z|x|/c). L -1 ab<il l L-1>a b c2(l + 2)(2l + 3) l+1 L-1>ac bc
These moments are the ones that are to be inserted into the linearized metric ab h 1 that represents the lowest approximation to the post-Minkowskian field ab sum hext = n>1 Gnhabn defined in Section 4.
In these formulas the notation is as follows: Some convenient source densities are defined from the post-Newtonian expansion of the pseudo-tensor ab t by
-- -- t00 +-t-ii- S = c2 , -0i S = t--, (81) i c --ij Sij = t
(where tii =_ dijtij). As indicated in Eqs. (80View Equation) these quantities are to be evaluated at the spatial point x and at time u + z|x|/c. Notice the presence of an extra integration variable z, ranging from - 1 to 1. The z-integration involves the weighting function11
(2l +-1)!! 2l dl(z) = 2l+1l! (1- z ), (82)
which is normalized in such a way that
integral 1 dz dl(z) = 1. (83) - 1
For completeness, we give also the formulas for the four auxiliary source moments WL, ...,ZL, which parametrize the gauge vector a f 1 as defined in Eqs. (28View Equation):
integral integral 1 { } W (u) = F P d3x dz ----2l +-1----d ^x S - ---------2l +-1--------d ^x S(1) , L - 1 (l + 1)(2l + 3) l+1 iL i 2c2(l + 1)(l + 2)(2l + 5) l+2 ijL ij (84) integral integral 1 { } 3 --------2l +-1-------- XL(u) = F P d x - 1dz 2(l + 1)(l + 2)(2l + 5)dl+2^xijLSij , (85) integral integral 1 { Y (u) = F P d3x dz - d^x S + ---3(2l +-1)--d ^x S(1) L - 1 lL ii (l + 1)(2l + 3) l+1 iL i 2(2l + 1) } - -2-------------------- dl+2^xijLS(2ij) , (86) c (l + 1)(l + 2)(2l + 5) integral 3 integral 1 { 2l + 1 } ZL(u) = F P d x dz eab<il - --------------dl+1^xL -1>bcSac . (87) - 1 (l + 2)(2l + 3)
As discussed in Section 4, one can always find two intermediate “packages” of multipole moments, ML and SL, which are some non-linear functionals of the source moments (80View Equation) and (84View Equation, 85View Equation, 86View Equation, 87View Equation), and such that the exterior field depends only on them, modulo a change of coordinates. See, e.g., Eq. (90View Equation) below.

In fact, all these source moments make sense only in the form of a post-Newtonian expansion, so in practice we need to know how to expand all the z-integrals as series when c-- > +o o. Here is the appropriate formula:

integral 1 + sum o o (2l + 1)!! ( |x| @ )2k dz dl(z)t(x,u + z |x|/c) = -k---------------- ------ t(x, u). (88) -1 k=0 2 k!(2l + 2k + 1)!! c @u
Since the right-hand side involves only even powers of 1/c, the same result holds equally well for the “advanced” variable u + z| x |/c or the “retarded” one u - z|x|/c. Of course, in the Newtonian limit, the moments IL and JL (and also ML, SL) reduce to the standard expressions. For instance, we have
( ) 1- IL(u) = QL(u) + O c2 , (89)
where QL is the Newtonian mass-type multipole moment (see Eq. (3View Equation)). (The moments WL, ..., ZL have also a Newtonian limit, but it is not particularly illuminating.)

Theorem 6 solves in principle the question of the generation of gravitational waves by extended post-Newtonian sources. However, note that this result has to be completed by the definition of an explicit algorithm for the post-Newtonian iteration, analogous to the post-Minkowskian algorithm we defined in Section 4, so that the source multipole moments, which contain the full post-Newtonian expansion of the pseudo-tensor t ab, can be completely specified. Such a systematic post-Newtonian iteration scheme, valid (formally) to any post-Newtonian order, has been recently implemented by Poujade and Blanchet [123] using matched asymptotic expansions (see Section 7 below for the metric developed explicitly up to the 3PN order). The solution of this problem yields, in particular, some general expression, valid up to any order, of the terms associated with the gravitational radiation reaction force inside the post-Newtonian source12. Needless to say, the formalism becomes prohibitively difficult to apply at very high post-Newtonian approximations. Some post-Newtonian order being given, we must first compute the relevant relativistic corrections to the pseudo stress-energy-tensor t ab (this necessitates solving the field equations inside the matter) before inserting them into the source moments (80View Equation, 81View Equation, 82View Equation, 83View Equation, 88View Equation, 84View Equation, 85View Equation, 86View Equation, 87View Equation). The formula (88View Equation) is used to express all the terms up to that post-Newtonian order by means of more tractable integrals extending over 3 R. Given a specific model for the matter source we then have to find a way to compute all these spatial integrals (we do it in Section 10 in the case of point-mass binaries). Next, we must substitute the source multipole moments into the linearized metric (26View Equation, 27View Equation, 28View Equation), and iterate them until all the necessary multipole interactions taking place in the radiative moments UL and VL are under control. In fact, we shall work out these multipole interactions for general sources in the next section up to the 3PN order. Only at this point does one have the physical radiation field at infinity, from which we can build the templates for the detection and analysis of gravitational waves. We advocate here that the complexity of the formalism reflects simply the complexity of the Einstein field equations. It is probably impossible to devise a different formalism, valid for general sources devoid of symmetries, that would be substantially simpler.

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