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5.2 General expression of the multipole expansion

Theorem 5 Under the hypothesis of matching, Equation (65View Equation), the multipole expansion of the solution of the Einstein field equation outside a post-Newtonian source reads
+ oo { } ab -1 B ab 4G- sum (--)l 1- ab M(h ) = F PB=0 [] ret[r M(/\ )]- c4 l! @L r H L (t- r/c) , (67) l=0
where the “multipole moments” are given by
integral ab 3 B -ab H L (u) = F PB=0 d x |x| xL t (x,u). (68)
Here, --ab t denotes the post-Newtonian expansion of the stress-energy pseudo-tensor defined by Equation (14View Equation).
Proof  [15Jump To The Next Citation Point20Jump To The Next Citation Point]: First notice where the physical restriction of considering a post-Newtonian source enters this theorem: the multipole moments (68View Equation) depend on the post-Newtonian expansion tab, rather than on ab t itself. Consider ab D, namely the difference between ab h, which is a solution of the field equations everywhere inside and outside the source, and the first term in Equation (67View Equation), namely the finite part of the retarded integral of the multipole expansion M(/\ab):
Dab =_ hab - F P []-re1t[M(/\ab)]. (69)
From now on we shall generally abbreviate the symbols concerning the finite-part operation at B = 0 by a mere F P. According to Equation (20View Equation), hab is given by the retarded integral of the pseudo-tensor tab. So,
Dab = 16pG--[]- 1tab- F P [] -1[M(/\ab)]. (70) c4 ret ret
In the second term the finite part plays a crucial role because the multipole expansion M(/\ab) is singular at r = 0. By contrast, the first term in Equation (70View Equation), as it stands, is well-defined because we are considering only some smooth field distributions: ab o o 4 t (- C (R ). There is no need to include a finite part F P in the first term, but a contrario there is no harm to add one in front of it, because for convergent integrals the finite part simply gives back the value of the integral. The advantage of adding “artificially” the F P in the first term is that we can re-write Equation (70View Equation) into the much more interesting form
ab 16pG-- - 1[ ab ab ] D = c4 FP []ret t - M(t ) , (71)
in which we have also used the fact that ab 4 ab M(/\ ) = 16pG/c .M(t ) because ab T has a compact support. The interesting point about Equation (71View Equation) is that Dab appears now to be the (finite part of a) retarded integral of a source with spatially compact support. This follows from the fact that the pseudo-tensor agrees numerically with its own multipole expansion when r > a (same equation as (63View Equation)). Therefore, ab M(D ) can be obtained from the known formula for the multipole expansion of the retarded solution of a wave equation with compact-support source. This formula, given in Appendix B of Ref. [28Jump To The Next Citation Point], yields the second term in Equation (67View Equation),
+o o { } ab 4G- sum (--)l 1- ab M(D ) = - c4 l! @L r H L (u) , (72) l=0
but in which the moments do not yet match the result (68View Equation); instead,
integral ab 3 [ ab ab ] H L = F P d x xL t - M(t ) . (73)
The reason is that we have not yet applied the assumption of a post-Newtonian source. Such sources are entirely covered by their own near zone (i.e. a « c), and, in addition, the integral (73View Equation) has a compact support limited to the domain of the source. In consequence, we can replace the integrand in Equation (73View Equation) by its post-Newtonian expansion, valid over all the near zone, i.e.
integral [-- --------] HabL = F P d3x xL tab - M(t ab) . (74)
Strangely enough, we do not get the expected result because of the presence of the second term in Equation (74View Equation). Actually, this term is a bit curious, because the object -------- M(t ab) it contains is only known in the form of the formal series whose structure is given by the first equality in Equation (66View Equation) (indeed t and h have the same type of structure). Happily (because we would not know what to do with this term in applications), we are now going to prove that the second term in Equation (74View Equation) is in fact identically zero. The proof is based on the properties of the analytic continuation as applied to the formal structure (66View Equation) of M(t--ab). Each term of this series yields a contribution to Equation (74View Equation) that takes the form, after performing the angular integration, of the integral integral +o o F PB=0 0 drrB+b(ln r)p, and multiplied by some function of time. We want to prove that the radial integral integral +o o drrB+b(ln r)p 0 is zero by analytic continuation (A B (- C). First we can get rid of the logarithms by considering some repeated differentiations with respect to B; thus we need only to consider the simpler integral integral + oo B+b 0 dr r. We split the integral into a “near-zone” integral integral R0 dr rB+b and a “far-zone” one integral R+ oo dr rB+b, where R is some constant radius. When R (B) is a large enough positive number, the value of the near-zone integral is B+b+1 R /(B + b + 1), while when R (B) is a large negative number, the far-zone integral reads the opposite, B+b+1 - R /(B + b + 1). Both obtained values represent the unique analytic continuations of the near-zone and far-zone integrals for any B (- C except - b - 1. The complete integral integral 0+ oo dr rB+b is equal to the sum of these analytic continuations, and is therefore identically zero (A B (- C, including the value -b - 1). At last we have completed the proof of Theorem 5:
integral ab 3 -ab H L = F P d x xLt . (75)

The latter proof makes it clear how crucial the analytic-continuation finite part F P is, which we recall is the same as in our iteration of the exterior post-Minkowskian field (see Equation (39View Equation)). Without a finite part, the multipole moment (75View Equation) would be strongly divergent, because the pseudo-tensor -ab t has a non-compact support owing to the contribution of the gravitational field, and the multipolar factor xL behaves like rl when r --> + oo. In applications (Part B of this article) we must carefully follow the rules for handling the F P operator.

The two terms in the right-hand side of Equation (67View Equation) depend separately on the length scale r 0 that we have introduced into the definition of the finite part, through the analytic-continuation factor rB = (r/r0)B (see Equation (36View Equation)). However, the sum of these two terms, i.e. the exterior multipolar field M(h) itself, is independent of r0. To see this, the simplest way is to differentiate formally M(h) with respect to r0. The independence of the field upon r0 is quite useful in applications, since in general many intermediate calculations do depend on r 0, and only in the final stage does the cancellation of the r0’s occur. For instance, we shall see that the source quadrupole moment depends on r0 starting from the 3PN level [45Jump To The Next Citation Point], but that this r0 is compensated by another r0 coming from the non-linear “tails of tails” at the 3PN order.


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