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4.2 Near zone contribution

We shall evaluate the near zone contribution as follows. First, we make a retarded expansion of Equation (75View Equation) and change the integral region to a τ = constant spatial hypersurface,
∑ n ( )n ∫ hμν = 4 (− ε)- ∂-- d3y|⃗x − ⃗y|n−1Λ μNν(τ, yk;ε). (76 ) n=0 n! ∂τ N
Note that the above integral depends on the arbitrary length ℛ in general. The cancellation between the ℛ dependent terms in the far zone contribution and those in the near zone contribution was shown by [129Jump To The Next Citation Point] through all the post-Newtonian order. In the following, we shall omit the terms which have negative powers of ℛ (ℛ −k;k > 0). In other words, we simply let ℛ → 0 whenever it gives a convergent result. On the other hand, we shall retain terms having positive powers of ℛ (k ℛ ;k > 0), and logarithmic terms (ln ℛ) to confirm that the final result, in the end of calculation, is independent of ℛ (and for logarithmic terms to keep the arguments of logarithm non-dimensional).

Second we split the integral into two parts: a contribution from the body zone BA, and from elsewhere, N ∕B. Schematically we evaluate the following two types of integrals (we omit indices of the field),

h = hB + hN ∕B, ∫ 2 6 ∑ 3 f-(τ,⃗zA-+-ε-⃗αA) hB = ε B d αA |⃗rA − ε2⃗αA |1−n , (77 ) ∫ A=1,2 A 3 f (τ,⃗y) hN ∕B = d y |⃗x-−-⃗y-|1−n-, N∕B
where ⃗rA ≡ ⃗x − ⃗zA. We shall deal with these two contributions successively.

4.2.1 Body zone contribution

As for the body zone contribution, we make a multipole expansion using the scaling of the integrand, i.e. Λ μν in the body zone. For example, the n = 0 part in Equation (76View Equation), hμBνn=0, gives

∑ ( τ k k ⟨kl⟩ k l ⟨klm⟩ k l m ) hττ = 4ε4 P-A + ε2D-ArA-+ ε4 3IA--rArA-+ ε65IA----rArArA- + 𝒪 (ε12), (78 ) B n=0 A=1,2 rA r3A 2r5A 2r7A ( i ki k kli k l ) τi 4∑ P-A 2JA-rA- 43J-A-rArA- 10 hB n=0 = 4ε rA + ε r3A + ε 2r5A + 𝒪 (ε ), (79 ) A=1,2( ) ∑ Zij Zkijrk 3Z ⟨kl⟩ijrkrl 5Z ⟨klm ⟩ijrkrlrm hiBjn=0 = 4ε2 --A-+ ε2--A-3A-+ ε4--A---5-A-A-+ ε6---A-----7A-A-A- + 𝒪(ε10). (80 ) A=1,2 rA rA 2rA 2rA
Here the operator ⟨...⟩ denotes a symmetric and tracefree (STF) operation on the indices in the brackets. See [129Jump To The Next Citation Point150165Jump To The Next Citation Point] for some useful formulas of STF tensors. Also we define rA ≡ |⃗rA |. To derive the 3 PN equations of motion, h ττ up to 𝒪 (ε10) and h τi as well as hij up to 𝒪 (ε8) are required.

In the above equations we define the multipole moments as

∫ IKl≡ ε2 d3αA Λ τταKl , (81 ) A BA N A ∫ τi K JKAli≡ ε4 d3αA Λ N αAl, (82 ) ∫BA Klij 8 3 ij Kl ZA ≡ ε d αA Λ N αA , (83 ) BA
where we introduced a collective multi-index Il ≡ i1i2...il and I i i i αAl ≡ αA1αA2...α lA. Then P τA ≡ IKA0, Dk1A ≡ IKA1, and PkA1 ≡ JAK1. We simply call P μA the four-momentum of the star A, P iA the momentum, and P τA the energy7. Also we call Dk A the dipole moment of the star A, and Ikl A the quadrupole moment of the star A.

Then we transform these moments into more convenient forms. By the conservation law (67View Equation), we have

τj ΛτNi = (ΛτNτyiA ),τ + (Λ N yiA),j + viAΛ τNτ, (84 ) ij τ(i j) k(i j) (i j)τ ΛN = (ΛN yA),τ + (ΛN yA),k + vAΛ N , (85 )
where viA ≡ ˙ziA, an overdot denotes a time derivative with respect to τ, and ⃗yA ≡ ⃗y − ⃗zA. Using these equations and noticing that the body zone remains unchanged (in the near zone coordinates), i.e. ˙ RA = 0, we have
dDi PAi= PτAviA + QiA + ε2---A-, (86 ) ( d)τ ij 1 ij 2dIiAj (i j) 1 −2 ij JA = -- M A + ε ---- + v AD A + --ε QA, (87 ) 2 dτ 2 1 d2Iij dDj) dv(i 1 dQij ZiAj= ε2P τAviAvjA + -ε6---A- + 2ε4v(Ai---A-+ ε4--A-DjA)+ ε2Q (iAvjA)+ ε2R (iAj)+ --ε2---A, (88 ) 2 dτ2 dτ dτ 2 dτ kij 3- kij (ij)k ZA = 2A A − A A , (89 )
∫ j]τ M iAj≡ 2ε4 d3αA α [iAΛ N , (90 ) ∮ BA Kli −4 ( τk k ττ) Kl i Q A ≡ ε dSk ΛN − vAΛ N yA yA, (91 ) ∂BA ∮ ( ) Klij −4 kj k τj Kl i RA ≡ ε ∂B dSk ΛN − vAΛ N yA yA, (92 ) A
k(ij) Akij ≡ ε2Jk(ivj) + ε2vkJ (ij) + Rk (ij) + ε4dJA---. (93 ) A A A A A A dτ
The operators [ ] and ( ) attached to the indices denote anti-symmetrization and symmetrization. M ij A is the spin of the star A and Equation (86View Equation) is the momentum-velocity relation. Thus our momentum-velocity relation is a direct analog of the Newtonian momentum-velocity relation (see Section 3.4). In general, we have
JKli = J(Kli)+ --2l-J(Kl−1[kl)i], (94 ) A A[ l + 1 A ] Klij 1 (Kli)j 2l (Kl−1[kl)i]j (Klj)i 2l (Kl−1[kl)j]i ZA = -- Z A + -----Z A + Z A + ----Z A , (95 ) 2 l + 1 l + 1
(Kli) --1-- 2dIKAli (iKl) --1-- −2l Kli JA = l + 1ε dτ + vAIA + l + 1ε Q A , (96 ) Kl(ij) Z(Kli)j+ Z(Klj)i= ε2v(iJ Kl)j+ ε2v(jJ Kl)i+ --2--ε4 dJA-----+ ε−2l+2RKl (ij), (97 ) A A A A A A l + 1 dτ A
where l is the number of indices in the multi-index Kl.

Now, from the above equations, especially Equation (88View Equation), we find that the body zone contributions, h μν Bn=0, are of order 𝒪 (ε4). Note that if we can not or do not assume the (nearly) stationarity of the initial data for the stars, then, instead of Equation (88View Equation) we have

2 ij Zij = ε2P τAviAvj + 1d-IA- + ..., (98 ) A A 2 dη2
where we used the dynamical time η (see Section 3.3). In this case the lowest order metric differs from the Newtonian form. From our (almost) stationarity assumption the remaining motion inside a star, apart from the spinning motion, is caused only by the tidal effect by the companion star and from Equation (88View Equation); it appears at 3 PN order [81Jump To The Next Citation Point].

To obtain the lowest order hμν B n=0, we have to evaluate the surface integrals QKli,RKlij A A. Generally, in h μν Bn=0 the moments JKli A and ZKlij A appear formally at the order ε2l+4 and ε2l+2. Thus QKli A and Klij RA appear as

Kl ⟨Kl⟩i hτi ∼ ⋅⋅⋅ + ε4rA-Q-A--+ ..., (99 ) Bn=0 r2Al+1 Kl ⟨Kl⟩(ij) hij ∼ ⋅⋅⋅ + ε4rA-R-A----+ ..., (100 ) Bn=0 r2Al+1
where we omitted irrelevant terms and numerical coefficients. Thus one may expect that Kli Q A and Klij RA appear at the order ε4 for any l and we have to calculate an infinite number of moments. In fact, this is not the case and it was shown in [91Jump To The Next Citation Point] that only l = 0,1 terms of RKlAij contribute to the 3 PN equations of motion. The important thing here is that ε4QKli A and ε4RKlij A are at most 𝒪 (ε4) in μν h Bn=0.

Finally, since the order of hμν(n ≥ 1) Bn is higher than that of h μν Bn=0, we conclude that h μν= 𝒪(ε4) B.

4.2.2 N ∕B contribution

As far as the N ∕B contribution is concerned, since the integrand Λμν = − gtμν+ χμναβ N LL ,αβ is at least quadratic in the small deviation field μν h, we make the post-Newtonian expansion in the integrand. Then, basically, with the help of (super-)potentials g(⃗x) which satisfy Δg (⃗x) = f(⃗x), Δ denoting the Laplacian, we have for each integral (see e.g. the n = 0 term in Equation (77View Equation))

∫ f (⃗y) ∮ [ 1 ∂g (⃗y) ∂ ( 1 )] d3y------- = − 4πg(⃗x) + dSk -----------k-− g(⃗y)--k- ------- . (101 ) N∕B |⃗x − ⃗y| ∂(N∕B) |⃗x − ⃗y| ∂y ∂y |⃗x − ⃗y|
Equation (101View Equation) can be derived without using Dirac delta distributions (see Appendix B of [91Jump To The Next Citation Point]). For the n ≥ 1 terms in Equation (77View Equation), we use potentials many times to convert all the volume integrals into surface integrals and “− 4 πg(⃗x)” terms8.

In fact finding the super-potentials is one of the most formidable tasks especially when we proceed to high post-Newtonian orders. Fortunately, up to 2.5 PN order, all the required super-potentials are available. At 3 PN order, there appear many integrands for which we could not find the required super-potentials. To obtain the 3 PN equations of motion, we devise an alternative method similar to the method employed by Blanchet and Faye [28Jump To The Next Citation Point]. The details of the method will be explained later.

Now the lowest order integrands can be evaluated with the body zone contribution h μBν, and since h μν B is 𝒪 (ε4), we find

ττ 6 Λ N = 𝒪 (ε ), (102 ) Λ τi= 𝒪 (ε6), (103 ) N ( ) Λij = (− g)tij + 𝒪 (ε8) = ε4--1- δi δj − 1-δijδ hττ,k hττ,l + 𝒪 (ε5), (104 ) N LL 64 π k l 2 kl 4 B 4 B
where we expanded μν hB in an ε series,
μν ∑ 4+n μν hB = ε nhB . (105 ) n=0
Similarly, in the following we expand h μν in an ε series. From these equations we find that the deviation field in N∕B, hμν, is 𝒪 (ε4). (It should be noted that in the body zone μν h is assumed to be of order unity and within our method we can not calculate μν h there explicitly. To obtain μν h in the body zone, we have to know the internal structure of the star.)
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