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4.1 Field equations

As discussed in the previous Section 3, we express our equations of motion in terms of surface integrals over the body zone boundary where it is assumed that the metric slightly deviates from the flat metric ημν = diag (− ε2,1, 1,1) (in the near zone coordinates (τ,xi)). Thus we define a deviation field hμν as
h μν ≡ ημν − √ −-g gμν, (60 )
where g is the determinant of the metric. Our h μν differs from the corresponding field in [30Jump To The Next Citation Point] in a sign. Indices are raised or lowered by the flat (auxiliary) metric ημν unless otherwise stated.

Now we impose the harmonic coordinate condition on the metric,

h μν,ν = 0, (61 )
where the comma denotes the partial derivative. In the harmonic gauge, we can recast the Einstein equations into the relaxed form,
μν μν □h = − 16πΛ , (62 )
where □ = ημν∂μ∂ν is the flat d’Alembertian and
Λ μν ≡ Θμν + χμναβ , (63 ) ,αμβν Θ μν ≡ (− g)(T μν + tLL), (64 ) μναβ 1 αν βμ αβ μν χ ≡ ----(h h − h h ). (65 ) 16π
Here, μν T and μν tLL denote the stress-energy tensor of the stars and the Landau–Lifshitz pseudotensor [115Jump To The Next Citation Point]. The explicit form of tμLνL in the harmonic gauge is
(− 16πg )tμν = g gγδhμα h νβ + 1gμνg hα γ hβδ − 2g gγ(μhν)α hδβ LL αβ ,γ ,δ 2 αβ ,δ ,γ αβ ,δ ,γ 1 ( 1 ) ( 1 ) + -- gμαgνβ − --gμνgαβ gγδgεζ − --gγεgδζ hγε,αh δζ,β. (66 ) 2 2 2
χ μναβ originates from our use of the flat d’Alembertian instead of the curved space d’Alembertian. In consistency with the harmonic condition, the conservation law is expressed as
μν Λ ,ν = 0. (67 )
Note that the divergence of χμναβ itself vanishes identically due to the symmetry of its indices.

Now we rewrite the relaxed Einstein equations into an integral form,

∫ μν k h μν(τ, xi) = 4 d3y Λ---(τ-−-ε|⃗x-−-⃗y|,y-;ε)+ hμν(τ,xi), (68 ) C(τ,xk) |⃗x − ⃗y| H
where k C(τ,x ) means the past light cone emanating from the event k (τ,x ). k C(τ,x ) is truncated at the τ = 0 hypersurface. μν hH is a homogeneous solution of the homogeneous wave equation in the flat spacetime, □h μHν = 0. hμHν evolves from a given initial data on the τ = 0 initial hypersurface. The explicit form of hμν H is available via the Poisson formula (see e.g. [144])
∮ [ ∮ ] h μν(τ,xi) = τ-- hμν (τ = 0,yi) dΩ + -1--∂- τ hμν(τ = 0, yi) dΩ . (69 ) H 4π ∂C(τ,xi) ,τ y 4π ∂τ ∂C (τ,xi) y

We solve the Einstein equations as follows. First we split the integral region into two zones: the near zone and the far zone.

The near zone is the region containing the gravitational wave source where the wave character of the gravitational radiation is not manifest. In other words, in the near zone the retardation effect on the field is negligible. The near zone covers the whole source system. The size of the near zone is about a little larger than one wave length of the gravitational wave emitted by the source. In this paper we take the near zone as a sphere centered at some fixed point and enclosing the binary system. The radius of this sphere is set to be ℛ ∕ε, where ℛ is arbitrary but larger than the size of the binary and the wave length of the gravitational radiation. The scaling of the near zone radius is derived from the ε dependence of the wavelength of the gravitational radiation emitted due to the orbital motion of the binary. Note that roughly speaking the frequency of such a wave is about twice the Keplerian frequency of the binary. The center of the near zone sphere would be determined, if necessary, for example, to be the center of mass of the near zone. The outside of the near zone is the far zone where the retardation effect of the field is crucial.

For the near zone field point PN, we write the field as

hμν(P ∈ N ) = hμν + hμν + hμν, (70 ) N P∫N(N) PN (F) H μν 3 Λ μν(τ − ε|⃗x − ⃗y|,yk;ε) h PN(N ) ≡ 4 d y--------|⃗x −-⃗y|-------, (71 ) ∫N= {y:|y|≤ℛ ∕ε} μν 3 Λ-μν(τ-−--ε|⃗x-−-⃗y|,yk;ε) h PN(F) ≡ 4 d y |⃗x − ⃗y | . (72 ) F ={y:|y|>ℛ∕ε}
h μPνN(N) is the near zone integral contribution to the near zone field, and hμPνN(F) is the far zone integral contribution to the near zone field.

The far zone contribution can be evaluated with the DIRE method developed in [129Jump To The Next Citation Point]. An explicit calculation shows that apparently there are the far zone contributions to the near zone field at 3 PN order. However, these 3 PN contributions are merely a gauge. Pati and Will showed that the far zone contribution does not affect the equations of motion up to 3 PN order inclusively and that the far zone contribution first appears at 4 PN order. This result is consistent with the earlier result of Blanchet and Damour [20] who used the multipolar-post-Minkowskian formalism. We follow the DIRE method and checke that the far zone contribution does not affect the equations of motion up to 3 PN order in Appendix A. Henceforth we shall focus our attention on the near zone contribution h μPν(N) N and do not write down the far zone contribution in the following calculation of the field.

As for the homogeneous solution, we shall ignore it for simplicity. If we take random initial data for the field [138] supposed to be of 1 PN order [79Jump To The Next Citation Point], they are irrelevant to the dynamics of the binary system up to the radiation reaction order [79]. As we have assumed in the previous Section 3.3 that the magnitude of the free data of the gravitational field on the initial hypersurface is 2.5 PN order, we expect that the homogeneous solution does not affect the equations of motion up to 3 PN order. We leave a full implementation of the initial value formulation on the field as future work.

It is worth noticing that when we let τ become sufficiently large, then the condition h μHν(τ, xi) = 0 corresponds to a no-incoming radiation condition at (Minkowskian) past null infinity (see e.g. [77Jump To The Next Citation Point]). Equation (69View Equation) can be written down as Kirchhoff’s formula,

| μν ∮ d Ωy [ ∂ ∂ ] | h H (τ,xi) = ---- ---(ρhμν(τ ′,yi)) + --′(ρhμν(τ′,yi)) || . (73 ) ∂C (τ,xi) 4π ∂ ρ ∂τ τ′=0,ρ= |⃗x−⃗y|=τ
Then the no-incoming radiation condition at (Minkowskian) past null-infinity,
[ ] ∂-- μν -∂- μν lτim=r, ∂r(rh ) + ∂τ (rh ) = 0, (74 ) r→ ∞
is a sufficient condition to h μHν(τ, xi) = 0 when τ goes to infinity.

Now we shall devote ourselves to the evaluation of the near zone contribution to the near zone field,

μν i ∫ 3 Λ μν(τ − ε|⃗x − ⃗y|,yk;ε) h (τ, x) = 4 d y ----------------------. (75 ) N (τ,xk) |⃗x − ⃗y |
Henceforth we shall omit the subscript PN (N ) of the field hμν for notational simplicity.
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