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7 Third Post-Newtonian Momentum-Velocity Relation

We now derive the 3 PN momentum-velocity relation by calculating the i QA integral at 3 PN order. From the definition of the i Q A integral, Equation (91View Equation),

∮ ( ) QiA = ε6 dSk 10ΛτNk − vkA10Λ ττN yiA, (158 ) ∂BA
we find that the calculation required is almost the same as that in the equation for dP τ∕dτ A. Namely, it turns out that we do not need to use the direct-integration method to compute the Qi A integral. Therefore it is straightforward to evaluate the surface integrals in the definition of i Q A. Here we split i Q A into QiAΘ and QiAχ as
i i i Q A = Q AΘ + Q Aχ, (159 )
∮ i −4 ( τk k ττ) i Q AΘ = ε dSk Θ N − vAΘ N yA, (160 ) ∮∂BA ( ) i −4 τkαβ k τταβ i QA χ = ε ∂B dSk χ N ,αβ − vAχ N ,αβ yA, (161 ) A
and we show only i Q AΘ:
3 ⟨ij⟩ j ( 3 ) ( ) ≤6Qi1Θ = − ε6m-1m2--n12-v12-= − ε6-d m-1m2-ri12 = ε6 d- 1m31ai1 , (162 ) 2r312 d τ 6r312 dτ 6
where f ≤n is the quantity f up to order n inclusively. Here it should be understood that ai A in the last expression is evaluated with the Newtonian acceleration.

QiA of 𝒪 (ε6) appears at the 4 PN or higher order field. Thus up to 3 PN order, 6QiA affects the equations of motion only through the 3 PN momentum-velocity relation. For this reason, not QiAχ but Qi AΘ is necessary to derive the 3 PN equations of motion. The explicit expression for Qi Aχ is given in [91Jump To The Next Citation Point].

Now with QiAΘ in hand, we obtain the momentum-velocity relation. It turns out that the χ part of the momentum velocity relation is a trivial identity12. Thus, defining the Θ parts of μ PA and i D A in the same way as for i Q A, we obtain

( ) i τ i 6 d- 1- 3 i 2dDi1Θ- P1Θ = P 1Θv1 + ε dτ 6m 1a1 + ε dτ . (163 )
As explained in the previous Sections 3.4 and 4.4, we define the representative point i zA of star A by choosing the value of DiA. In other words, one can freely choose DiA in principle13. One may set DiA equal to zero up to 2.5 PN order. Alternatively, one may find it “natural” to see a three-momentum proportial to a three-velocity and take another choice,
i 4 1 3 i 4 i D AΘ = − ε -m AaA = ε δAΘ. (164 ) 6
Henceforth, we shall define i zA by this equation.

Finally, it is important to realize that the nonzero dipole moment DiA of order ε4 affects the 3 PN field and the 3 PN equations of motion in essentially the same manner as the Newtonian dipole moment affects the Newtonian field and equations of motion. From Equations (78View Equation, 79View Equation, 80View Equation) we see that i δAΘ appears only at ττ 10h as

ττ 10 ∑ δkAΘrkA 11 h |δAΘ = 4 ε ---3--+ 𝒪(ε ). (165 ) A=1,2 r A
Then the corresponding acceleration becomes
3m1 δi ⟨ik⟩ 3m2 δi ⟨ik⟩ d2δi m1ai1|δAΘ = − ε6---32Θ-n12 + ε6---3-1Θ-n12 − ε6---12Θ. (166 ) r12 r12 dτ
The last term compensates the QiA integral contribution appearing through the momentum-velocity relation (163View Equation).

Note that this change of the acceleration does not affect the existence of the conservation of the (Newtonian-sense) energy,

∑ [ k ] m1ai1|δAΘvi1 + m2ai2|δAΘvi2 = ε6-d- δkA ΘdvA-− vkA-d-δkAΘ . (167 ) dτ A=1,2 dτ d τ

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