Third Post-Newtonian Momentum-Velocity Relation

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 (91 ),

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

with

&#x222E; i &#x2212;4 &#x0028; &#x03C4;k k &#x03C4;&#x03C4;&#x0029; i Q A&#x0398; = &#x03B5; dSk &#x0398; N &#x2212; vA&#x0398; N yA, &#x0028;160 &#x0029; &#x222E;&#x2202;BA &#x0028; &#x0029; i &#x2212;4 &#x03C4;k&#x03B1;&#x03B2; k &#x03C4;&#x03C4;&#x03B1;&#x03B2; i QA &#x03C7; = &#x03B5; &#x2202;B dSk &#x03C7; N ,&#x03B1;&#x03B2; &#x2212; vA&#x03C7; N ,&#x03B1;&#x03B2; yA, &#x0028;161 &#x0029; A

and we show only $i Q AΘ$ :

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 [91 ].

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

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,

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 (78 , 79 , 80 ) we see that $i δAΘ$ appears only at $ττ 10h$ as

Then the corresponding acceleration becomes

The last term compensates the $QiA$ integral contribution appearing through the momentum-velocity relation (163

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