On the arbitrary constant RA

4.5 On the arbitrary constant $RA$

Since we have introduced the body zone by hand, the arbitrary body zone size $RA$ seems to appear in the metric, the multipole moments of the stars, and the equations of motion. More specifically $RA$ appears in them because of (i) the splitting of the deviation field into two parts (i.e. $B$ and $N ∕B$ contributions), the definition of the moments, and (ii) the surface integrals that we evaluate to derive the equations of motion.

4.5.1 $RA$ dependence of the field

$B$ and $N ∕B$ contributions to the field depend on the body zone boundary $εR A$ . But $hμν$ itself does not depend on $εRA$ . Thus it is natural to expect that there are renormalized multipole moments which are independent of $RA$ since we use nonsingular matter sources. This renormalization would absorb the $εRA$ dependence occuring in the computation of the $N ∕B$ field (see Section 4.8 for an example of such a renormalization). One possible practical obstacle for this expectation might be the $ln(εRA )$ dependence of multipole moments. Although at 3 PN order there appear such logarithmic terms, it is found that we could remove them by rechoosing the value of the dipole moment $i D A$ of the star.

Though we use the same symbol for the moments henceforth as before for notational simplicity, it should be understood that they are the renormalized ones. For instance, we use the symbol “ $Pμ A$ ” for the renormalized $μ PA$ .

4.5.2 $RA$ dependence of the equations of motion

Since we compute integrals over the body zone boundary, in general the resulting equations of motion seem to depend on the size of the body zone boundary, $εRA$ . Actually this is not the case.

In the derivation of Equation (111 ), if we did not use the conservation law (67 ) until the final step, we have

dvi &#x222E; &#x222E; &#x0028; &#x222E; &#x222E; &#x0029; P&#x03C4;A --A-+ &#x03B5;&#x2212;4 dSk &#x039B;kiN &#x2212; &#x03B5;&#x2212; 4vkA dSk &#x039B; &#x03C4;Ni&#x2212; &#x03B5;&#x2212;4viA dSk &#x039B;kN&#x03C4;&#x2212; vkA dSk &#x039B; &#x03C4;N&#x03C4; + d&#x03C4; &#x2202;BA &#x2202;BA &#x2202;BA &#x2202;BA dQiA 2d2DiA -d&#x03C4;--+ &#x03B5; -d&#x03C4;2-- &#x222B; &#x222B; &#x0028; &#x222B; &#x0029; = &#x03B5;&#x2212; 4 d3y &#x039B;i&#x03BD;,&#x03BD; &#x2212; &#x03B5;&#x2212;4vi d3y&#x039B; &#x03C4;&#x03BD;,&#x03BD; + &#x03B5;&#x2212;4-d- d3y&#x039B; &#x03C4;&#x03BD;,&#x03BD;yi . &#x0028;112 &#x0029; BA N A BA N d&#x03C4; BA N A

Now the conservation law is satisfied for whatever value we take for $RA$ , then the right hand side of the above equation is zero for any $RA$ . Hence the equations of motion (111

) do not depend on $RA$ (a similar argument can be found in [73 ]).

Along the same line, the momentum-velocity relation (86 ) does not depend on $RA$ .

In Section 4.8 we shall explicitly show the irrelevance of the field and the equations of motion to $R A$ by checking the cancellation among the $RA$ dependent terms up to 0.5 PN order.

4.5 On the arbitrary constant

4.5.1 dependence of the field

4.5.2 dependence of the equations of motion

4.5 On the arbitrary constant $RA$

4.5.1 $RA$ dependence of the field

4.5.2 $RA$ dependence of the equations of motion