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

dvi ∮ ∮ ( ∮ ∮ ) PτA --A-+ ε−4 dSk ΛkiN − ε− 4vkA dSk Λ τNi− ε−4viA dSk ΛkNτ− vkA dSk Λ τNτ + dτ ∂BA ∂BA ∂BA ∂BA dQiA 2d2DiA -dτ--+ ε -dτ2-- ∫ ∫ ( ∫ ) = ε− 4 d3y Λiν,ν − ε−4vi d3yΛ τν,ν + ε−4-d- d3yΛ τν,νyi . (112 ) BA N A BA N dτ 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 (111View Equation) do not depend on RA (a similar argument can be found in [73]).

Along the same line, the momentum-velocity relation (86View Equation) 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.

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