### 4.5 On the arbitrary constant

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

#### 4.5.1 dependence of the field

and contributions to the field depend on the body zone boundary . But itself does
not depend on . Thus it is natural to expect that there are renormalized multipole moments which
are independent of since we use nonsingular matter sources. This renormalization would absorb the
dependence occuring in the computation of the field (see Section 4.8 for an example of such
a renormalization). One possible practical obstacle for this expectation might be the
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 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 “” for the
renormalized .

#### 4.5.2 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, . 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

Now the conservation law is satisfied for whatever value we take for , then the right hand side of the
above equation is zero for any . Hence the equations of motion (111) do not depend on (a
similar argument can be found in [73]).
Along the same line, the momentum-velocity relation (86) does not depend on .

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