### 4.8 Body zone boundary dependent terms

As explained in Section 4.5 we discard the body zone boundary dependent terms in the field, since we
expect that they cancel out between the body zone contribution and the contribution.
Before moving on to the higher order calculations, however, it is instructive to see that such a
cancellation really occurs in the field and consequently the equations of motion up to 0.5 PN
order.
First, we show the independence of the body zone radius in the 0.5 PN field. Returning back to
the derivation of the 1 PN with care for the dependence (see Equation (128)), we get

Now we split as where does not depend on while does. In order to
evaluate the dependent part of , , we use the fact that the integral of over the near
zone does not depend on the size of the body zone. (Notice that is defined as the volume integral of
over .) In fact, from the relevant expression of the pseudotensor, we obtain at the lowest order
Hence we find The above equation shows us that the 0.5 PN field is
independent of and fully expressed by the independent energy , or mass up to 0.5 PN
order.
In the similar manner we split as and obtain
from the definition of . Thus from the fact that , we find
that the 0.5 PN momentum-velocity relation does not depend on : .
Finally evaluating the surface integrals in the general form of equations of motion using the
“renormalized” (barred) moments, we find that the equations of motion are independent of as was
expected.

As remarked in Section 4.5 from now on we use the same symbol for the renormalized moments as for
the “bare” moments henceforth as before for notational simplicity.