### 8.3 Consistency relation

Our new choice of the dipole moments of the stars is in a sense natural. To see this, let us consider the
harmonic condition:
where “” are irrelevant terms. These equations are a manifestation of the fact that the harmonic
condition is consistent with the evolution equation for , the momentum-velocity relation, and the
equations of motion (and relations among higher multipole moments, hidden in “”). Thus if the
logarithmic dependence of arises from the second term of Equation (178), must
have the same logarithmic dependence (times minus sign) to ensure harmonicity. This and the
momentum velocity relation in turn mean , , or have corresponding logarithmic
dependence. Since we already know that the have no logarithm up to 3 PN order, or
should have logarithms. This is consistent with the fact that a choice of determines .
depends on logarithms if the old choice is taken, while it does not if our new choice is
taken.
There is yet another fact which supports our interpretation. Let us retain for a while. We then
find that the near zone dipole moment defined by a volume integral of becomes

Then if we take the old choice of , the volume integral becomes
where terms denoted by “” have no logarithmic dependence. Notice that the near zone dipole moment
can be freely determined, say, , because we can define the origin of the near zone freely. By
taking temporal derivatives twice of , we see that gives a natural definition of
the center of the mass in terms of which the 3 PN equations of motion are independent of
.