### 9.3 Equations of motion in the center-of-mass frame

In this section we translate the origin of coordinates to the binary’s center-of-mass by imposing that the
binary’s dipole (notation of Part A). Actually the dipole moment is computed as the center-of-mass
conserved integral associated with the boost symmetry of the 3PN equations of motion and
Lagrangian [103, 43]. This condition results in the (3PN-accurate, say) relationship between the individual
positions in the center-of-mass frame and , and the relative position
and velocity (formerly denoted and ). We shall also use
the orbital separation , together with and . Mass parameters
are the total mass ( in the notation of Part A), the mass difference
, the reduced mass , and the very useful symmetric mass ratio
The usefulness of this ratio lies in its interesting range of variation: , with in the
case of equal masses, and in the “test-mass” limit for one of the bodies.
The 3PN and even 3.5PN center-of-mass equations of motion are obtained by replacing in the
general-frame 3.5PN equations of motion (168) the positions and velocities by their center-of-mass
expressions, applying as usual the order-reduction of all accelerations where necessary. We write the relative
acceleration in the center-of-mass frame in the form

and find [43] that the coefficients and are
Up to the 2.5PN order the result agrees with the calculation of [155]. The 3.5PN term is issued from
Refs. [136, 137, 138, 174, 148, 164]. At the 3PN order we have some gauge-dependent logarithms
containing a constant which is the “logarithmic barycenter” of the two constants and :
The logarithms in Equations (182, 183), together with the constant therein, can be removed by
applying the gauge transformation (169), while still staying within the class of harmonic coordinates. The
resulting modification of the equations of motion will affect only the coefficients of the 3PN order in
Equations (182, 183), let us denote them by and . The new values of these coefficients, say
and , obtained after removal of the logarithms by the latter harmonic gauge transformation,
are then [161]
These gauge-transformed coefficients are useful because they do not yield the usual complications associated
with logarithms. However, they must be handled with care in applications such as [161], since one must
ensure that all other quantities in the problem (energy, angular momentum, gravitational-wave fluxes, etc.)
are defined in the same specific harmonic gauge avoiding logarithms. In the following we shall no
longer use the coordinate system leading to Equations (185, 186). Therefore all expressions we
shall derive below, notably all those concerning the radiation field, are valid in the “standard”
harmonic coordinate system in which the equations of motion are given by Equation (168) or (182,
183).