### 4.3 Lorentz contraction and multipole moments

In Equations (81, 82, 83), we have defined multipole moments of a star. The definition of those
multipole moments are operational ones and are not necessarily equal to “intrinsic” multipole moments of
the star. This is clear, for example, if we remember that a moving ball has spurious multipole moments due
to Lorentz contraction.
We have adopted the generalized Fermi normal coordinates (GFC) [13] as the star’s reference
coordinates to address this problem. A question specific to our formalism is whether the differences between
the multipole moments defined in Equations (81, 82, 83) and the multipole moments in GFC give purely
monopole terms. If so, we of course have to take into account such terms in the field to compute the
equations of motion for two intrinsically spherical star binaries. This problem is addressed in the
Appendix C of [91] and the differences are mainly attributed to the shape of the body zone. The body zone
which is spherical in the near zone coordinates (NZC) is not spherical in the GFC mainly because of a
kinematic effect (Lorentz contraction). To derive the 3 PN equations of motion, it turns out that it is
sufficient to compute the difference in the STF quadrupole moment up to 1 PN order. The result is

where . are the quadrupole moments defined in the generalized Fermi normal
coordinates. Note that the difference is expressed in a surface integral form.
As is obvious from Equation (106), this difference appears even if the companion star does not exist. We
note that we could derive the 3 PN metric for an isolated star moving at a constant velocity
using our method explained in this section by simply letting the mass of the companion star
be zero. Actually, the above is a necessary term which makes the so-obtained 3 PN
metric equal to the Schwarzschild metric boosted at the constant velocity in harmonic
coordinates.

The reason why the influence of a body’s Lorentz contraction appears starting from 3 PN order is as
follows. The body’s Lorentz contraction appears as an apparent deformation of the body, namely,
apparent quadrupole moment as the leading order in the frame where the body is moving.
The body’s (real) quadrupole affects the field from 2 PN order through (see Equation (78))

The radius of a compact body is of order of its mass , and the Lorentz contraction deforms the
body so that the radius of the body changes by the amount . So the apparent quadrupole
moments due to Lorentz contraction is of order of
This field is the 3 PN field and thus the Lorentz contraction of a body affects the equations of motion
starting from 3 PN order.