## 3. Projections of BD moments in the co-rotating coordinate systemThe BD moments are the basic physical quantities in DSX theory.
They generate the gravitational field in place of the stress-energy
tensor of the matter. They are defined in the local DSX coordinate
system (e.g. dynamical non-rotating coordinate system) of a body. Due
to the rotation of celestial bodies, BD moments generally change with
time According to Newtonian mechanics, the mass multipole moments of a rigid body in its mass-centered and co-rotating reference system are time-independent. This prompts that the projections of the 1PN mass multipole moments of a body in its co-rotating system would become time-slowly-changing. Moreover, it is easy to infer that the projections of the time-derivatives of ( and ) in the co-rotating system are also time-slowly-changing. Two facts make things more complicated. They are: (a) There is no " rigid body" in general relativity; (b) The real celestial body such as the Earth is not rigid under tidal influence and geophysical mass redistribution even in Newtonian mechanics. So there is no unique co-rotating system. We must define the co-rotating system of a body to assure that the spin of the body with respect to this system is as small as possible. Let be the co-rotating coordinate system of the Earth, which is connected with the local DSX coordinate system of the Earth by an orthogonal matrix . The relation between the two coordinate systems is , or . The matrix satisfies Apparently, . The rotational angular velocity reads where is the projection of
on the spatial axes of the co-rotating system,
= ,
= . Making use of
and Eq. (15), where If the standard PN gauge is adopted, corresponding to
Eqs. (13) and (14), respectively. In Eqs. (18)-(22), , , and are the projections of , , and , respectively. They are all time-slowly-changing physical quantities. It should be noted that and are in general not small quantities, though and are small indeed. One can find that the right-hand side of Eq. (21) is simpler
than that of Eq. (18), which is just the reason to choose the
standard PN gauge (Eq. (12)) other than the harmonic gauge
. The merits of choosing the gauge
(Eq. (12)) have been stated in Sect. 2. One more argument is
: if we take up the harmonic gauge , © European Southern Observatory (ESO) 1998 Online publication: April 28, 1998 |