Astron. Astrophys. 333, 1100-1106 (1998)

3. Projections of BD moments in the co-rotating coordinate system

The 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 T. Only physical quantities that are time-independent or slowly-changing with time can be considered as astronomical or physical constants. Therefore, a set of new multipole moments that are time-independent or slowly-changing with time should be defined.

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), W and can be projected into the co-rotating system , and obtain

where w and have expansions formally similar to those of W and . If the harmonic gauge is adopted: , corresponding to Eqs. (10) and (11), w and can be written as

and

respectively, where

If the standard PN gauge is adopted, corresponding to Eqs. (13) and (14), w and read

and

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 , w is expressed by Eq. (18), and it can be formally expanded in terms of spherical harmonics, but, the spherical harmonic coefficients and must be related to r. This would cause that and are not constants even in the rigid case.

© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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