RELATION OF CARTESIAN AND SPHERICAL MULTIPOLE MOMENTS IN GENERAL-RELATIVITY

The Earth's gravitational field is represented by its multipole moment s. Multipole moments have two kinds of equivalent forms, that is, the Cartesian symmetric and trace-free tensors and the spherical harmonic coefficients. The relation between these two forms is interesting and useful for some practical problems. Under Newtonian approximation, the re exists a simple relation between the aforesaid two kinds of multipo le moments (see Hartmann et al., 1994, for details). But in the IPN ap proximation of general relativity, the relation mentioned above become s complicated. This paper discusses how to turn the expansion of the I PN Earth's gravitational potential, which consists of a scalar potenti al and a vector potential, in terms of BD moments into that in terms o f a set of time-slowly-changing, observable multipole moments. Under a specific standard PN gauge, we derive the corresponding expansion of the potential in terms of spherical harmonics, obtain the relation bet ween the IPN spherical harmonic coefficients and the Cartesian multipo le moments, and compute the expressions of the lowest order spherical harmonic coefficients including the relation between the 1PN Earth dyn amical form-factor J(2) and the ED mass quadrupole moment of the Earth . As for the IPN vector potential, we also discuss its expansion in te rms of Cartesian multipole moments under the rigidity approximation. I n this paper, we emphasize the choice of the coordinate gauge. Under o ur ad hoc standard PN gauge, the results have simpler form and clearer physical meaning.