Relation of Cartesian and spherical multipole moments in general relativity

The Earth’s gravitational field is represented by its multipole moments. Multipole moments have two kinds of equivalent forms, that is, the Cartesian symmetric and tracefree tensors and the spherical harmonic coefficients. The relation between these two forms is interesting and useful for some practical problems. Under Newtonian approximation, there exists a simple relation between the aforesaid two kinds of multipole moments (see Hartmann et al., 1994, for details). But in the 1PN approximation of general relativity, the relation mentioned above becomes complicated. This paper discusses how to turn the expansion of the 1PN Earth’s gravitational potential, which consists of a scalar potential and a vector potential, in terms of BD moments into that in terms of a set of time-slowlychanging, 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 between the 1PN spherical harmonic coefficients and the Cartesian multipole moments, and compute the expressions of the lowest order spherical harmonic coefficients including the relation between the 1PN Earth dynamical form-factor J2 and the BD mass quadrupole moment of the Earth. As for the 1PN vector potential, we also discuss its expansion in terms of Cartesian multipole moments under the rigidity approximation. In this paper, we emphasize the choice of the coordinate gauge. Under our ad hoc standard PN gauge, the results have simpler form and clearer physical meaning.