About the MacCullagh relations in relativity

The Newtonian MacCullagh relations relate two different aspects of the global Earth: its rotational motion and its external gravitational field. They relate components of the tensor of inertia with l = 2 potential coefficients.These two concepts can be generalized to the first post-Newtonian approximation to Einstein's theory of gravity and one can ask if the usual form of the MacCullagh relation still holds if 1/c2 are taken into account. To answer this question a simple Newtonian model for the Earth was employed: a uniformly rotating homogenous oblate spheroid. For this model a violation of the usual MacCullagh relations was found. Implications for a post-Newtonian nutation model are shown to be negligible.