General-relativistic celestial mechanics. III. Rotational equations of motion

The rotational laws of motion for arbitrarily shaped, weakly self-gravitating bodies, members of gravitationally interacting N-body systems, are obtained at the first post-Newtonian approximation of general relativity. The derivation uses our previously introduced framework, characterized by the combined use of N local (body-attached) reference systems with one global reference system, and by the introduction of new sets of relativistic multipole moments, and relativistic tidal moments. We show how to associate with each body (considered in its corresponding local frame) a first-post-Newtonian-accurate spin vector, whose local-time evolution is entirely determined by the coupling between the multipole moments of that body and the tidal moments it experiences. The leading relativistic effects in the spin motion are discussed: gravitational Larmor theorem (de Sitter--Fokker--Eddington precession) and post-Newtonian contributions to the torque associated with the quadrupole moment and the quadrupole tidal tensor.