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

Abstract

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 SitterFokker -Eddington precession) and post-Newtonian contributions to the torque associated with the quadrupole moment and the quadrupole tidal tensor.PACS number(s): 95.10.Ce, 04.20.Me

Angular velocity of rotation of extended bodies in general relativity

“…The local reference systems are physically adequate for describing local physical processes related to the body under consideration. The local reference systems are proved to be suitable to describe rotational motion of the individual bodies (Damour, Soffel, Xu, 1993). Bearing in mind that the Earth is the primary body whose motion and rotation is to be investigated we will designate the local reference system of the body under consideration as GRS (Geocentric Reference System), although "GRS" can be related not only to the Earth, but to any other body.…”

“…Note also that our form of the rotational equations of motion (11)- (17) being written for a Damour-Soffel-Xu local reference system coincides with Eq. (2.4)-(2.8) of Damour, Soffel, Xu (1993).…”

“…In the framework of general relativity the spin (angular momentum) of an arbitrarily composed and shaped, weakly self-gravitating deformable body being a member of a system of Ν arbitrarily composed and shaped, weakly self-gravitating deformable bodies is a well-defined notion (see, e.g., Damour, Soffel, Xu, 1993). However, from the astronomical (observational) point of view not only the spin, but also the angular velocity of rotation of celestial bodies plays an important role.…”

“…Later the equations derived by Fock have been thoroughly investigated in Brumberg (1972) assuming the rigid-body distribution of the velocity of the matter of each body relative to a global reference system. Recently the result has been significantly improved by Damour, Soffel and Xu (1993). In that paper the post-Newtonian rotational equations of motion of an arbitrarily shaped, weakly self-gravitating bodies being members of a system of Ν bodies have been derived.…”

Relativistic aspects of Earth's rotation

“…In a series of papers, Damour, Soffel & Xu (1991, 1992, 1993 have constructed an improved post-Newtonian celestial mechanics with the BCRS and GCRS as basis. The dynamical equation for rotational motion of the Earth is given explicitly with full post-Newtonian expression for the torque in the GCRS.…”