Post-Newtonian Mechanics Of A Gravitational N-body System

It has been almost a century since Einstein published the general relativity. But the relativistic N-body problem is still an unsolved problem and far from well-studied. Focusing on the solar system and the Earth-Moon system, this thesis tries to establish the post-Newtonian (PN) mechanics of a gravitational N-body system for future high-precision astrometry and experiments, especially the lunar laser ranging (LLR).A set of PN reference frames are introduced for a comprehensive study of the or-bital dynamics and rotational motion of Moon and Earth by LLR with the precision of 1 millimeter. We employ a framework of a scalar-tensor theory of gravity depending on two parameters,βandγ, of the parameterized post-Newtonian (PPN) formalism and utilize the concepts of the relativistic resolutions on reference frames adopted by the International Astronomical Union (IAU) in 2000. We assume that the solar system is isolated and space-time is asymptotically fiat at infinity. The primary reference frame covers the entire space-time, has its origin at the solar-system barycenter (SSB) and spatial axes stretching up to infinity. The SSB frame is not rotating with respect to a set of distant quasars that are assumed to be at rest on the sky forming the International Celestial Reference Frame (ICRF). The secondary reference frame has its origin at the Earth-Moon barycenter (EMB). The EMB frame is locally-inertial with its spatial axes spreading out up to the orbits of Venus and Mars, and is not rotating dynamically in the sense that equation of motion of a test particle moving with respect to the EMB frame, does not contain the Coriolis and centripetal forces. Two other local frames-geocentric (GRF) and selenocentric (SRF)-have their origins at the center of mass of Earth and Moon respectively and do not rotate dynamically. Each local frame is sub-ject to the geodetic precession both with respect to other local frames and with respect to the ICRF because of their relative motion with respect to each other. Theoretical advantage of the dynamically non-rotating local frames is in a more simple mathemat-ical description. Each local frame can be aligned with the axes of ICRF after applying the matrix of the relativistic precession. The set of one global and three local frames is introduced in order to fully decouple the relative motion of Moon with respect to Earth from the orbital.motion of the Earth-Moon barycenter as well as to connect the coordinate description of the lunar motion, an observer on Earth, and a retro-reflector on Moon to directly measurable quantities such as the proper time and the round-trip laser-light distance. We solve the gravitational field equations and find out the metric tensor and the scalar field in all frames, which description includes the PN definition of the multipole moments of the gravitational field of Earth and Moon. We also de-rive the PN coordinate transformations between the frames and analyze the residual gauge freedom imposed by the scalar-tensor theory of gravity on the metric tensor and equations of motion. The residual gauge freedom is used for removing the spurious, coordinate-dependent PN effects from the equations of motion of Earth and Moon.Based on the previous reference frames, the equations of motion are derived. With the law of motion of the origin of the EMB frame in the SSB frame given by the matching procedure and the condition that the origin of the EMB frame coincides with the center of mass of the Earth-Moon system at any instant, the equations of the motion of the center of mass of the Earth-Moon system in the SSB frame are obtained. The 3D coordinate accelerations in these equations are expressed by the local multipole moments. Through extension of multipole moments of external gravitational fields, masses and spins from Cartesian symmetry-trace-free tensors to 4D covariant tensors, it is shown that the equations of motion of the center of mass of the Earth-Moon system in the SSB frame can be written in a covariant form and the higher moments (l> 1) of the Earth-Moon system and the violation of the strong equivalence principle cause the world line of its center of mass to deviate from the geodesics in the background gravitational field. The equations of motion of the Earth or the Moon in the EMB frame are given, and then they lead to the equations of motion of the Moon with respect to the Earth, which could be used in the next generation of LLR. All of the results could be easily extended and applied to a gravitational N-body system.