Gravitational field of one uniformly moving extended body and N arbitrarily moving pointlike bodies in post-Minkowskian approximation

High precision astrometry, space missions and certain tests of General Relativity, require the knowledge of the metric tensor of the solar system, or more generally, of a gravitational system of N extended bodies. Presently, the metric of arbitrarily shaped, rotating, oscillating and arbitrarily moving N bodies of finite extension is only known for the case of slowly moving bodies in the post-Newtonian approximation, while the post-Minkowskian metric for arbitrarily moving celestial objects is known only for pointlike bodies with mass-monopoles and spin-dipoles. As one more step towards the aim of a global metric for a system of N arbitrarily shaped and arbitrarily moving massive bodies in post-Minkowskian approximation, two central issues are on the scope of our investigation. (i) We first consider one extended body with full multipole structure in uniform motion in some suitably chosen global reference system. For this problem a co-moving inertial system of coordinates can be introduced where the metric, outside the body, admits an expansion in terms of Damour–Iyer moments. A Poincare transformation then yields the corresponding metric tensor in the global system in post-Minkowskian approximation. (ii) It will be argued why the global metric, exact to post-Minkowskian order, can be obtained by means of an instantaneous Poincare transformation for the case of pointlike mass-monopoles and spin-dipoles in arbitrary motion.