Celestial coordinate reference systems in curved space-time

Abstract

In the weak-field and slow-motion approximation of the General Relativity theory the relativistic theory of construction of nonrotating harmonic coordinate reference systems is developed. The general case of an isolated astronomicalN-body system is considered. The bodies are assumed to consist of a perfect fluid and to possess any number of the internal mass and current multipole moments characterizing the internal structure and own gravitational field of the bodies. The description of the coordinate reference systems and gravitational field is realized by the specific forms of the metric tensor. A metric is determined by the method of the Post-Newtonian Approximations (PNA) from the inhomogeneous Einstein equations under the harmonic coordinate condition. We have obtained two different specific forms of a metric, which are related to the inertial and quasiinertial coordinate reference systems. The metric in inertial coordinates is a near-zone solution of the Einstein equations for anN-body system. The metric in quasi-inertial coordinates is a solution of Einstein equations in the body's neighborhood, which is a world tube surrounding the body under consideration and extending up to another nearest body. The coordinate transformation between the inertial and quasiinertial reference systems is derived by a matching of both solutions in the body's neighborhood. A new method is proposed for construction of the body's proper reference system. This coordinate system has an origin which moves along a nongeodesic (in the general case) worldline of the body's center of inertia. The proper reference system is used for derivation of the Newtonian equations of translational and rotational motion of the body. The equations give us exhaustive information about the nonlinear Newtonian interaction between gravitational fields of the bodies in terms of internal mass multipole moments. Finally, the coordinate transformation between the inertial and proper reference systems is discussed in the first PNA.