Relativistic Navigation: A Theoretical Foundation

We present a theoretical foundation for relativistic astronomical measurements in curved space-time. In particular, we discuss a new iterative approach for describing the dynamics of an astronomical N-body system. To do this, we generalize the Fock-Chandrasekhar method of the weak-field and slow-motion approximation (WFSMA) and develop a theory of relativistic reference frames (RF) for a gravitationally bounded many-extended-body problem. In any proper RF constructed in the immediate vicinity of an arbitrary body, the N-body solutions of the gravitational field equations are presented as a sum of the Riemann-flat inertial space-time, the gravitational field generated by the body itself, the unperturbed solutions for each body in the system transformed to the coordinates of this proper RF, and the gravitational interaction term. We develop the basic concept of a general theory of the celestial RFs applicable to a wide class of metric theories of gravity with an arbitrary model of matter distribution. We apply the proposed method to general relativity. Bodies are described using a perfect fluid model; as such, they possess any number of internal mass and current multipole moments which explicitly characterize their internal structure. The obtained relativistic equations of motion contain an explicit information about the coupling of the bodies proper multiple moments to the surrounding gravitational field. We further generalize the proposed method and include two Eddington parameters $(\gamma,\beta)$. This generalized approach was used to derive the relativistic equations of satellite motion in the vicinity of the extended bodies. We discuss the possible future implementation of the proposed formalism in software codes developed for solar-system orbit determination.