Slava G. Turyshev, Viktor T. Toth
We present a new approach to describe the dynamics of an isolated, gravitationally bound astronomical $N$-body system in the weak field and slow-motion approximation of general relativity. Celestial bodies are described using an arbitrary energy-momentum tensor and assumed to possess any number of internal multipole moments. The solution of the gravitational field equations in any reference frame is presented as a sum of three terms: i) the inertial flat spacetime in that frame, ii) unperturbed solutions for each body in the system boosted to the coordinates of this frame, and iii) the gravitational interaction term. We use the harmonic gauge conditions to impose a significant constraint on the structure of the post-Galilean coordinate transformation functions that relate global coordinates in the inertial reference frame to the local coordinates of the non-inertial frame associated with a particular body. The remaining parts of these functions are constrained using dynamical conditions, which are obtained by constructing the relativistic proper reference frame associated with a particular body. In this frame, the effect of external forces acting on the body is balanced by the fictitious frame-reaction force that is needed to keep the body at rest with respect to the frame, conserving its relativistic linear momentum. We find that this is sufficient to determine explicitly all the terms of the coordinate transformation. The resulting post-Galilean coordinate transformations have an approximate group structure that extends the Poincar\'e group of global transformations to the case of a gravitational $N$-body system. We present and discuss the structure of the metric tensors corresponding to the reference frames involved, the rules for transforming relativistic gravitational potentials, the coordinate transformations between frames and the resulting relativistic equations of motion.
View original:
http://arxiv.org/abs/1304.8122
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