New perturbative method for solving the gravitational N-body problem in the general theory of relativity

TL;DR

This paper introduces a new perturbative approach to model the dynamics of gravitational N-body systems in general relativity, accounting for internal multipole moments and providing explicit coordinate transformations and equations of motion.

Contribution

It develops a novel perturbative method for solving the gravitational N-body problem in general relativity, including detailed coordinate transformations and relativistic equations of motion.

Findings

Derived explicit post-Galilean coordinate transformations.

Formulated relativistic equations of motion for N-body systems.

Presented metric tensor structures for various reference frames.

Abstract

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 the general theory of 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 that is covariantly transformed to the coordinates of this frame, and iii) the gravitational interaction term. We use the harmonic gauge conditions that allow reconstruction of a significant part of the structure of the post-Galilean coordinate transformation functions relating global coordinates of 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 determined from dynamical conditions, 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 three-momentum. The resulting post-Galilean coordinate transformations have an approximate group structure that extends the Poincar'e group of global transformations to the case of accelerating observers in a gravitational field of $N$-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.