1.3 Plan of this paper

This article is organized as follows: In Section 2 we introduce the Newtonian limit in general relativity and present how to construct the post-Newtonian hierarchy. There we mention how to avoid divergent integrals which appear at a higher order in the previous treatments. We shall then give a formulation of the dynamics of a Newtonian perfect fluid as the lowest order post-Newtonian approximation in harmonic coordinates.

We focus our attention on the derivation of the 3 PN equations of motion based on the work by the authors of this article from Section 3 to Section 8.

In Section 3, we discuss the basics of our derivation of the post-Newtonian equations of motion for relativistic compact binaries. There we discuss how to incorporate strong internal gravity in the post-Newtonian approximation.

In Section 4, we formulate our method in more detail. As an application, we derive the Newtonian and the 1 PN equations of motion in our formalism.

In Section 5, we briefly explain how to derive the 3 PN gravitational field when possible, and how to derive equations of motion when the gravitational field is unavailable in an explicit closed form.

In Sections 6 and 7, we derive the mass-energy relation and the momentum-velocity relation up to 3 PN order, and finally in Section 8, we derive the 3 PN equations of motion. The resulting equations of motion respect Lorentz invariance, admit a conserved energy (modulo the 2.5 PN radiation reaction effect), and are free from any ambiguity. Section 8 ends with a summary and some remarks on possible future study of the 4 PN order iteration.

In Appendix A, we give a brief sketch of the direct integration of the relaxed Einstein equations (DIRE) method by Will and Wiseman [129162165] which we have used in this article.

Although in the main body of this article we focus our attention on spherical massive bodies (or point particles), we shall apply our formalism to extended bodies in Appendix B.

In Appendix C, using the strong field point particle limit and the surface integral approach, we discuss that a particle with strong internal gravity moves on a geodesic of some smooth metric part which is produced by the particle itself. This work is done by the authors in the collaboration with Takashi Fukumoto.