Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. II. Two-body equations of motion to second post-Newtonian order, and radiation-reaction to 3.5 post-Newtonian order

TL;DR

This paper derives precise equations of motion and radiation-reaction effects for binary systems in general relativity using the direct integration of relaxed Einstein equations, extending previous results to higher post-Newtonian orders.

Contribution

It introduces a method to compute binary system dynamics up to 3.5PN order, including radiation reaction, using a divergence-free potential framework, and confirms consistency with energy and momentum loss.

Findings

Equations of motion up to 2PN order match previous results.

Radiation-reaction effects are computed at 2.5PN and 3.5PN orders.

The method is free of divergences and applicable to non-rotating spherical bodies.

Abstract

We derive the equations of motion for binary systems of compact bodies in the post-Newtonian (PN) approximation to general relativity. Results are given through 2PN order (order (v/c)^4 beyond Newtonian theory), and for gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method is based on a framework for direct integration of the relaxed Einstein equations (DIRE) developed earlier, in which the equations of motion through 3.5PN order can be expressed in terms of Poisson-like potentials that are generalizations of the instantaneous Newtonian gravitational potential, and in terms of multipole moments of the system and their time derivatives. All potentials are well defined and free of divergences associated with integrating quantities over all space. Using a model of the bodies as spherical, non-rotating fluid balls whose characteristic size s is small compared to the bodies' separation r, we develop a method for carefully extracting only terms that are independent of the parameter s, thereby ignoring tidal interactions, spin effects, and internal self-gravity effects. Through 2.5PN order, the resulting equations agree completely with those obtained by other methods; the new 3.5PN back-reaction results are shown to be consistent with the loss of energy and angular momentum via radiation to infinity.