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

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.

Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries

“…[422, 335, 336]), after earlier attempts by Epstein & Wagoner [185] and Thorne [403]. This formalism has exactly the same scope as the one of Refs.…”

“…The latter works have been extended to 2PN order [226] and to include the leading spin-orbit effects [428]. The result has been then established from first principles (i.e., not relying on balance equations) in various works at 1PN order [260, 336, 278, 322, 254]. The 1PN radiation reaction force has also been obtained for general extended fluid systems in a particular gauge [43, 47].…”

Binary Systems as Test-Beds of Gravity Theories

“…) in the case of observational relevance where one neglects the self-gravity of the white-dwarf companion] but rather 23 [33]…”

“…(76). 23 This refinement is given here for pedagogical completeness. However, in practice, the lowest-order result…”

The Confrontation Between General Relativity and Experiment

“…The method provides a natural "regularization" technique to control potentially divergent integrals (see [51] for a thorough review). The "Direct Integration of the Relaxed Einstein Equations" (DIRE) approach of Will, Wiseman, and Pati [432,316,317] retains Eq. (79) as the global solution, but splits the integration into one over the near zone and another over the far zone, and uses different integration variables to carry out the explicit integrals over the two zones.…”