Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. IV. Radiation reaction for binary systems with spin-spin coupling

TL;DR

This paper extends the post-Newtonian framework to include spin-spin interactions in radiation-reaction terms at $3.5PN$ for binary systems of spinning, finite-size bodies. Using hydrodynamic equations and the conserved baryonic density, it derives the 3.5PN spin-spin contributions to both the relative motion and the evolution of spins, showing that radiation reaction induces a pure precession of the spins without altering their magnitudes, and that this effect vanishes if a spin is aligned with the orbital angular momentum. The authors define the proper spin to maintain consistent dynamics under radiation reaction and demonstrate exact energy and angular momentum balance with the gravitational-wave fluxes calculated by Kidder et al., modulo total time derivatives that can be absorbed into redefinitions of conserved quantities. The results, applicable to quasi-circular and highly eccentric inspirals, provide instantaneous equations of motion suitable for numerical evolution and set the stage for future inclusion of higher-order spin terms beyond $S_1S_2$ products.

Abstract

Using post-Newtonian equations of motion for fluid bodies that include radiation-reaction terms at 2.5 and 3.5 post-Newtonian (PN) order O[(v/c)^5] and O[(v/c)^7] beyond Newtonian order), we derive the equations of motion for binary systems with spinning bodies, including spin-spin effects. In particular we determine the effects of radiation-reaction coupled to spin-spin effects on the two-body equations of motion, and on the evolution of the spins. We find that radiation damping causes a 3.5PN order, spin-spin induced precession of the individual spins. This contrasts with the case of spin-orbit coupling, where there is no effect on the spins at 3.5PN order. Employing the equations of motion and of spin precession, we verify that the loss of total energy and total angular momentum induced by spin-spin effects precisely balances the radiative flux of those quantities calculated by Kidder et al.