Radiation-reaction driven dynamics at the third-and-a-half post-Newtonian order in different gauges
We consider the coordinate-dependent definition of the radiation-reaction force at the third-and-a-half post-Newtonian order for general orbits. In order to (partially) determine its expression we refer to the balance method, involving the energy and angular momentum lost by the system, the Schott terms, and the energy and angular momentum fluxes at infinity. Only the latter are gauge-invariant quantities when passing from a coordinate system to another. The gauge dependence of both radiation-reaction force and Schott terms is encoded in a set of gauge parameters entering their definitions. We show how to relate the harmonic-coordinate losses of mechanical energy and angular momentum by the system with the radial and azimuthal components of the radiation-reaction force in a different coordinate system expressed in terms of phase-space variables in a Hamiltonian framework. The advantage of this approach is that only the coordinate transformation between harmonic coordinates and coordinates and momenta in the new coordinate system is needed, without solving again the balance equations. We derive such a transformation for both Arnowitt-Deser-Misner and Effective-One-Body coordinates. In the latter case we also discuss some simplifying choices of the gauge parameters adopted in current waveform models. Finally, we show how to obtain the solution for the radiation-reaction correction to the orbit in the new coordinate system simply by transforming the harmonic-coordinate solution known in the literature. This is a remarkable simplification, since one can avoid to solve again for the radiation-reacted dynamics.