Gravitational radiation reaction for compact binary systems at the fourth-and-a-half post-Newtonian order

We compute the gravitational radiation–reaction (RR) force on a compact binary source at the fourth-and-a-half post-Newtonian (4.5PN) order of general relativity, i.e. 2PN order beyond the leading 2.5PN radiation reaction. The calculation is valid for general orbits in a general frame, but in a particular coordinate system which is an extension of the Burke–Thorne coordinate system at the lowest order. With the RR acceleration, we derive (from first principles) the flux-balance laws associated with the energy, the angular and linear momenta, and the center-of-mass (CM) position, in a general frame and up to 4.5PN order. Restricting our attention to the frame of the center of mass, we point out that the equations of motion (EOM) acquire a non-local-in-time contribution at the 4.5PN order, made of the integrated flux of linear momentum (responsible for the recoil of the source) together with the instantaneous flux of CM position. The non-local contribution was overlooked in the past literature, which assumed locality of the RR force in the center of mass frame at 4.5PN order. We discuss the consequences of this non-local effect and obtain consistent non-local EOM and flux balance laws at 4.5PN order in the CM frame.