Gravitational radiation reaction and balance equations to post-Newtonian order

TL;DR

This work advances the theoretical treatment of gravitational radiation reaction by deriving the first post-Newtonian (1PN) corrections to the Burke-Thorne reaction potential and establishing 1.5PN balance equations for energy, linear momentum, and angular momentum in general weakly self-gravitating systems. The authors perform a careful matching between a near-zone interior PN solution and a far-zone exterior post-Minkowskian solution, expressing the reaction potentials in terms of interior multipoles and showing consistency with the exterior multipole description. They demonstrate that, at 1PN order, the system’s energy and momenta losses coincide with known far-field fluxes, and they extend the formalism to include tail effects at 1.5PN order. The results provide a robust foundation for accurate modeling of inspiralling binaries and cross-checks with the Iyer–Will framework for binary systems. The tail terms yield a well-defined, tail-modified radiative quadrupole moment that preserves energy balance with the wave-zone flux, reinforcing the validity of the balance approach at these orders.

Abstract

Gravitational radiation reaction forces and balance equations for energy and momenta are investigated to 3/2 post-Newtonian (1.5PN) order beyond the quadrupole approximation, corresponding to the 4PN order in the equations of motion of an isolated system. By matching a post-Newtonian solution for the gravitational field inside the system to a post-Minkowskian solution (obtained in a previous work) for the gravitational field exterior to the system, we determine the 1PN relativistic corrections to the ``Newtonian" radiation reaction potential of Burke and Thorne. The 1PN reaction potential involves both scalar and vectorial components, with the scalar component depending on the mass-type quadrupole and octupole moments of the system, and the vectorial component depending in particular on the current-type quadrupole moment. In the case of binary systems, the 1PN radiation reaction potential has been shown to yield consistent results for the 3.5PN approximation in the binary's equations of motion. Adding up the effects of tails, the radiation reaction is then written to 1.5PN order. In this paper, we establish the validity to 1.5PN order, for general systems, of the balance equations relating the losses of energy, linear momentum, and angular momentum in the system to the corresponding fluxes in the radiation field far from the system.