Far-Zone Gravitational Interactions at the Sixth Post-Newtonian Order

Abstract

We compute the nonlinear far-zone dynamics of compact binaries at the sixth post-Newtonian (6PN) order in the effective field theory of general relativity. We derive the complete $\mathcal{O}(G^2)$ far-zone action of a generic multipolar source in both the causal and the conservative prescriptions. Besides the known tails and failed tails, it contains five new sectors generated by the nonlinear memory of the emitted waves. On scattering orbits, they contain a boundary term, nonlocal in time, fixed by the balance of angular momentum. The same construction yields the emitted waveforms, whose energy and angular-momentum fluxes balance the losses of the action. For the binary, the 6PN dynamics also requires PN corrections to the lower-order sectors, the iterated radiation reaction, and a coupling quadratic in the curvature of the composite source, whose coefficient we fix by matching the gravitational Compton amplitude to that of the constituents. We compute the two-radiation contributions to the total scattering angle at fourth, fifth and sixth post-Minkowskian (PM) order and to the conservative angle at 4PM and 5PM, retaining the integer-PN terms through 6PN. For the conservative angle, we give both the local and $\gamma$-3 prescriptions. Combined with the tail and the known potential contributions, our results reproduce the conservative and total 4PM angles and the second-self-force part of the conservative 5PM angle. Including the dissipative tail, we recover the total 5PM angle through 5PN and determine its new 6PN coefficient. At 6PM, the two-radiation contribution to the total angle extends the known 5PN result to 6PN.