Sixth post-Newtonian local-in-time dynamics of binary systems

Using a recently introduced method [D. Bini, T. Damour, and A. Geralico, Phys. Rev. Lett. 123, 231104 (2019)], which splits the conservative dynamics of gravitationally interacting binary systems into a nonlocal-in-time part and a local-in-time one, we compute the local part of the dynamics at the sixth post-Newtonian (6PN) accuracy. Our strategy combines several theoretical formalisms: post-Newtonian, post-Minkowskian, multipolar-post-Minkowskian, effective-field-theory, gravitational self-force, effective one-body, and Delaunay averaging. The full functional structure of the local 6PN Hamiltonian (which involves 151 numerical coefficients) is derived, but contains four undetermined numerical coefficients. Our 6PN-accurate results are complete at orders $G^3$ and $G^4$, and the derived $O(G^3)$ scattering angle agrees, within our 6PN accuracy, with the computation of [Z. Bern, C. Cheung, R. Roiban, C. H. Shen, M. P. Solon, and M. Zeng, Phys. Rev. Lett. 122, 201603 (2019)]. All our results are expressed in several different gauge-invariant ways. We highlight, and make a crucial use of, several aspects of the hidden simplicity of the mass-ratio dependence of the two-body dynamics.

Figure 1

Schematic representation of the irreducible information contained, at each post-Minkowskian level (keyed by a power of $u=GM/r$), in the local dynamics. Each vertical column of dots describes the post-Newtonian expansion (keyed by powers of $p^2$) of an energy-dependent function parametrizing the scattering angle. The various columns at a given post-Minkowskian level correspond to increasing powers of the symmetric mass-ratio $\nu$. See text for details.