Black Hole Dynamics at Fifth Post-Newtonian Order

TL;DR

This paper derives the 5PN order two-body dynamics including radiation effects, introducing a new isotropic-like description that clarifies the conservative Hamiltonian and scattering observables.

Contribution

It provides a systematic framework for characterizing two-body dynamics at 5PN order using scattering data and Feynman's prescription, fixing key coefficients and clarifying nonlocal contributions.

Findings

Derived 5PN impulse, scattering angle, and time delay including radiation-reaction.

Fixed the conservative Hamiltonian and EOB coefficients at 5PN order.

Compared Feynman's and $ ext{-}3$ prescriptions, finding exact agreement with recent PM results.

Abstract

Using the worldline action in [2409.05860], we derive the total even-in-velocity (relative) impulse, scattering angle, and time delay at fifth post-Newtonian (5PN) order, including radiation-reaction and hereditary contributions at ${\cal O}(G^5\nu^2)$ and ${\cal O}(G^6\nu^2)$. We introduce an isotropic-like description which, together with the associated losses of energy and angular momentum, fixes the evolution of the system from scattering data. This framework opens the door to an unambiguous characterization of the underlying two-body dynamics solely in terms of scattering observables. Following [2409.05860], we isolate a conservative component using Feynman's $i0^+$ prescription. This sector contains both "tail-like" and "memory-like" contributions, the latter being nonlocal in time and described by a double Principal-Value integral. Owing to the local-in-time character of the corresponding (in-in) action, we establish a systematic procedure that is consistent with Feynman's prescription while preserving the complete local dynamics. This provides a universal contribution to the conservative (isotropic) Hamiltonian at 5PN order and, as a byproduct, also fixes the value of the Effective One Body coefficients $\{{\bar d}_{5{\rm loc}}, a_{6{\rm loc}}\}$ consistently with the Tutti-Frutti framework. For completeness, we analyse the "$\gamma\text{-}3$" prescription introduced in recent post-Minkowskian computations. When implemented in our formalism, we find exact agreement over the overlapping regime of validity. In contrast, Feynman's prescription yields a (local) memory-like contribution with the opposite sign at ${\cal O}(G^5\nu^2)$. We also find that an analogous $\gamma\text{-}3$ rerouting at ${\cal O}(G^6\nu^2)$ would be incompatible with the conjecture that all $\pi^2$ terms arise solely from the potential region, while Feynman's formulation preserves this expectation.