Gravitational scattering, post-Minkowskian approximation and Effective One-Body theory
Thibault Damour
TL;DR
This paper introduces a post-Minkowskian (PM) version of the Effective One-Body (EOB) framework that uses scattering-state dynamics rather than bound-state PN expansions. By computing the real two-body scattering function to first order in $G$ and comparing it with an effective-one-body description, the authors prove that, to $O(G)$, the two-body dynamics is equivalent to a test particle in a Schwarzschild metric with a uniquely determined, exactly quadratic energy map ${\mathcal{E}}_{\rm eff} = \frac{{\mathcal{E}}_{\rm real}^2 - m_1^2 c^4 - m_2^2 c^4}{2 (m_1+m_2) c^2}$. The equivalence holds in tensor-scalar gravity as well, and the framework naturally extends to higher PM orders and spinning bodies. Overall, the PM-based dictionary provides a velocity-agnostic route to refine analytical models of gravitational dynamics and waveforms for strong-field binaries.
Abstract
A novel approach to the Effective One-Body description of gravitationally interacting two-body systems is introduced. This approach is based on the post-Minkowskian approximation scheme (perturbation theory in G, without assuming small velocities), and employs a new dictionary focussing on the functional dependence of the scattering angle on the total energy and the total angular momentum of the system. Using this approach, we prove to all orders in v/c two results that were previously known to hold only to a limited post-Newtonian accuracy: (i) the relativistic gravitational dynamics of a two-body system is equivalent, at first post-Minkowskian order, to the relativistic dynamics of an effective test particle moving in a Schwarzschild metric; and (ii) this equivalence requires the existence of an exactly quadratic map between the real (relativistic) two-body energy and the (relativistic) energy of the effective particle. The same energy map is also shown to apply to the effective one-body description of two masses interacting via tensor-scalar gravity.