Post-Minkowskian Scattering Angle in Einstein Gravity

TL;DR

This work establishes a deep equivalence between the classical Lippmann-Schwinger problem and the relativistic two-body energy equation in the Post-Minkowskian regime. By leveraging the implicit (Dini) function theorem and Damour’s non-relativistic mapping, it expresses the fully relativistic scattering angle entirely in terms of the classical part of the scattering amplitude via an effective radial potential $V_{\rm eff}(r)$ with coefficients $f_n(E)$. The authors derive an all-orders formula for the PM scattering angle, show manifest $r_m$-independence, and reveal vanishing patterns (notably the absence of an $f_1^2$ contribution at 2PM in 4D) that align with eikonal exponentiation. The results provide a universal, amplitude-driven framework to compute classical gravitational observables directly from amplitudes, confirming and extending known PM results and offering computational simplifications for high-order calculations.

Abstract

Using the implicit function theorem we demonstrate that solutions to the classical part of the relativistic Lippmann-Schwinger equation are in one-to-one correspondence with those of the energy equation of a relativistic two-body system. A corollary is that the scattering angle can be computed from the amplitude itself, without having to introduce a potential. All results are universal and provide for the case of general relativity a very simple formula for the scattering angle in terms of the classical part of the amplitude, to any order in the post-Minkowskian expansion.