Mining the Geodesic Equation for Scattering Data

TL;DR

This paper develops a method to connect geodesic equations with scattering amplitudes, enabling the analysis of gravitational interactions with small perturbations like tidal effects and higher derivatives at all orders in the post-Minkowskian expansion.

Contribution

It introduces an algebraic mapping between perturbed geodesic equations and scattering amplitudes, and provides formulas for tidal operators and higher derivative corrections at arbitrary orders.

Findings

Derived formulas for tidal operators and higher derivative corrections.

Established a general method for all-order PM calculations in the test-particle limit.

Connected geodesic motion with scattering amplitudes for small perturbations.

Abstract

The geodesic equation encodes test-particle dynamics at arbitrary gravitational coupling, hence retaining all orders in the post-Minkowskian (PM) expansion. Here we explore what geodesic motion can tell us about dynamical scattering in the presence of perturbatively small effects such as tidal distortion and higher derivative corrections to general relativity. We derive an algebraic map between the perturbed geodesic equation and the leading PM scattering amplitude at arbitrary mass ratio. As examples, we compute formulas for amplitudes and isotropic gauge Hamiltonians for certain infinite classes of tidal operators such as electric or magnetic Weyl to any power, and for higher derivative corrections to gravitationally interacting bodies with or without electric charge. Finally, we present a general method for calculating closed-form expressions for amplitudes and isotropic gauge Hamiltonians in the test-particle limit at all orders in the PM expansion.