Splitting gravity's self-force into conservative and dissipative parts gets complicated

When a body moves through its own field, its motion is altered by a self-force that can be split into a part that conserves energy and a part that dissipates it. Using a nonlinear scalar field as a controlled stand-in for gravity, this work shows that the standard first-order splitting is unique, but multiple inequivalent definitions emerge at second order, each yielding a different Hamiltonian. The analysis generalizes the Detweiler-Whiting framework to include a three-point function and clarifies which choices are physically motivated. Results apply to unbound (scattering) trajectories; bound orbits remain out of reach due to infrared divergences.