Conservative and dissipative sectors in a nonlinear scalar model for the gravitational self-force problem

TL;DR

This paper explores the decomposition of the scalar self-force into conservative and dissipative parts within a nonlinear scalar model, analyzing higher-order implications and Hamiltonian properties for unbound trajectories.

Contribution

It introduces multiple definitions of the conservative second-order self-force and discusses their properties and Hamiltonian formulations in a scalar toy model.

Findings

Different criteria for splitting self-force agree at first order

Higher-order conservative self-force definitions vary in implications

Hamiltonian formulations are identified for conservative sectors

Abstract

When considering how self-interaction affects an object's motion, it can be convenient to decompose the self-force into conservative and dissipative pieces. As a toy model for understanding such decompositions of the gravitational self-force, we consider objects that do not affect the spacetime, but are instead coupled to a nonlinear scalar field. There is then a standard splitting of the first-order scalar self-force into conservative and dissipative components. Multiple criteria can be used to obtain this splitting, all of which imply the same result. However, the implications of these criteria generically differ at higher orders. Demanding that any reasonable conservative sector be Hamiltonian, we identify multiple possible definitions of the conservative second-order self-force. Motivations for these possibilities and their properties are discussed and relevant Hamiltonians are obtained. We assume the existence of a three-point function with certain properties that is a generalization of the Detweiler-Whiting two-point function. These results apply to the two-body problem but are restricted to unbound scattering trajectories, due to infrared divergences that arise for bound orbits.