Second-order perturbation theory: the problem of infinite mode coupling

TL;DR

This paper discusses the challenge of infinite mode coupling in second-order perturbation theory for self-force calculations, proposing a strategy to address the issue using a scalar toy model.

Contribution

It identifies the problem of infinite mode coupling due to singularities in second-order perturbations and offers a robust solution demonstrated with a scalar toy model.

Findings

Infinite mode coupling arises from singularities in the field.

A robust strategy can mitigate the infinite mode coupling problem.

The approach is demonstrated using a scalar toy model in flat space.

Abstract

Second-order self-force computations, which will be essential in modeling extreme-mass-ratio inspirals, involve two major new difficulties that were not present at first order. One is the problem of large scales, discussed in [Phys. Rev. D 92, 104047 (2015)]. Here we discuss the second difficulty, which occurs instead on small scales: if we expand the field equations in spherical harmonics, then because the first-order field contains a singularity, we require an arbitrarily large number of first-order modes to accurately compute even a single second-order mode. This is a generic feature of nonlinear field equations containing singularities, allowing us to study it in the simple context of a scalar toy model in flat space. Using that model, we illustrate the problem and demonstrate a robust strategy for overcoming it.