Second-order perturbation theory: problems on large scales

TL;DR

This paper addresses large-scale issues in second-order perturbation theory, proposing methods to manage secular growth and divergences, with applications to gravitational self-force problems in general relativity.

Contribution

Develops a multiscale expansion and matching techniques to overcome large-distance pathologies in second-order perturbation theory, applicable to gravitational self-force calculations.

Findings

Secular growth is controlled with multiscale expansion.

Infrared divergences are eliminated through matching to retarded solutions.

Methods are adaptable to black hole orbit scenarios.

Abstract

In general-relativistic perturbation theory, a point mass accelerates away from geodesic motion due to its gravitational self-force. Because the self-force is small, one can often approximate the motion as geodesic. However, it is well known that self-force effects accumulate over time, making the geodesic approximation fail on long timescales. It is less well known that this failure at large times translates to a failure at large distances as well. At second perturbative order, two large-distance pathologies arise: spurious secular growth and infrared-divergent retarded integrals. Both stand in the way of practical computations of second-order self-force effects. Utilizing a simple flat-space scalar toy model, I develop methods to overcome these obstacles. The secular growth is tamed with a multiscale expansion that captures the system's slow evolution. The divergent integrals are eliminated by matching to the correct retarded solution at large distances. I also show how to extract conservative self-force effects by taking local-in-time "snapshots" of the global solution. These methods are readily adaptable to the physically relevant case of a point mass orbiting a black hole.