Second Order Gravitational Self-Force

TL;DR

This work develops a first-principles treatment of the second-order gravitational self-force for extreme mass-ratio inspirals by constructing a perturbative framework with distinct far-zone and near-zone limits. It introduces the $P$-gauge, a mass-centered, near-zone consistent gauge, and an effective-source method to compute global metric perturbations without relying on singular sources, enabling a consistent definition of motion at second order. The authors derive explicit forms for the first- and second-order singular and regular fields in a $P$-smooth class of gauges and provide a concrete prescription to determine the metric perturbations and the corresponding motion, including the gauge freedom that influences second-order quantities. They discuss the implications for long-term waveform generation, connecting their approach to existing schemes for self-force evolution and outlining how to extend these results to inspiral calculations via a sequence of perturbative steps and adiabatic evolution.

Abstract

The second-order gravitational self-force on a small body is an important problem for gravitational-wave astronomy of extreme mass-ratio inspirals. We give a first-principles derivation of a prescription for computing the first and second perturbed metric and motion of a small body moving through a vacuum background spacetime. The procedure involves solving for a "regular field" with a specified (sufficiently smooth) "effective source", and may be applied in any gauge that produces a sufficiently smooth regular field.