Nonlinear gravitational self-force. I. Field outside a small body

TL;DR

This paper develops a method to compute the nonlinear gravitational self-force on small bodies by deriving the metric perturbation at all orders in a generalized gauge, enabling more accurate modeling of compact binary dynamics.

Contribution

It introduces a framework for obtaining the metric perturbation outside a small body at all orders, including nonlinear effects and spin, using a buffer region and numerical puncture schemes.

Findings

Derived the form of the perturbation in the buffer region analytically.

Extended the puncture scheme to second order including spin effects.

Defined generalized singular and regular fields for higher-order self-force analysis.

Abstract

A small extended body moving through an external spacetime $g_{\alpha\beta}$ creates a metric perturbation $h_{\alpha\beta}$, which forces the body away from geodesic motion in $g_{\alpha\beta}$. The foundations of this effect, called the gravitational self-force, are now well established, but concrete results have mostly been limited to linear order. Accurately modeling the dynamics of compact binaries requires proceeding to nonlinear orders. To that end, I show how to obtain the metric perturbation outside the body at all orders in a class of generalized wave gauges. In a small buffer region surrounding the body, the form of the perturbation can be found analytically as an expansion for small distances $r$ from a representative worldline. Given only a specification of the body's multipole moments, the field obtained in the buffer region suffices to find the metric everywhere outside the body via a numerical puncture scheme. Following this procedure at first and second order, I calculate the field in the buffer region around an arbitrarily structured compact body at sufficiently high order in $r$ to numerically implement a second-order puncture scheme, including effects of the body's spin. I also define $n$th-order (local) generalizations of the Detweiler-Whiting singular and regular fields and show that in a certain sense, the body can be viewed as a skeleton of multipole moments.