A Rigorous Derivation of Gravitational Self-force

TL;DR

This paper rigorously derives the gravitational self-force equations in general relativity by analyzing a family of scaled metrics, establishing the worldline as a geodesic and identifying self-force effects as first-order perturbations.

Contribution

It provides a systematic, assumption-free derivation of the MiSaTaQuWa equations using a one-parameter family of metrics, clarifying the foundations of gravitational self-force theory.

Findings

Proves the worldline is a background geodesic in the limit.

Identifies self-force as a first-order correction in the scaling parameter.

Clarifies the status and derivation of the MiSaTaQuWa equations.

Abstract

There is general agreement that the MiSaTaQuWa equations should describe the motion of a "small body" in general relativity, taking into account the leading order self-force effects. However, previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features. For example, all previous derivations have invoked, without proper justification, the step of "Lorenz gauge relaxation", wherein the linearized Einstein equation is written down in the form appropriate to the Lorenz gauge, but the Lorenz gauge condition is then not imposed--thereby making the resulting equations for the metric perturbation inequivalent to the linearized Einstein equations. In this paper, we analyze the issue of "particle motion" in general relativity in a systematic and rigorous way by considering a one-parameter family of metrics, $g_{ab} (\lambda)$, corresponding to having a body (or black hole) that is "scaled down" to zero size and mass in an appropriate manner. We prove that the limiting worldline of such a one-parameter family must be a geodesic of the background metric, $g_{ab} (\lambda=0)$. Gravitational self-force--as well as the force due to coupling of the spin of the body to curvature--then arises as a first-order perturbative correction in $\lambda$ to this worldline. No assumptions are made in our analysis apart from the smoothness and limit properties of the one-parameter family of metrics. Our approach should provide a framework for systematically calculating higher order corrections to gravitational self-force, including higher multipole effects, although we do not attempt to go beyond first order calculations here. The status of the MiSaTaQuWa equations is explained.