Motion of Small Objects in Curved Spacetimes: An Introduction to Gravitational Self-Force

Abstract

In recent years, asymptotic approximation schemes have been developed to describe the motion of a small compact object through a vacuum background to any order in perturbation theory. The schemes are based on rigorous methods of matched asymptotic expansions, which account for the object’s finite size, require no “regularization” of divergent quantities, and are valid for strong fields and relativistic speeds. Up to couplings of the object’s multipole moments to the external background curvature, these schemes have established that at least through second order in perturbation theory, the object’s motion satisfies a generalized equivalence principle: it moves on a geodesic of a certain smooth metric satisfying the vacuum Einstein equation. I describe the foundations of this result, particularly focusing on the fundamental notion of how a small object’s motion is represented in perturbation theory. The three common representations of perturbed motion are (i) the “Gralla-Wald” description in terms of small deviations from a reference geodesic, (ii) the “self-consistent” description in terms of a worldline that obeys a self-accelerated equation of motion, and (iii) the “osculating geodesics” description, which utilizes both (i) and (ii). Because of the coordinate freedom in general relativity, any coordinate desscription of motion in perturbation theory is intimately related to the theory’s gauge freedom. I describe asymptotic solutions of the Einstein equations adapted to each of the three representations of motion, and I discuss the gauge freedom associated with each. I conclude with a discussion of how gauge freedom must be refined in the context of long-term dynamics.