Perspective on gravitational self-force analyses

Abstract

A point particle of mass $\mu$ moving on a geodesic creates a perturbation $h_{ab}$, of the spacetime metric $g_{ab}$, that diverges at the particle. Simple expressions are given for the singular $\mu/r$ part of $h_{ab}$ and its distortion caused by the spacetime. This singular part $h^\SS_{ab}$ is described in different coordinate systems and in different gauges. Subtracting $h^\SS_{ab}$ from $h_{ab}$ leaves a regular remainder $h^\R_{ab}$. The self-force on the particle from its own gravitational field adjusts the world line at $\Or(\mu)$ to be a geodesic of $g_{ab}+h^\R_{ab}$; this adjustment includes all of the effects of radiation reaction. For the case that the particle is a small non-rotating black hole, we give a uniformly valid approximation to a solution of the Einstein equations, with a remainder of $\Or(\mu^2)$ as $\mu\to0$. An example presents the actual steps involved in a self-force calculation. Gauge freedom introduces ambiguity in perturbation analysis. However, physically interesting problems avoid this ambiguity.