Perspective on gravitational self-force analyses

TL;DR

The paper develops a rigorous framework for gravitational self-force, showing how a small mass moves on a geodesic of a perturbed spacetime via a singular/regular field decomposition $h^{\mathrm{act}}_{ab}=h^{\mathrm{S}}_{ab}+h^{\mathrm{R}}_{ab}$, where the regular part $h^{\mathrm{R}}_{ab}$ accounts for the self-force at $\mathcal{O}(\mu)$. It builds this picture through matched asymptotic expansions (inner Schwarzschild region and outer background), THZ/Fermi locally inertial coordinates, and a Hadamard-based Green’s function analysis, ensuring a gauge-consistent, physically meaningful interpretation of the self-force as geodesic motion in $g_{ab}+h^{\mathrm{R}}_{ab}$. The work shows how to compute $h^{\mathrm{R}}_{ab}$ using mode-sum regularization in concrete settings (e.g., circular Schwarzschild orbits) and discusses gauge invariance and the role of S/R decomposition in ensuring finite, well-defined results. It also outlines practical prospects, including phase evolution in quasi-circular inspirals, connections to PN theory, and extensions to more general spacetimes and higher-order perturbations, which are crucial for accurate gravitational waveform predictions in extreme mass-ratio binaries.

Abstract

A point particle of mass $μ$ 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 $μ/r$ 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 $\Or(μ)$ 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 $\Or(μ^2)$ as $μ\to0$. 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.