Self-force via a Green's function decomposition

TL;DR

The paper addresses the gravitational self-force on a small mass $\mu$ moving in curved spacetime by introducing a Green's-function decomposition that splits the perturbation into a local inhomogeneous $\mu/r$ piece and a global homogeneous remainder. It defines Hadamard-based S and R fields, showing the self-force is entirely governed by the interaction with the homogeneous field $\psi^{\mathrm{R}}$ (or $h^{\mathrm{R}}_{ab}$ in the gravitational case), which is differentiable at the particle and satisfies the field equations; the $S$-field encodes the tidal $\mu/r$ distortion. Consequently, the particle’s motion is a geodesic of the effective metric $g_{ab}+h^{\mathrm{R}}_{ab}$, with no local radiation reaction at $O(\mu)$, while the full description remains consistent with Dirac’s flat-space intuition and curved-space generalizations via tail terms. This framework, applicable to scalar, electromagnetic, and gravitational fields, provides a local, covariant method for practical self-force calculations by isolating the homogeneous, radiation-reaction–driving component from the inhomogeneous near-particle field.

Abstract

The gravitational field of a particle of small mass μmoving through curved spacetime is naturally decomposed into two parts each of which satisfies the perturbed Einstein equations through O(μ). One part is an inhomogeneous field which, near the particle, looks like the μ/r field distorted by the local Riemann tensor; it does not depend on the behavior of the source in either the infinite past or future. The other part is a homogeneous field and includes the ``tail term''; it completely determines the self force effects of the particle interacting with its own gravitational field, including radiation reaction. Self force effects for scalar, electromagnetic and gravitational fields are all described in this manner.