Elementary development of the gravitational self-force

The gravitational field of a particle of small mass $\mu$ moving through curved spacetime, with metric $g_{ab}$, is naturally and easily decomposed into two parts each of which satisfies the perturbed Einstein equations through $O(\mu)$. One part is an inhomogeneous field $h^S_{ab}$ which, near the particle, looks like the Coulomb $\mu/r$ field with tidal distortion from the local Riemann tensor. This singular field is defined in a neighborhood of the small particle and does not depend upon boundary conditions or upon the behavior of the source in either the past or the future. The other part is a homogeneous field $h^R_{ab}$. In a perturbative analysis, the motion of the particle is then best described as being a geodesic in the metric $g_{ab}+h^R_{ab}$. This geodesic motion includes all of the effects which might be called radiation reaction and conservative effects as well.