ar X iv : 0 90 8 . 43 63 v 1 [ gr - q c ] 3 1 A ug 2 00 9 Elementary development of the gravitational self - force

The gravitational field of a particle of small mass m moving through curved spacetime, with metric gab, is naturally and easily decomposed into two parts each of which satisfies the perturbed Einstein equations through O(m). One part is an inhomogeneous field hab which, near the particle, looks like the Coulomb m/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 hab. In a perturbative analysis, the motion of the particle is then best described as being a geodesic in the metric gab + hab. This geodesic motion includes all of the effects which might be called radiation reaction and conservative effects as well.