Self-force via a Green’s function decomposition

The gravitational field in a neighborhood of a particle of small mass $\mu$ moving through curved spacetime is naturally decomposed into two parts each of which satisfies the perturbed Einstein equations through $O\left(\mu \right).\mathrm{}$ One part is an inhomogeneous field which looks like the $\mu /r$ field tidally distorted by the local Riemann tensor. The other part is a homogeneous field that completely determines the self-force of the particle interacting with its own gravitational field, which changes the worldline at $O\left(\mu \right)$ and includes the effects of radiation reaction. Surprisingly, a local observer measuring the gravitational field in a neighborhood of a freely moving particle sees geodesic motion of the particle in a perturbed vacuum geometry and would be unaware of the existence of radiation at $O\left(\mu \right).\mathrm{}$ In the light of all previous work this is quite an unexpected result.