A Rigorous Derivation of Gravitational Self-force

There is general agreement that the MiSaTaQuWa equations should describe the motion of a "small body" body in general relativity, taking into account the leading order self-force effects. However, previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features. For example, all previous derivations have invoked, without proper justification, the step of ``Lorenz gauge relaxation'', wherein the linearized Einstein equation is written down in the form appropriate to the Lorenz gauge, but the Lorenz gauge condition is then not imposed--thereby making the resulting equations for the metric perturbation inequivalent to the linearized Einstein equations. In this paper, we analyze the issue of ``particle motion'' in general relativity in a systematic and rigorous way by considering a one-parameter family of metrics, $g_{ab} (\lambda)$, corresponding to having a body (or black hole) that is ``scaled down'' to zero size and mass in an appropriate manner. We prove that the limiting worldline of such a one-parameter family must be a geodesic of the background metric, $g_{ab} (\lambda=0)$. Gravitational self-force--as well as the force due to coupling of the spin of the body to curvature--then arises as a first-order perturbative correction in $\lambda$ to this worldline. No assumptions are made in our analysis apart from the smoothness and limit properties of the one-parameter family of metrics. Our approach should provide a framework for systematically calculating higher order corrections to gravitational self-force, including higher multipole effects, although we do not attempt to go beyond first order calculations here. The status of the MiSaTaQuWa equations is explained.