A Rigorous Derivation of Gravitational Self-force

TL;DR

The authors present a rigorous derivation of gravitational self-force by constructing a smooth one-parameter family of spacetimes $g_{ab}(\lambda)$ in which a body shrinks to a worldline $\gamma$. They prove that $\gamma$ is a geodesic of the background metric $g_{ab}(0)$ at zeroth order, and derive the first-order correction to motion, including spin-curvature coupling and the gravitational self-force arising from the tail of the Green's function, yielding the MiSaTaQuWa equations in a Lorenz-gauge framework. Their approach uses distinct far-zone and near-zone expansions, a scaled limit, and a uniformity condition to obtain a universal equation of motion for the displacement $Z^a(t)$, independent of the body's microscopic nature. This framework eliminates prior ad hoc assumptions (notably Lorenz gauge relaxation), clarifies gauge issues, and provides a path to higher-order self-force corrections and self-consistent evolution, with important implications for extreme-mass-ratio inspirals. Overall, the paper offers a rigorous, gauge-conscious foundation for gravitational self-force theory and connects geometric perturbations to observable deviations from geodesic motion.

Abstract

There is general agreement that the MiSaTaQuWa equations should describe the motion of a "small 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, $g_{ab} (λ)$, 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, $g_{ab} (λ=0)$. 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 $λ$ 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.

Figures (2)

• Figure 1: A spacetime diagram illustrating the type of one-parameter family we wish to consider, as well as the two limits we define. As $\lambda \rightarrow 0$, the body shrinks and finally disappears, leaving behind a smooth background spacetime with a preferred world-line, $\gamma$, picked out. The solid lines illustrate this "ordinary limit" of $\lambda \rightarrow 0$ at fixed $r$, which is taken along paths that terminate away from $\gamma$ (i.e., $r>0$). By contrast, the "scaled limit" as $\lambda \rightarrow 0$, shown in dashed lines, is taken along paths at fixed $\bar{r}$ that converge to a point on $\gamma$.
• Figure 2: The two limits