Self-consistent gravitational self-force

I review the problem of motion for small bodies in general relativity, with an emphasis on developing a self-consistent treatment of the gravitational self-force. An analysis of the various derivations extant in the literature leads me to formulate an asymptotic expansion in which the metric is expanded while a representative worldline is held fixed. I discuss the utility of this expansion for both exact point particles and asymptotically small bodies, contrasting it with a regular expansion in which both the metric and the worldline are expanded. Based on these preliminary analyses, I present a general method of deriving self-consistent equations of motion for arbitrarily structured (sufficiently compact) small bodies. My method utilizes two expansions: an inner expansion that keeps the size of the body fixed, and an outer expansion that lets the body shrink while holding its worldline fixed. By imposing the Lorenz gauge, I express the global solution to the Einstein equation in the outer expansion in terms of an integral over a worldtube of small radius surrounding the body. Appropriate boundary data on the tube are determined from a local-in-space expansion in a buffer region where both the inner and outer expansions are valid. This buffer-region expansion also results in an expression for the self-force in terms of irreducible pieces of the metric perturbation on the worldline. Based on the global solution, these pieces of the perturbation can be written in terms of a tail integral over the body’s past history. This approach can be applied at any order to obtain a self-consistent approximation that is valid on long time scales, both near and far from the small body. I conclude by discussing possible extensions of my method and comparing it to alternative approaches.

Figure 1

Regular limit of a family of spacetimes. The dashed lines indicate a limit process in which $ϵ\to 0$ at fixed coordinate values. As $ϵ\to 0$, the worldtube of the body shrinks to zero size, leaving a geodesic remnant curve. The remnant curve lies in the manifold ${\mathcal{M}}_0$ defined by $ϵ=0$.

Figure 2

Singular limits of a family of spacetimes. The dashed lines correspond to the external limit, which lets the body shrink to zero size but keeps its motion fixed. The dotted lines correspond to the internal limit, which keeps the size of the body fixed. The worldline lies in the manifold ${\mathcal{M}}_0$, but it does not correspond to the remnant curve defined by the regular limit; instead, it is allowed to have $ϵ$ dependence, and it is determined by the particular value of $ϵ$ (in this case, $ϵ={ϵ}^{*}$) at which an approximate solution is sought.

Figure 3

The spacetime region $\Omega$ is bounded by the union of the spacelike surface $\Sigma$, the timelike worldtube $\Gamma$, and the null surface $\mathcal{J}$.

Figure 4

The two-dimensional hypersurface $\mathcal{S}$ is defined by the intersection of the worldtube $\Gamma$ with the past light cone of the point $x$. $x$ is linked to a point $x^{\prime }\in \mathcal{S}$ by a null geodesic ${\alpha }^{\prime }$. $x$ and $x^{\prime }$ are separately linked to points $x^{\prime \prime }=\gamma (t)$ and $\overline{x}=\gamma (t^{\prime })$ by spacelike geodesics $\beta$ and ${\beta }^{\prime }$, each of which is perpendicular to $\gamma$.