The gravitational self-force formalism addresses the regime of the
relativistic two body problem where one body (called the primary) is much more massive than the other body (the secondary). It aims to solve the dynamics as a systematic expansion in powers of the mass-ratio q = m_2/m_1. In typical situations both bodies are assumed to be
black holes, but this assumption is not strictly necessary and most results hold for general bodies, as long as the secondary is compact, meaning that the length scales associated with the secondary are proportional to Gm_2/c^2. The gravitational self-force formalism exploits these systems will exhibit a hierarchy of length scales with the typical length scale of the secondary much smaller than the curvature length scale associated with the primary. As a consequence the metric "far" away from the secondary can be described as a perturbation of the metric that would have been produced by the primary alone :g_{\mu\nu}^{\rm far} = g_{\mu\nu}^{\rm primary} + q h_{\mu\nu}^{(1)} + q^2 h_{\mu\nu}^{(2)} + \mathcal{O}(q^3). At the same time, the
equivalence principle implies that if we zoom in on the secondary that in some sufficiently small neighborhood "near" the secondary the metric is described as :g_{\mu\nu}^{\rm near} = g_{\mu\nu}^{\rm secondary} + q H_{\mu\nu}^{(1)} + q^2 H_{\mu\nu}^{(2)} + \mathcal{O}(q^3). Taking general solutions in each regime and matching both in some intermediate regime using the
method of matched asymptotic expansions leads to an effective description of the binary in terms of an effective spacetime :g_{\mu\nu}^{\rm eff} = g_{\mu\nu}^{\rm primary} + q h_{\mu\nu}^{R1} + q^2 h_{\mu\nu}^{R2} + \mathcal{O}(q^3), and the motion of the secondary being represented as a worldline in this spacetime. If the secondary is spherically metric (e.g. a
Schwarzschild black hole) then its worldline will be a
geodesic in the effective spacetime. Our equivalently, expressed relative to the background spacetime g_{\mu\nu}^{\rm primary} the 4-velocity of worldline u^\musatisfies :u^\nu \nabla_\nu u^\mu = 0 + q F^\mu_{(1)}[ h_{\mu\nu}^{R1}] + q^2 F^\mu_{(2)}[ h_{\mu\nu}^{R1},h_{\mu\nu}^{R2}] + \mathcal{O}(q^3), where the left-hand side is simply the geodesic equation in the background spacetime, and the right-hand side representing an effective force term correcting the motion, the
gravitational self-force. Specifically, F^\mu_{(1)} is known as the first-order gravitational self-force and F^\mu_{(2)} as the second-order gravitational self-force.
General secondary If the secondary is not spherical, then the compactness of the secondary implies that the above result gets modified by the
multipole moments of the gravitational field of the secondary, with higher order multipole moments showing up at higher order in the mass-ratio. At zeroth order in the mass-ratio, the worldline in still given by a geodesic in the background spacetime. At linear order, the motion gets corrected by a force term coming from the current-dipole moment (a.k.a. the spin) of the secondary coupling to the background curvature, the (linear)
Mathisson–Papapetrou–Dixon force, while the first order effective metric perturbation h_{\mu\nu}^{R1} depends only on the monopole moment (a.k.a. the mass) of the secondary. At second order we get the quadractic Mathisson–Papapetrou–Dixon force in the first-order effective metric, a coupling of the secondary quadrupole moment to the background curvature, and the second order effective metric perturbation h_{\mu\nu}^{R2} picks up a contribution sourced by the secondary spin. And so forth at higher orders. ==The Capra meetings on Radiation Reaction in General Relativity==