Gravitational radiation reaction and second-order perturbation theory

A point particle of small mass $m$ moves in free fall through a background vacuum spacetime metric $g_{ab}^0$ and creates a first-order metric perturbation $h_{ab}^{1\mathrm{ret}}$ that diverges at the particle. Elementary expressions are known for the singular $m/r$ part of $h_{ab}^{1\mathrm{ret}}$ and for its tidal distortion determined by the Riemann tensor in a neighborhood of $m$. Subtracting this singular part $h_{ab}^{1\mathrm{S}}$ from $h_{ab}^{1\mathrm{ret}}$ leaves a regular remainder $h_{ab}^{1\mathrm{R}}$. The self-force on the particle from its own gravitational field adjusts the world line at $O(m)$ to be a geodesic of $g_{ab}^0+h_{ab}^{1\mathrm{R}}$. The generalization of this description to second-order perturbations is developed and results in a wave equation governing the second-order $h_{ab}^{2\mathrm{ret}}$ with a source that has an $O(m^2)$ contribution from the stress-energy tensor of $m$ added to a term quadratic in $h_{ab}^{1\mathrm{ret}}$. Second-order self-force analysis is similar to that at first order: The second-order singular field $h_{ab}^{2\mathrm{S}}$ subtracted from $h_{ab}^{2\mathrm{ret}}$ yields the regular remainder $h_{ab}^{2\mathrm{R}}$, and the second-order self-force is then revealed as geodesic motion of $m$ in the metric $g_{ab}^0+h^{1\mathrm{R}}+h^{2\mathrm{R}}$.