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 gab⁰ and creates a first-order metric perturbation hab¹ʳᵉᵗ that diverges at the particle. Elementary expressions are known for the singular $m/r$ part of hab¹ʳᵉᵗ and for its tidal distortion determined by the Riemann tensor in a neighborhood of m. Subtracting this singular part hab1S from hab¹ʳᵉᵗ leaves a regular remainder hab1R. The self-force on the particle from its own gravitational field adjusts the world line at $O(m)$ to be a geodesic of gab⁰+hab1R. The generalization of this description to second-order perturbations is developed and results in a wave equation governing the second-order hab²ʳᵉᵗ with a source that has an O(m²) contribution from the stress-energy tensor of m added to a term quadratic in hab¹ʳᵉᵗ. Second-order self-force analysis is similar to that at first order: The second-order singular field hab2S subtracted from hab²ʳᵉᵗ yields the regular remainder hab2R, and the second-order self-force is then revealed as geodesic motion of m in the metric gab⁰+h1R+h2R.