The difference between the “adiabatic” and “self-consistent” evolutions is in the treatment of the past history
of the particle, which is what determines the self-force. In the “adiabatic” approximation the past history is
assumed to be geodesic, so the self-force is computed for objects on geodesics and then the final trajectory
is evolved through a sequence of “osculating” geodesics using this geodesic self-force. The “self-consistent”
approach uses the actual trajectory to describe the past history of the particle. The two approaches differ at
second order in the mass ratio, which is the same order as other corrections that have been ignored in both
approaches.