Self-force driven motion in curved spacetimes
  We adopt the Dirac-Detweiler-Whiting radiative and regular effective field in curved spacetime. Thereby, we derive straightforwardly the first order perturbative correction to the geodesic of the background in a covariant form, for the extreme mass ratio two-body problem. The correction contains the self-force contribution and a background metric dependent term. 
Alessandro D.A.M. Spallicci, P. Ritter, S. Aoudia
gr-qc