On the perturbed Schwarzschild geometry for determination of particle   motion
  A novel method for calculation of the motion and radiation reaction for the two-body problem (body plus particle, the small parameter m/M being the ratio of the masses) is presented. In the background curvature given by the Schwarzschild geometry rippled by gravitational waves, the geodesic equations insure the presence of radiation reaction also for high velocities and strong field. The method is generally applicable to any orbit, but radial fall is of interest due to the non-adiabatic regime (equality of radiation reaction and fall time scales), in which the particle locally and immediately reacts to the emitted radiation. The energy balance hypothesis is only used (emitted radiation equal to the variation in the kinetic energy) for determination of the 4-velocity via the Lagrangian and normalization of divergencies. The solution in time domain of the Regge-Wheeler-Zerilli-Moncrief radial wave equation determines the metric tensor expressing the polar perturbations, in terms of which the geodesic equations are written and shown herein. 
Alessandro D.A.M. Spallicci (Univ. Salerno at Benvento, Gravitation   Research Group) & (Univ. Leiden, Fac. Math. & Nat. Sc....