  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.
