Motion of a thin spherically symmetric Shell of Dust in the   Schwarzschild field
  The equation of motion announced in the title was already deduced for the cases the inner metric being flat and the shell being negligibly small (test matter), using surface layers and geodesic trajectories resp. Here we derive the general equation of motion and solve it in closed form for the case of parabolic motion. Especially the motion near the horizon and near the singularity are examined. 
H.-J. Schmidt
gr-qc