An Exact General-Relativity Solution for the Motion and Intersections of   Self-Gravitating Shells in the Field of a Massive Black Hole
  The motion with intersections of relativistic gravitating shells in the Schwarzschild gravitational field of a central body is considered. Formulas are derived for calculating parameters of the shells after intersection via their parameters before intersection. Such special cases as the Newtonian approximation, intersections of light shells, and intersections of a test shell with a gravitating shell are also considered. The ejection of one of the shells to infinity in the relativistic region is described. The equations of motion for the shells are analyzed numerically. 
M. V. Barkov, V. A. Belinski, and G. S. Bisnovatyi-Kogan