Acceleration of colliding shells around a black hole: Validity of the
  test particle approximation in the Banados-Silk-West process
  Recently, Banados, Silk and West (BSW) showed that the total energy of two
colliding test particles has no upper limit in their center of mass frame in
the neighborhood of an extreme Kerr black hole, even if these particles were at
rest at infinity in the infinite past. We call this mechanism the BSW mechanism
or BSW process. The large energy of such particles would generate strong
gravity, although this has not been taken into account in the BSW analysis. A
similar mechanism is seen in the collision of two spherical test shells in the
neighborhood of an extreme Reissner-Nordstr\"om black hole. In this paper, in
order to draw some implications concerning the effects of gravity generated by
colliding particles in the BSW process, we study a collision of two spherical
dust shells, since their gravity can be exactly treated. We show that the
energy of two colliding shells in the center of mass frame observable from
infinity has an upper limit due to their own gravity. Our result suggests that
an upper limit also exists for the total energy of colliding particles in the
center of mass frame in the observable domain in the BSW process due the
gravity of the particles.
