Crossing of spherical massive shells in vacuum space-time

We investigate collisions between two spherical massive shells in vacuum sp ace-time by assuming that the shells are transparent. We give an expression for the energy transfer and the change in the 3-momentum due to the collis ion. Compared with the Newtonian situation, the relativistic effect always increases the energy transfer. In particular, in a collision close to the s ingularity and the horizon defined in the region between the shells, the en ergy transfer can be arbitrarily large. However, in such cases, the collisi on is not observable for the asymptotic observer. Further, we discuss a sit uation similar to the shell crossing on a spherical void wall of the horizo n scale. In this case, for some parameter choices, the outer shell initiall y going outward may suddenly change its motion to contraction due to a rela tivistic effect when it crosses with the inner shell.