Ultrahigh energy collision with neither black hole nor naked singularity

We explore the collision between two concentric spherical thin shells. The inner shell is charged, whereas the outer one is either neutral or charged. In the situation we consider, the charge of the inner shell is larger than its gravitational mass, and the inside of it is empty and regular. Hence, the domain just outside it is described by the overcharged Reissner-Nordström geometry, whereas the inside of it is Minkowskian. First, the inner shell starts to shrink from infinity with finite kinetic energy, and then the outer shell starts to shrink from infinity with vanishing kinetic energy. The inner shell bounces on the potential wall and collides with the ingoing outer shell. The energy of collision between these shells at “their center-of-mass frame” does not exceed the total energy of the system. By contrast, by virtue of the very large gamma factor of the relative velocity of the shells, the energy of collision between two of the constituent particles of these shells at their center-of-mass frame can be much larger than the Planck scale. This result suggests that the black hole or naked singularity is not necessary for the ultrahigh energy collision of particles.