Acceleration of particles and shells by Reissner-Nordstr\"{o}m naked singularities

We explore the Reissner-Nordstr\"{o}m naked singularities with a charge $Q$ larger than its mass $M$ from the perspective of the particle acceleration. We first consider a collision between two test particles following the radial geodesics in the Reissner-Nordstr\"{o}m naked singular geometry. An initially radially ingoing particle turns back due to the repulsive effect of gravity in the vicinity of naked singularity. Such a particle then collides with an another radially ingoing particle. We show that the center of mass energy of collision taking place at $r \approx M$ is unbound, in the limit where the charge transcends the mass by arbitrarily small amount $0<1-M/Q\ll1$.The acceleration process we described avoids fine tuning of the parameters of the particle geodesics for the unbound center of mass energy of collisions and the proper time required for the process is also finite. We show that the coordinate time required for the trans-Plankian collision to occur around one solar mass naked singularity is around million years while it is many orders of magnitude larger than Hubble time in the black hole case. We then study the collision of the neutral spherically symmetric shells made up of dust particles. In this case, it is possible to treat the situation by exactly taking into account the gravity due to the shells using Israel`s thin shell formalism, and thus this treatment allows us to go beyond the test particle approximation. The center of mass energy of collision of the shells is then calculated in a situation analogous to the test particle case and is shown to be bounded above. However, we find thatthe energy of a collision between two of constituent particles of the shells at the center of mass frame can exceed the Planck energy.