Acceleration of particles and shells by Reissner-Nordström naked singularities

Abstract

We explore the Reissner-Nordstr\"om naked singularities with a charge $Q$ larger than its mass $M$ from the perspective of particle acceleration. We first consider a collision between two test particles following the radial geodesics in the Reissner-Nordstr\"om naked singular geometry. An initially radially ingoing particle turns back due to the repulsive effect of gravity in the vicinity of a naked singularity. Such a particle then collides with another radially ingoing particle. We show that the center of mass energy of collision taking place at $r\ensuremath{\approx}M$ is unbound, in the limit where the charge transcends the mass by an arbitrarily small amount $0<1\ensuremath{-}M/Q\ensuremath{\ll}1$. The acceleration process we describe 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 as observed by the distant observer required for the trans-Planckian collisions to occur around the naked singularity with one solar mass is merely of the order of million years, which is much smaller than the Hubble time. On the contrary, the time scale for collisions associated with an extremal black hole in an analogous situation is many orders of magnitude larger than the age of the Universe. 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 that the energy of a collision between two constituent particles of the shells at the center of mass frame can exceed the Planck energy.