Stationary solutions of second-order equations for fermions in Reissner-Nordstr\"{o}m space-time

TL;DR

This paper investigates stationary bound states of fermions in Reissner-Nordström space-time, revealing conditions for their existence, the absence in extreme cases, and proposing dark matter candidates based on naked singularities.

Contribution

It demonstrates the existence of bound states in non-extreme RN fields, analyzes the absence in extreme cases, and introduces naked RN singularities as potential dark matter particles.

Findings

Bound states exist near RN horizons for charged and uncharged fermions.

No bound states with E<mc^2 in extreme RN fields.

Naked RN singularities can have discrete energy spectra and are potential dark matter candidates.

Abstract

Existence of degenerate stationary bound states with square integrable radial wave functions was proved when second-order equations are used with the effective potential of the Reissner-Nordstr\"{o}m (RN) field with two event horizons for charged and uncharged fermions. The fermions in such states are localized near event horizons within the ranges from zero to several fractions of Compton wave length of fermions versus the values of gravitational and electromagnetic coupling constants and the values of angular and orbital momenta $j,l$. In case of extreme RN fields, absence of stationary bound states of fermions with the energies of $E<mc^{2}$ is shown for solutions of the second-order equation for any value of gravitational and electromagnetic coupling constants. Existence of the discrete energy spectrum is shown for the naked RN singularity due to solution of the second-order equation at definite values of physical parameters. The discrete spectrum exists for both charged and uncharged fermions. The naked RN singularity in quantum mechanics with the second-order equation for half-spin particles poses no threat to cosmic censorship since it is covered with an infinitely large potential barrier. Electrically neutral systems of atomic type (RN collapsars with the definite number of fermions in degenerate bound states) are proposed to consider as particles of dark matter.