Motion of spin-half particles in the axially symmetric field of naked singularities of the static q-metric

Abstract

Quantum-mechanical motion of a spin-half particle is examined in the axially symmetric fields of static naked singularities formed by a mass distribution with a quadrupole moment (q-metric). The analysis is performed by means of the method of effective potentials of the Dirac equation generalized to the case where radial and angular variables are not separated. If −1 < q < qlim, |qlim| ≪ 1, where q is the quadrupolemoment in proper units, the naked singularities do not exclude the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For an oblate mass distribution, the naked singularities of the q-metric are separated from a Dirac particle by infinitely large repulsive barriers followed by a potential well which deepens while moving apart from the equator (from θ = θ_min or θ = π − θ_min) toward the poles. The poles make an exception, and at 0 < q < q*, there are some points θ_i for particle states with j ≥ 3/2.