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

TL;DR

This paper investigates the quantum motion of half-spin particles in the axially symmetric field of static naked singularities described by the q-metric, revealing conditions for bound states and potential barriers depending on mass distribution shape.

Contribution

It introduces a generalized effective potential method for Dirac particles in non-separable variables within the q-metric, analyzing bound states and barriers for different quadrupole moments.

Findings

Bound states exist for prolate distributions with small q.

Oblate distributions create potential barriers near the poles.

Certain angles and states allow particles to approach the poles for specific q values.

Abstract

Quantum-mechanical motion of a half-spin particle was examined in the axially symmetric field of static naked singularities formed by mass distribution with quadrupole moment (q-metric). The analysis was performed by means of the method of effective potentials of the Dirac equation generalized for the case when radial and angular variables are not separated. As $-1<q<q_{lim}, |q_{lim}|<<1 $ the naked singularities do not except the existence of stationary bound states of Dirac particles for a prolate mass distribution in the q-metric along the axial axis. For the oblate mass distribution, the naked singularities of the q-metric are separated from the Dirac particle by infinitely large repulsive barriers with the subsequent potential well deepening while moving along the angle from the equator (or from $\theta=\theta_{min}, \theta=\pi-\theta_{min}$) towards poles. The exception are the poles and, as $0<q<q*$, some points $\theta_{i}$ for the states of the particle with j>=3/2.