Some Aspects of Quantum Mechanics of Particle Motion in Static Centrally Symmetric Gravitational Fields

TL;DR

This paper investigates the behavior of quantum particles in static spherically symmetric gravitational fields, revealing conditions for particle fall into horizons and the potential for bound states near naked singularities.

Contribution

It provides a detailed analysis of wave functions and effective potentials in various Reissner-Nordström metrics, highlighting the existence or absence of bound states under different conditions.

Findings

Particles cannot have bound stationary states outside the horizon in the extreme Reissner-Nordström case.

Wave functions vanish at the event horizon, satisfying Hilbert causality.

Bound states may exist near naked singularities of the Reissner-Nordström field.

Abstract

The domain of wave functions and effective potentials of the Dirac and Klein-Gordon equations for quantum-mechanical particles in static centrally symmetric gravitational fields are analyzed by taking into account the Hilbert causality condition. For all the explored metrics, assuming existence of event horizons, the conditions of a "fall" of a particle to the appropriate event horizons are implemented. The exclusion is one of the solutions for the Reissner-Nordstroem extreme field with the single event horizon. In this case, while fulfilling the condition found by V.I.Dokuchaev, Yu.N.Yeroshenko, the normalization integral is convergent and the wave functions become zero on the event horizon. This corresponds to the Hilbert causality condition. In our paper, due to the analysis of the effective potential for the Reissner-Nordstroem extreme field with real radial wave functions of the Dirac equation, the impossibility is demonstrated for the bound stationary state existence of quantum-mechanical particles, with positive energy, off the event horizon. The analysis of the effective potential of the Dirac equation for the case of the naked singularity of the Reissner-Nordstroem field shown the possible existence of stationary bound states of quantum-mechanical half-spin particles.