Bound states of spin-half particles in a static gravitational field close to the black hole field

TL;DR

This paper investigates the energy levels of spin-1/2 particles in the gravitational field of near-black hole objects, showing how the spectrum collapses in the singular limit and differs from electromagnetic cases in particle pair production.

Contribution

It provides detailed calculations of bound-state energies for specific interior metrics and analyzes the spectrum collapse as the metric approaches singularity, highlighting differences from Coulomb systems.

Findings

Energy spectrum collapses near singularity

No particle pair production occurs before singularity

Spectrum becomes quasi-continuous in the singular limit

Abstract

We consider the bound-state energy levels of a spin-1/2 fermion in the gravitational field of a near-black hole object. In the limit that the metric of the body becomes singular, all binding energies tend to the rest-mass energy (i.e. total energy approaches zero). We present calculations of the ground state energy for three specific interior metrics (Florides, Soffel and Schwarzschild) for which the spectrum collapses and becomes quasi-continuous in the singular metric limit. The lack of zero or negative energy states prior to this limit being reached prevents particle pair production occurring. Therefore, in contrast to the Coulomb case, no pairs are produced in the non-singular static metric. For the Florides and Soffel metrics the singularity occurs in the black hole limit, while for the Schwarzschild interior metric it corresponds to infinite pressure at the centre. The behaviour of the energy level spectrum is discussed in the context of the semi-classical approximation and using general properties of the metric.