Non-existence of Fermionic Bound States in Schwarzschild Geometry

TL;DR

This paper proves that there are no bound fermionic states in Schwarzschild black hole spacetime for energies below the fermion mass, using the Dirac equation in Newman-Penrose formalism and polynomial solutions analysis.

Contribution

It provides a rigorous proof of the non-existence of bound states for fermions in Schwarzschild geometry, extending understanding of quantum fields in curved spacetime.

Findings

No bound states for {} < me in Schwarzschild spacetime.

Solutions decay over time, indicating resonant states.

Mathematical proof using generalized Heun equation analysis.

Abstract

Dirac equation in Newman-Penrose formalism is comprehensively introduced to prove the non-existence of bound states for {\omega} < me in Schwarzschild geometry. The proof by contradiction is drawn in the context of finding polynomial solutions for the generalized Heun equation, thus the solutions are resonant states decaying over time. This work is divided into five parts. The first part is about the Dirac equation and the circumstances in which this equation emerged. Followed by two brief subsections on the mathematical substratum of the Dirac equation and its solution, i.e., Dirac algebra and Dirac spinor respectively. For the sake of elaborating on the mathematical tools used in this subject, the second part consists of a brief motivation for the general theory of relativity and its mathematical methods. The third part is dedicated to treating the Dirac equation in a more advanced mathematical framework, the Newman-Penrose formalism, which is a special case of the tetrad formalism of general relativity. The proof is elaborated in part 4, followed by conclusions in the fifth part.