# Massless fermions on static general prolate metrics and their Heun solutions

## TL;DR

This paper derives solutions to the massless Dirac equation in various static prolate spacetimes using Heun functions, providing a unified approach to understanding fermions in these geometries.

## Contribution

It introduces a coordinate-free method to solve the Dirac equation in static prolate metrics and expresses solutions in terms of Heun functions, expanding analytical tools for these spacetimes.

## Key findings

- Solutions expressed in terms of Heun functions and their confluent forms
- Unified approach applicable to multiple static prolate spacetimes
- Explicit solutions for Dirac particles in black hole and black hole-like geometries

## Abstract

Employing a pseudo-orthonormal coordinate-free approach we write down the Dirac equation in spacetimes with static generally prolate metrics. As examples, we consider the electrically charged C-metric, the vacuum C-metric, the Reissner-Nordstrom and Schwarszchild spacetimes and the BBMB black hole and show that the solutions to the Dirac equations for particles these spacetimes can be derived in terms of Heun's general functions and their confluent and double confluent forms.