Heun polynomials and exact solutions for the massless Dirac particle in the C-metric

TL;DR

This paper derives exact solutions for a massless Dirac particle in the C-metric using Heun polynomials and a novel algebraic approach involving $su(1,1)$ generators with differential operators of fractional degrees.

Contribution

It introduces a new algebraic framework with $su(1,1)$ generators of fractional degree to construct Heun polynomials and exact solutions for the Dirac equation in the C-metric.

Findings

Derived new Heun polynomials using the algebraic structure.

Constructed explicit exact solutions for the Dirac equation.

Demonstrated the applicability of the method to the C-metric geometry.

Abstract

The equation of motion of a massless Dirac particle in the C-metric leads to the general Heun equation (GHE) for the radial and the polar variables. The GHE, under certain parametric conditions, has been cast in terms of a new set of $su(1,1)$ generators involving differential operators of \emph{degrees} $\pm 1/2$ and $0$. Additional \emph{Heun polynomials} are obtained using this new algebraic structure and are used to construct some exact solutions for the radial and the polar parts of the Dirac equation.