On Challenges to Separability of the Dirac Equation in Kerr Geometry under Compact Hyperboloidal Coordinates

TL;DR

This paper investigates the separability of the Dirac equation in Kerr geometry using compact hyperboloidal coordinates, revealing unexpected challenges in decoupling the equation into radial and angular parts.

Contribution

It introduces a new coordinate system for Kerr black holes and demonstrates the difficulties in achieving Dirac equation separability within this framework.

Findings

Standard separability methods fail in hyperboloidal coordinates

The Dirac equation does not decouple into radial and angular parts

Implications for analytical and numerical studies of spin-1/2 particles in Kerr spacetime

Abstract

The Dirac equation governs the behaviour of spin-1/2 particles. The equation's separability into decoupled radial and angular differential equations is a crucial step in analytical and numerical computations of quantities like eigenvalues, quasinormal modes and bound states. However, this separation has been performed in co-ordinate systems that are not well-behaved in either limiting regions of $r \rightarrow r_{horizon}$, $r \rightarrow r_\infty$ or both. In particular, the extensively used Boyer-Lindquist co-ordinates contains unphysical features of spacetime geometry for both $r_{horizon}$ and $r_\infty$. Therefore, motivated by the recently developed compact hyperboloidal co-ordinate system for Kerr Black Holes that is well behaved in these limiting regions, we attempt the separation of the Dirac equation. We first construct a null tetrad suitable for the separability analysis under the Newman-Penrose formalism. Then, an unexpected result is shown that by using the standard separability procedure based on the mode ansatz under this tetrad, the Dirac equation does not decouple into radial and angular equationsPossible reasons for this behaviour as well as importance of proving separability for various computations are discussed.