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

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 $r\to r_{horizon}$, $r\to r_{\infty }$ 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.

arXiv comment10 pages, Submitted to IJP. Comments on the validity of the result are especially encouraged

References in corpus (13)

• Quasinormal modes of black holes and black branes
• Hyperboloidal foliations and scri-fixing
• Signature of a pair of Majorana zero modes in superconducting gold surface states
• A geometric framework for black hole perturbations
• Spectral decomposition of black-hole perturbations on hyperboloidal slices
• Non-Existence of Time-Periodic Solutions of the Dirac Equation in an Axisymmetric Black Hole Geometry
• Quasinormal modes of massive fermions in Kerr spacetime: Long-lived modes and the fine structure
• A Pseudospectral Method for Gravitational Wave Collapse
• Distinguishing Dirac and Majorana neutrinos by their gravi-majoron decays
• Decay Rates and Probability Estimates for Massive Dirac Particles in the Kerr-Newman Black Hole Geometry
• Asymptotic Spectroscopy of Rotating Black Holes
• The Massive Dirac Equation in the Kerr Geometry: Separability in Eddington-Finkelstein-type Coordinates and Asymptotics
• A No-Boundary Method for Numerical Relativity