  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.
