Black Hole versus Naked Singularity via Axial Perturbation

Abstract

We differentiate non-extremal black hole, \emph{extremal} black hole and \emph{naked singularity} via metric perturbations for Reissner-Nordstr\"{o}m spacetime. First we study the axial perturbations for \emph{extremal} Reissner-Nordstr\"{o}m black hole and compute the effective potential due to these perturbations. Then we study the axial perturbations for the naked singularity case and compute the effective potential. We show that for the non-extremal black hole, \emph{the effective potential outside the event horizon~($r_{+}$) is real and positive. While in between Cauchy horizon~($r_{-}$) and event horizon~($r_{-}<r<r_{+}$) the effective potential is negative.} For the \emph{extremal black hole, the effective potential is always positive}. Also for \emph{naked singularity, the effective potential is positive.} From the effective potential diagram, we show that the structure of effective potentials for extremal BH looks like a potential barrier outside the horizon. While for non-extremal BH, the structure of the effective potentials look like a \emph{potential well} rather than a potential barrier. For NS the structure of the effective potentials is \emph{neither a potential barrier nor a potential well. Preferably it looks like an exponential decay function}. We observe that the geometric construction of an effective potential barrier due to axial perturbations could allow us to distinguish between the non-extremal black hole, extremal black hole, and naked singularity. Stability of extremal BH has been discussed.