Black Hole versus Naked Singularity via Axial Perturbation
  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.
