  We study the massless Vlasov equation on the exterior of the subextremal and
extremal Reissner-Nordstr\"om spacetimes. We prove that moments decay at an
exponential rate in the subextremal case and at a polynomial rate in the
extremal case. This polynomial rate is shown to be sharp along the event
horizon. In the extremal case we show that transversal derivatives of certain
components of the energy momentum tensor do not decay along the event horizon
if the solution and its first time derivative are initially supported on a
neighbourhood of the event horizon. The non-decay of transversal derivatives in
the extremal case is compared to the work of Aretakis on instability for the
wave equation. Unlike Aretakis' results for the wave equation, which exploit a
hierarchy of conservation laws, our proof is based entirely on a quantitative
analysis of the geodesic flow and conservation laws do not feature in the
present work.
