  The characteristic damping times of the natural oscillations of a Kerr black
hole become arbitrarily large as the extremal limit is approached. This
behavior is associated with the so-called zero damped modes (ZDMs), and
suggests that extremal black holes are characterized by quasinormal modes whose
frequencies are purely real. Since these frequencies correspond to oscillations
whose angular phase velocity matches the horizon angular velocity of the black
hole, they are sometimes called "synchronous frequencies". Several authors have
studied the ZDMs for near-extremal black holes. Recently, their correspondence
to branch points of the Green's function of the wave equation was linked to the
Aretakis instability of extremal black holes. Here we investigate the existence
of ZDMs for extremal black holes, showing that these real-axis resonances of
the field are unphysical as natural black hole oscillations: the corresponding
frequency is always associated with a scattering mode. By analyzing the
behavior of these modes near the event horizon we obtain new insight into the
transition to extremality, including a simple way to understand the Aretakis
instability.
