  It was first pointed out by Press and Teukolsky that a system composed of a
spinning Kerr black hole surrounded by a reflecting mirror may develop
instabilities. The physical mechanism responsible for the development of these
exponentially growing instabilities is the superradiant amplification of
bosonic fields confined between the black hole and the mirror. A remarkable
feature of this composed black-hole-mirror-field system is the existence of a
critical mirror radius, $r^{\text{stat}}_{\text{m}}$, which supports {\it
stationary} (marginally-stable) field configurations. This critical
(`stationary') mirror radius marks the boundary between stable and unstable
black-hole-mirror-field configurations: composed systems whose confining mirror
is situated in the region $r_{\text{m}}<r^{\text{stat}}_{\text{m}}$ are stable
(that is, all modes of the confined field decay in time), whereas composed
systems whose confining mirror is situated in the region
$r_{\text{m}}>r^{\text{stat}}_{\text{m}}$ are unstable (that is, there are
confined field modes which grow exponentially over time). In the present paper
we explore this critical (marginally-stable) boundary between stable and
explosive black-hole-mirror-field configurations. It is shown that the
innermost ({\it smallest}) radius of the confining mirror which allows the
extraction of rotational energy from a spinning Kerr black hole approaches the
black-hole horizon radius in the extremal limit of rapidly-rotating black
holes. We find, in particular, that this critical mirror radius (which marks
the onset of superradiant instabilities in the composed system) scales linearly
with the black-hole temperature.
