  Under conditions prevailing in certain classes of compact astrophysical
systems, the active magnetosphere of a rotating black hole becomes
charge-starved, giving rise to formation of a spark gap in which plasma is
continuously produced. The plasma production process is accompanied by
curvature and inverse Compton emission of gamma rays in the GeV-TeV band, that
may be detectable by current and future experiments. The properties of the gap
emission have been studied recently using a fully general relativistic model of
a local steady gap. However, this model requires artificial adjustment of the
electric current which is determined, in reality, by the global properties of
the magnetosphere. In this paper we map the parameter regime in which steady
gap solutions exist, using a steady-state gap model in Kerr geometry, and show
that such solutions are allowed only under restrictive conditions that may not
apply to most astrophysical systems. We further argue that even the allowed
solutions are inconsistent with the global magnetospheric structure. We
conclude that magnetospheric gaps are inherently intermittent, and point out
that this may drastically change their emission properties.
