  The rotational energy of a Kerr black hole is often invoked as the free
energy source that powers compact astrophysical systems An important question
concerning the energy extraction mechanism is whether a Kerr black hole can
communicate stresses to a distant load via a surrounding, force-free
magnetosphere, and whether such structures are stable. In this paper we address
this question. We first re-examine the properties of short wavelength
force-free waves, and show that contrary to earlier claims, fast magnetosonic
disturbances do affect the global electric current flowing in the system, even
in flat spacetime; beyond the light cylinder the perturbed charge density is on
the order of the perturbed Goldreich-Julian charge density. This implies that
the fast magnetosonic surface is a causal wind boundary. We then go on to study
the evolution of a force-free system driven by a spinning-up black hole, by
solving the time dependent Maxwell equations near the horizon using
perturbation method. We find that the electromagnetic field, current and charge
density grow exponentially in the linear regime. We conclude that after time
$t=2M\ln(1/2\alpha^2)$, where $\alpha$ is the lapse function, the evolution
will reach an adiabatic stage, whereby the solution is to a good approximation
the static one, with the angular momentum of the hole being the adiabatic
parameter. This implies that black hole magnetospheres are stable. A brief
discussion on the relation between global current closure and causality in
force-free systems is given at the end.
