  [Abridged] Electromagnetic emission can be produced as a precursor to the
merger, as a prompt emission during the collapse of a NS and at the spin-down
stage of the resulting BH. We demonstrate that the time evolution of the
axisymmetric force-free magnetic fields can be expressed in terms of the
hyperbolic Grad-Shafranov equation. We find exact non-linear time-dependent
split-monopole structure of magnetosphere driven by spinning and collapsing NS
in Schwarzschild geometry. Based on this solution, we argue that the collapse
of a NS into the BH happens smoothly, without natural formation of current
sheets or other dissipative structures on the open field lines and, thus, does
not allow the magnetic field to become disconnected from the star and escape to
infinity. Thus, as long as an isolated Kerr BH can produce plasma and currents,
it does not lose its open magnetic field lines, its magnetospheric structure
evolved towards a split monopole and the BH spins down electromagnetically. The
"no hair theorem", which assumes that the outside medium is a vacuum, is not
applicable in this case: highly conducting plasma introduces a topological
constraint forbidding the disconnection of the magnetic field lines from the
BH. Eventually, a single random large scale spontaneous reconnection event will
lead to magnetic field release, shutting down the electromagnetic BH engine
forever. We also discuss the nature of short Gamma Ray Bursts and suggest that
the similarity of the early afterglows properties of long and short GRBs can be
related to the fact that in both cases a spinning BH can retains magnetic field
for sufficiently long time to extract a large fraction of its rotation energy
and produce high energy emission via the internal dissipation in the wind.
