  A tiny Lorentz symmetry breaking can be mediated in Electrodynamics by means
of the Chern-Simons (CS) interaction polarized along a constant CS vector. Its
presence makes the vacuum optically active that has been recently estimated
from astrophysical data. We examine two possibilities for the CS vector to be
time-like or space-like, under the assumption that it originates from v.e.v. of
some pseudoscalar matter. It is shown that: a) a time-like CS vector makes the
vacuum unstable under pairs creation of tachyonic photon modes with the finite
vacuum decay rate, i.e. it is unlikely realized at macroscopic time scales; b)
on the contrary, the space-like CS vector does not yield any tachyonic modes
and, moreover, if its dynamical counterpart is substantially described by a
scale invariant interaction, then the QED radiation effects induce the
dynamical breaking of Lorentz symmetry, i.e. the occurrence of space-like CS
vector appears to be rather natural.
