  Modified theories of electrodynamics that include violations of Lorentz
symmetry often allow for the possibility of vacuum Cerenkov radiation. This
phenomenon has previously been studied in a number of Lorentz-violating
theories, but none of the methods that have previously been developed are
sufficient to study a theory with a timelike Chern-Simons term $k_{AF}$,
because such a term may generate exponentially growing solutions to the field
equations. Searching for vacuum Cerenkov radiation in a theory with a purely
timelike Chern-Simons term using only elementary methods, we find that, despite
the presence of the runaway modes, a charge in uniform nonrelativistic motion
does not radiate energy, up to second order in the velocity.
