Absence of Long-Wavelength Cerenkov Radiation With Isotropic Lorentz and CPT Violation

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.