Electromagnetic Instabilities in the Nonthermal Relativistic Plasma

The results of a previous investigation of growing electromagnetic waves in a gyrotropic electron plasma are now extended to relativistic electron energies. If γ is the ratio of the longitudinal to the transverse electron thermal energy, then a nonrelativistic analysis predicts instabilities for both γ < 1 and γ > 1. However, for γ > 1 the phase velocity of the growing waves exceeds the velocity of light in vacuo and it is inconceivable that there are electrons that interact strongly with such a wave to give a finite growth rate. The relativistic analysis clears up this difficulty by predicting no instability for γ > 1; the error lies in the use of a Galilean rather than the correct Lorentz transformation to go from the wave frame to the frame in which the electrons are at rest. A criterion for the existence of such instabilities is given. The case of a beam of electrons with γ « 1 drifting at a relativistic velocity v0 through a cold isotropic plasma is also analyzed to be unstable. The growth rates, frequencies and wave vectors of the unstable waves are derived.