Instability of Transverse Waves in a Relativistic Plasma

Linearized equations are set up to describe disturbances in an infinite, spatially uniform, relativistic plasma without an ambient magnetic field. It is shown that, as well as the usual electrostatic waves, there also exists a class of electromagnetic waves. The two sets of waves are coupled in general, but can still be classified as mainly longitudinal or mainly transverse. Under the assumption that the system is stable against the longitudinal disturbances it is shown that the relativistic plasma will be unstable to the transverse waves unless it is virtually isotropic.