Radiation from a Point Source in a Plasma

Abstract

The fundamental solution for the linearized relativistic Vlasov equation with Maxwell's equations is obtained. The unperturbed state is assumed to be spatially homogeneous and time independent, with no electric or magnetic fields. The unperturbed distribution function depends only on the magnitude of relativistic momentum. An examination of the asymptotic behavior of the electric field for large time permits a general description of the solution. The electric field may be expressed as the sum of transverse and longitudinal parts. The transverse part looks like an ordinary electromagnetic wave extremely close to the light cone, and within the light cone it oscillates rapidly with varying frequency and wave number and a local phase velocity greater than the speed of light. Near the origin of the disturbance there is a transition to an ordinary plasma oscillation. In addition to a weak precursor wave, the longitudinal wave propagates outward with a definite speed and behind its wavefront the electric field oscillates with local phase velocity greater than the speed of light. Another ordinary plasma oscillation is left near the origin of the disturbance. The longitudinal speed of propagation goes to zero in the nonrelativistic case and to the speed of light in the extreme relativistic limit. The transverse and longitudinal fields defined according to the Fourier transform modes are not each zero outside the light cone, but their sum is zero. Thus, no signal travels faster than the speed of light. This peculiarity illustrates some unnatural aspects of the separation of a solution into modes of Fourier components.