Initial-Value Problem for Relativistic Plasma Oscillations

Abstract

The solution of the collisionless, relativistic Vlasov‐Maxwell set of equations is given for the case where neither ambient electric nor magnetic fields are present and a smeared‐out negative charge background preserves over‐all space‐charge neutrality with a relativistic proton plasma. The method of solution is based upon the eigenfunction expansion method invented by van Kampen. It is shown that, besides the relativistically modified modes of Case and Zelazny, there exists a new discrete set of modes whose phase velocities are greater than the speed of light in vacuo. The solution also exhibits the coupling of electrostatic and electromagnetic modes and the existence of complex phase velocities. This initial‐value problem (or the problem with the electron and proton roles reversed) is of interest because of the existence of cosmic rays, relativistic electrons emitting synchrotron radiation in nonthermal radio sources, solar noise generation, where electron velocities may approach c (Bailey, 1951), and high‐energy electrons in laboratory machines (Post, 1960). In these cases the nonrelativistic treatment is clearly inadequate.