The dispersion relation of electrostatic waves with phase velocities smaller than the electron thermal velocity is investigated in relativistic temperature plasmas. The model equations are the electron relativistic collisionless hydrodynamic equations and the ion non-relativistic Vlasov equation, coupled to the Poisson equation. The complex frequency of electrostatic modes are calculated numerically as a function of the relevant parameters kλ De and ZT e /T i where k is the wavenumber, λ De , the electron Debye length, T e and T i the electron and ion temperature, and Z, the ion charge number. Useful analytic expressions of the real and imaginary parts of frequency are also proposed. The non-relativistic results established in the literature from the kinetic theory are recovered and the role of the relativistic effects on the dispersion and the damping rate of electrostatic modes is discussed. In particular, it is shown that in highly relativistic regime the electrostatic waves are strongly damped.

The longitudinal dielectric function is derived analytically from the relativistic Vlasov equation for arbitrary values of the relevant parameters z=mc2/T, where m is the rest electron mass, c is the speed of light, and T is the electron temperature in energy units. A new analytical approach based on the Legendre polynomial expansion and continued fractions was used. Analytical expression of the electron distribution function was derived. The real part of the dispersion relation and the damping rate of electron plasma waves are calculated both analytically and numerically in the whole range of the parameter z. The results obtained improve significantly the previous results reported in the literature. For practical purposes, explicit expressions of the real part of the dispersion relation and the damping rate in the range z>30 and strongly relativistic regime are also proposed.

The dispersion relation of longitudinal electrostatic waves in unmagnetized isotropic thermal plasmas is investigated in the collisionless limit using fully relativistic kinetic plasma theory. Specifically, an electron-positron plasma is studied in the limits of nonrelativistic and ultrarelativistic plasma temperatures. In the limit of nonrelativistic plasma temperatures, the earlier dispersion relation, based on nonrelativistic dispersion theory, is reproduced. In the limit of ultrarelativistic plasma temperatures, we analyze analytically and numerically the dispersion relation of the longitudinal electrostatic mode which occurs in the subluminal and superluminal frequency regime. We correct some earlier results in the literature on the subluminal dispersion relation.