Equation of state for relativistic plasma waves

It is shown that in a fully relativistic plasma an isothermal equation of state must be assumed in order to obtain from fluid equations the correct dispersion equation of longitudinal plasma waves. This isothermal equation of state contrasts with the adiabatic equation that applies to a non‐relativistic plasma. For arbitrary values of the ratio α between the particle rest mass and particle temperature, a polytropic equation of state must be introduced. The dependence of the polytropic index γ on α can be derived from the relativistic Vlasov equation and can be shown to join the adiabatic (α≫1) and the isothermal (α≪1) regimes smoothly.