A relativistic two-stream instability in an extremely low-density plasma

A linear analysis based on two-fluid equations in the approximation of a cold plasma, wherein the plasma temperature is assumed to be zero, demonstrates that a two-stream instability occurs in all cases. However, if this were true, the drift motion of electrons in an electric current over a wire would become unstable, inducing an oscillation in an electric circuit with ions bounded around specific positions. To avoid this peculiar outcome, we must assume a warm plasma with a finite temperature when discussing the criterion of instability. The two-stream instability in warm plasmas has typically been analyzed using kinetic theory to provide a general formula for the instability criterion from the distribution function of the plasma. However, the criteria based on kinetic theory do not have an easily applicable form. Here, we provide an easily applicable criterion for the instability based on the two-fluid model at finite temperatures, extensionally in the framework of special relativity. This criterion is relevant for analyzing two-stream instabilities in low-density plasmas in the universe and in Earth-based experimental devices.