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The aim of this work is to provide a relativistically correct characterization for the stability of counterstreaming plasma structures ubiquitous in fusion plasma experiments and astrophysical sources of nonthermal radiation. Here, in the first part of this work, a new relativistically correct approach is formulated for the counterstreaming plasmas in thermal equilibrium, on the basis of the relativistic Jüttner-Maxwell distribution function and a correct representation of this distribution in the laboratory frame of reference by using the appropriate Lorentz transformations for momentum and energy. The particle velocity resulting from the thermal motion and the bulk displacement of plasma particles is thus limited according to the relativistic theory to less than c (the speed of light in vacuum). New criteria are derived for the existence of counterstreams conditioned by the magnitude of their bulk velocity with respect to the thermal speed. Accurate simplified forms of the distribution functions derived here for different limits of the streaming velocity and the plasma temperature, will be invoked in the second part of this work as input to the stability analysis of these systems. PACS Numbers: 52.25.Dg, 52.35.-g; 52.35.Qz; 52.27.-h; 52.27.Ny.

Towards a Relativistically Correct Characterization of Counterstreaming Plasmas. I. Distribution Functions