Theory of relativistic electron holes in hot plasmas

We present a theory for finite-amplitude relativistic electron holes, which are localized Bernstein–Greene–Kruskal (BGK) solutions characterized by a trapped population of electrons moving with the electron hole. We find that the relativistic effects can drastically modify the electron hole, which becomes wider and is associated with a larger electrostatic potential, accelerating the electrons to highly relativistic energies. The theory has relevance for understanding the properties of strong localized electric fields in high-energy laser-plasma experiments and in supernovae remnants, where relativistic electron holes are a natural product of streaming instabilities.