Neutrino beam plasma instability

We derive relativistic fluid set of equations for neutrinos and electrons from relativistic Vlasov equations with Fermi weak interaction force. Using these fluid equations, we obtain a dispersion relation describing neutrino beam plasma instability, which is little different from normal dispersion relation of streaming instability. It contains new, nonelectromagnetic, neutrino-plasma (or electroweak) stable and unstable modes also. The growth of the instability is weak for the highly relativistic neutrino flux, but becomes stronger for weakly relativistic neutrino flux in the case of parameters appropriate to the early universe and supernova explosions. However, this mode is dominant only for the beam velocity greater than 0.25c and in the other limit electroweak unstable mode takes over.