Negative-energy longitudinal wave instability in the interaction of relativistic electron beam with ultra-relativistic hot plasma

The investigation of waves and instabilities in ultra-relativistic charged particle beam-plasma systems is considerable for a wide class of astrophysical environments. Hence, in this study, using relativistic kinetic plasma theory and Lorentz transformations, we study the interaction of a relativistic electron beam with an ultra-relativistic hot plasma. First, we obtain the dispersion relation of fast and slow longitudinal waves and consequently the growth rate of the slow wave (negative-energy wave) instability. Then, we plot the instability growth rate in terms of different physical parameters, such as the Lorentz factor, the electron beam density, the thermal velocity of electrons, and the temperature of the beam particles, and discuss them further. Our results in the ultra-relativistic regime show that the growth rate of instability decreases by increasing the Lorentz factor and thermal velocity of particles, while the beam density enhances the growth rate of instability. Moreover, a comparison with the results obtained from previous studies shows that the growth rate of instability in the cold electron beam-cold plasma interaction is higher than the instability growth rate in the case of the relativistic electron beam and ultra-relativistic hot plasma.