Zero Sound in a Relativistic Fermi Gas and Zel'dovich's Model

A relativistic Fermi gas with short-range repulsive interaction is adequately represented by a model where the interaction is due to a vector field: This is Zel'dovich's model. We consider the zero-sound excitation in such a system, which, for low tempeatures, is the collective mode having physical significance, in contrast to the ordinary sound contempated by Zel'dovich. The quantity of main interest is the sound velocity $u$, which is shown to approach the $u\to c$ behavior in the very high-density limit, but for reasons different from those manifested in the case of ordinary sound.