The Landau damping of an electrostatic wave is analyzed in relativistic plasmas by solving the Vlasov–Poisson system numerically. The Jüttner–Synge distribution function is perturbed in order to obtain the numerical dispersion relation for short wavelenghts and for vph lesssim c, vph being the phase velocity of the wave. This is compared with the analytical results available in the literature. The ordering of the time-scales characterizing the system and the corresponding formation of vortices in phase space, as a function of the initial temperature, are investigated.