We examine the phenomenon of Landau Damping in relativistic plasmas via a study of the relativistic Vlasov-Poisson system (both on the torus and on $\mathbb{R}^3$) linearized around a sufficiently nice, spatially uniform kinetic equilibrium. We find that exponential decay of spatial Fourier modes is impossible under modest symmetry assumptions. However, by assuming the equilibrium and initial data are sufficiently regular functions of velocity for a given wavevector (in particular that they exhibit a kind of Gevrey regularity), we show that it is possible for the mode associated to this wavevector to decay sub-exponentially if its magnitude exceeds a certain critical size. We also give a heuristic argument why one should not expect such rapid decay for modes with wavevectors below this threshold.

Accepted for publication in J. Diff. Eqns. April 2015