  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.
