  This paper is concerned with the relativistic Boltzmann equation without
angular cutoff. We establish the global-in-time existence, uniqueness, and
asymptotic stability for solutions nearby the relativistic Maxwellian. We work
in the case of a spatially periodic box. We assume the generic hard-interaction
and soft-interaction conditions on the collision kernel that were derived by
Dudy\'nski and Ekiel-Je$\dot{\text{z}}$ewska (Comm. Math. Phys.
\textbf{115}(4):607--629, 1985), and our assumptions include the case of Israel
particles (J. Math. Phys. \textbf{4}:1163--1181, 1963). In this physical
situation, the angular function in the collision kernel is not locally
integrable, and the collision operator behaves like a fractional diffusion
operator. The coercivity estimates that are needed rely crucially on the sharp
asymptotics for the frequency multiplier that has not been previously
established. We further derive the relativistic analogue of the Carleman dual
representation for the Boltzmann collision operator. This resolves the open
question of perturbative global existence and uniqueness without the Grad's
angular cut-off assumption.
