  This paper is concerned with the relativistic Boltzmann equation without
angular cutoff. The non-cutoff theory for the relativistic Boltzmann equation
has been rarely studied even under a smallness assumption on the initial data
due to the lack of understanding on the spectrum and the need for coercivity
estimates on the linearized collision operator. Namely, it is crucial to obtain
the sharp asymptotics for the frequency multiplier to obtain this coercivity
that has never been established before. In this paper, we prove the sharp
asymptotics for the frequency multiplier for a general relativistic scattering
kernel without angular cutoff. As a consequence of our calculations, we further
explain how the well known change of variables $p' \to p$ is not well defined
in the special relativistic context.
