  A relativistic analysis of the polarization properties of light elastically
scattered by atomic hydrogen is performed, based on the Dirac equation and
second order perturbation theory. The relativistic atomic states used for the
calculations are obtained by making use of the finite basis set method and
expressed in terms of $B$ splines and $B$ polynomials. We introduce two
experimental scenarios in which the light is circularly and linearly polarized,
respectively. For each of these scenarios, the polarization-dependent angular
distribution and the degrees of circular and linear polarization of the
scattered light are investigated as a function of scattering angle and photon
energy. Analytical expressions are derived for the polarization-dependent
angular distribution which can be used for scattering by both hydrogenic as
well as many-electron systems. Detailed computations are performed for Rayleigh
scattering by atomic hydrogen within the incident photon energy range 0.5 to 10
keV. Particular attention is paid to the effects that arise from higher
(nondipole) terms in the expansion of the electron-photon interaction.
