  Position and momentum information measures are evaluated for the ground state
of the \emph{relativistic} hydrogen-like atoms. Consequences of the fact that
the radial momentum operator is not self-adjoint are explicitly studied,
exhibiting fundamental shortcomings of the conventional uncertainty measures in
terms of the radial position and momentum variances. The Shannon and R\'enyi
entropies, the Fisher information measure, as well as several related
information measures, are considered as viable alternatives. Detailed results
on the onset of relativistic effects for low nuclear charges, and on the
extreme relativistic limit, are presented. The relativistic position density
decays exponentially at large $r$, but is singular at the origin.
Correspondingly, the momentum density decays as an inverse power of $p$. Both
features yield divergent R\'enyi entropies away from a finite vicinity of the
Shannon entropy. While the position space information measures can be evaluated
analytically for both the nonrelativistic and the relativistic hydrogen atom,
this is not the case for the relativistic momentum space. Some of the results
allow interesting insight into the significance of recently evaluated
Dirac-Fock vs. Hartree-Fock complexity measures for many-electron neutral
atoms.
