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Position and momentum information measures are evaluated for the ground stateof the \emph{relativistic} hydrogen-like atoms. Consequences of the fact thatthe radial momentum operator is not self-adjoint are explicitly studied,exhibiting fundamental shortcomings of the conventional uncertainty measures interms of the radial position and momentum variances. The Shannon and R\'enyientropies, the Fisher information measure, as well as several relatedinformation measures, are considered as viable alternatives. Detailed resultson the onset of relativistic effects for low nuclear charges, and on theextreme relativistic limit, are presented. The relativistic position densitydecays exponentially at large $r$, but is singular at the origin.Correspondingly, the momentum density decays as an inverse power of $p$. Bothfeatures yield divergent R\'enyi entropies away from a finite vicinity of theShannon entropy. While the position space information measures can be evaluatedanalytically for both the nonrelativistic and the relativistic hydrogen atom,this is not the case for the relativistic momentum space. Some of the resultsallow interesting insight into the significance of recently evaluatedDirac-Fock vs. Hartree-Fock complexity measures for many-electron neutralatoms.

Relativistic effects on information measures for hydrogen-like atoms