A study of the characterizations under which regions of a four-dimensional Einstein-Riemann space admit contemporaneous and Born rigid motions. These characterizations are given in terms of the existence or nonexistence of coordinate systems for which the fundamental metric tensor exhibits specifically-stated structure and dependence on the coordinates involved. These results show that contemporaneous rigid motions are a proper subset of Born rigid motions. Conditions for the reduction of the former to the latter are given directly in terms of the components of the fundamental metric tensor. The results indicate that neither definition of rigidity is immune to criticism.
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