A Constructive-Axiomatic Approach to Physical Space and Spacetime Geometries of Constant Curvature by the Principle of Reproducibility

Abstract

The so-called space-problem, wellknown for more than hundred years, is based on the famous works of B. Riemann, H.v.Helm-holtz and S. Lie. It concerns questions about an axiomatic characterization of physical space, precisely to the question what topological and grouptheoretical assumptions may suffice to deduce the classification of Euclidean and non-Euclidean geometries of constant curvature. About fifty years later a new complex of questions arose which offers much analogy to the space-problem, the axiomatic characterization of the special relativistic spacetime - the so-called Poincaré- Einstein-Minkowski spacetime-problem. Both sets of problems have distinct issues in common, because both are invariance problems of groups of automorphisms. In the space-problem the invariance structures appear in the congruence or rigidity of spatial regions being conserved under displacements. In the case of spacetime, the invariant structures enter into the Minkowskian metric and into the (Poincaré-invariant) straight lines of events which are physically interpreted by light rays and orbits of freely falling test particles.Both invariance structures are closely interrelated; for instance, spatial congruence und rigidity (rods) may be represented in spacetime by orbits of freely falling particles (Ehlers 1973).