Abstract : Busemann's work on timelike spaces had a counterpart in the Soviet school, a subject that was started by A. D. Alexandrov in the 1950s, known under the name ``chronogeometry", and which is the geometry underlying the theory of relativity. Alexandrov was motivated by the axiomatization of physics, and he was interested in particular in the problem of deducing the geometric structures that are behind the theory of relativity from the least possible amount of axioms. One of his objectives was to characterize the spacetime geometries that appears as the model space for Einstein's relativity theory, in terms of the metric geometry. The model spaces considered include the Minkowski and de Sitter spacetimes, there are many associated rigid geometric structures. This led him to several mathematical questions, including the characterization of homeomorphisms preserving order structures, or, equivalently, cone structures (sets of timelike rays in the space), with various sorts of homogeneity assumptions on the spaces involved. From the physical point of view, the particles follow straight lines in these cones, and the mathematical order relation reflects the physical notion of precedence: every point in the space is the vertex of a cone consisting of the set of points that follow it. Chronogeometry is mainly concerned with two topics: (1) to provide an axiomatic setting for the physical theory of special relativity; (2) to study the geometry of affine space $A^4$ equipped of a translation-invariant partial order. It also includes topics such as group actions of order-preserving mappings or of cone-preserving mappings, mappings that preserve a distance function or a volume in a metric space, mappings that preserve convex sets, etc. This article, we shall give a quick review of some works of Alexandrov's school on chronogeometry in relation with Busemann's timelike spaces.
Type de document :
Chapitre d'ouvrage
In: Herbert Busemann, Selected Works, Vol. I, Springer Verlag, 2018, p. 133-141., 2018, 978-3-319-64294-9
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Contributeur : Athanase Papadopoulos <>
Soumis le : samedi 4 août 2018 - 04:22:56
Dernière modification le : dimanche 5 août 2018 - 01:11:36


  • HAL Id : hal-01853741, version 1



Athanase Papadopoulos. Chronogeometry. In: Herbert Busemann, Selected Works, Vol. I, Springer Verlag, 2018, p. 133-141., 2018, 978-3-319-64294-9. 〈hal-01853741〉



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