Different representations of Euclidean geometry and their application to the space-time geometry

 Yuri A. Rylov

Institute for Problems in Mechanics, Russian Academy of Sciences
 101-1 ,Vernadskii Ave., Moscow, 119526, Russia
 email: rylov@ipmnet.ru
 Web site: http://rsfq1.physics.sunysb.edu/~rylov/yrylov.htm
or mirror Web site: http://gasdyn-ipm.ipmnet.ru/~rylov/yrylov.htm

Updated March 3, 2011


Three different representation of the proper Euclidean geometry are considered. They differ in the number basic elements, from which the geometrical objects are constructed. In E-representation there are three basic elements (point, segment, angle) and no additional structures. Vrepresentation contains two basic elements (point, vector) and additional structure: linear vector space. In $\sigma $-representation there are only one basic element and additional structure: world function $\sigma =\rho ^{2}/2$, where $\rho $ is the distance. The concept of distance  appears in all representations. However, as a structure, determining the geometry, the distance appears only in the $\sigma $-representation. The $\sigma $-representation is most appropriate for modification of the proper Euclidean geometry. Practically any modification of the proper Euclidean geometry turns it into multivariant geometry, where there are many vectors $\mathbf{Q}_{0}\mathbf{Q}_{1},\mathbf{Q}_{0}\mathbf{Q}_{1}^{\prime },...$, which are equal to the vector \mathbf{P}_{0}\mathbf{P}_{1}$, but they are not equal between themselves, in general. Concept of multivariance is very important in application to the space-time geometry. The real space-time geometry is multivariant. Multivariance of the space-time geometry is responsible for quantum effects.

There is text of the paper in English (pdf, ps) and in Russian (ps, pdf)