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

abstract

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)