The R.I. Pimenov unified gravitation and electromagnetism field theory as semi-Riemannian geometry
- PDF / 615,887 Bytes
- 7 Pages / 612 x 792 pts (letter) Page_size
- 104 Downloads / 178 Views
ELEMENTARY PARTICLES AND FIELDS Theory
The R.I. Pimenov Unified Gravitation and Electromagnetism Field Theory as Semi-Riemannian Geometry* N. A. Gromov** Department of Mathematics, Komi Science Center UrD RAS, Syktyvkar, Russia Received October 14, 2008
Abstract—More than forty years ago R.I. Pimenov introduced a new geometry—semi-Riemannian one —as a set of geometrical objects consistent with a fibering pr: Mn → Mm . He suggested the heuristic principle according to which the physically different quantities (meter, second, Coulomb, etc.) are geometrically modelled as space coordinates that are not superposed by automorphisms. As there is only one type of coordinates in Riemannian geometry and only three types of coordinates in pseudo-Riemannian one, a multiple-fibered semi-Riemannian geometry is the most appropriate one for the treatment of more than three different physical quantities as unified geometrical field theory. Semi-Euclidean geometry 3 R54 with 1-dimensional fiber x5 and 4-dimensional Minkowski space–time as a base is naturally interpreted as classical electrodynamics. Semi-Riemannian geometry 3 V54 with the general relativity pseudo-Riemannian space–time 3 V4 , and 1-dimensional fiber x5 , responsible for the electromagnetism, provides the unified field theory of gravitation and electromagnetism. Unlike Kaluza–Klein theories, where the fifth coordinate appears in nondegenerate Riemannian or pseudo-Riemannian geometry, the theory based on semiRiemannian geometry is free from defects of the former. In particular, scalar field does not arise. PACS numbers: 04.50.Cd, 02.40.-k, 11.10.Kk DOI: 10.1134/S106377880905007X
1. INTRODUCTION The old problem of geometrical unification of gravity and electromagnetism goes back to the Kaluza– Klein theory [1, 2], where electromagnetism is described by curvature in an extra spacelike dimension, i.e., the fifth coordinate is introduced in pseudoRiemanian geometry. However, the physical dimension of electromagnetism is different from the dimension of space, furthermore the fifth dimension is not observed as part of our everyday lives. To overcome last difficulty, the fifth dimension is supposed to be cyclic with very small radius. In addition there is also restriction on the set of transformations under which the theory is invariant. Indeed, rotation in a plane spanning the fifth coordinate x5 and some space coordinate x x = x cos ϕ + x5 sin ϕ,
x5 = x5 cos ϕ − x sin ϕ,
immediately leads to the dependence of space–time coordinates on the fifth dimension what is obviously unsatisfactory. Moreover, an additional scalar field arises in Kaluza–Klein type theories [3, 4], which makes Coulomb potential everywhere regular. ∗ **
The text was submitted by the author in English. E-mail: [email protected]
Nevertheless, the problem of unification of gravitation with electromagnetism in the framework of one geometry is still actual. This may help to incorporate some other interactions (weak interaction is the first candidate) in one geometrical picture. To overcome the disadventages of Kaluza–Klein t
Data Loading...
