Geometrical structure of gravitation and matter fields
type: Item
instance of: scholarly article
title: Geometrical structure of gravitation and matter fields
zbMATH Open document ID: 0809.53088
DOI: 10.1007/BF00844980
author: Leopold Halpern
published in: International Journal of Theoretical Physics
publication date: 1994-07-12T00:00:00Z
review text: The paper discusses the possibility to make Einstein's requirement of a unified geometrical description of gravitational fields and their matter sources, relaxing the requirement of a minimal interaction of gravitation with matter. In order to perform this relaxation inevitable, the author proposes a gauge theory in which the Yang-Mills field can be interpreted as being composed of Riemannian curvatures and a tensor formed out of torsion. The field equations of the author's theory are derived from a gauge principle associated to the structure of the principal fiber bundle \(P(G, H, G/H, \pi)\) with \(G\) a semisimple Lie group and \(H\) its semisimple Lie subgroup, which is isomorphic to the typical fiber. The space of left \(G/H\) cosets forms the base manifold and \(\pi G \rightarrow G/H\) is the natural projection. For \(G\) the choice of the model is the anti-de Sitter group \(\text{SO} (3,2)\) and for \(H\) the proper Lorentz group \(\text{SO}(3,1)\) leads to special features which pertain to elementary particle quantization and spin-statistics. The metric \(\gamma\) of the \(r\)-dimensional group \(G\) fulfils Einstein's equations with a cosmological member \[ R_{UV} - {1 \over 2} \gamma_{VU} R + \gamma_{UV} {r-2 \over 8} = 0.\tag{1} \] The author considers a large range of solutions for the equation (1). One of them results in the metric of the de Sitter universe, \(\rho = \sqrt {6}\). Other solutions, due to gravitational waves, give rise to torsion on the base manifold. The most general admissible solution of eq. (1) is a metric \(\gamma\) which has six orthonormal Killing vector fields \(A_ M\) which lie on the fiber through every point on the manifold of \(G\) and have the commutation relations of the Lie subalgebra of \(H\). It is shown that the Riemannian curvature tensor on \(G\) can be expressed on the base manifold by the Riemannian curvature tensor and its first covariant derivatives with the projected metric \(g = \pi' \gamma\) plus expressions for…
Mathematics Subject Classification ID: 53Z05
Mathematics Subject Classification ID: 83E15
zbMATH DE Number: 575537
zbMATH Keywords: gauge theory
zbMATH Keywords: Yang-Mills field
zbMATH Keywords: elementary particle quantization
zbMATH Keywords: spin- statistics
zbMATH Keywords: de Sitter universe
reviewed by: Serban Misicu
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 575537