Geometrical structure of gravitation and matter fields
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 formed out of the contraction tensor \(K\) and its first covariant derivatives. The field equations of the author's theory are the ``homogeneous'' Einstein equations in ten dimensions with a cosmological member, the solutions being subjected to the restrictions imposed by the vertical Killing vectors. The author substantiates the fact that this theory offers no explanation for the units of horizontal line elements and cosmological lengths. The consequence of their large ratio is that the torsion tensor, describing the energy density of matter assumes values much larger than the curvature related to it. Formally, the presented theory is able to separate curvature from torsion and to use Riemannian geometry for parallel transfer. Finally the author remarks that his treatment does not yield a general realistic description of matter but there is a base to accomplish Einstein's requirement to geometrize the right-hand member of his field equations.
scientific article; zbMATH DE number 575537
scientific article