The mathematical and geometrical structure of the spacetime and the concept of unification, matter and energy
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he Mathematical and Geometrical Structure of the Spacetime and the Concept of Unification, Matter and Energy1 Diego Julio CiriloLombardo International Institute of Physics, Federal University of Rio Grande do Norte, 59078400—NatalRN, Brazil Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna (Moscow region), 141980 Russian Federation email: [email protected] Abstract—Geometrical analysis of a new type of Unified Field Theoretical models follow the guidelines of previous works of the authors is presented. These new unified theoretical models are characterized by an underlying hypercomplex structure, zero nonmetricity and the geometrical action is determined fundamen tally by the curvature provenient of the breaking of symmetry of a group manifold in higher dimensions. This mechanism of CartanMacDowellMansouri type, permits us to construct geometrical actions of determi nantal type leading a non topological physical Lagrangian due the splitting of a reductive geometry. Our goal is to take advantage of the geometrical and topological properties of this theory in order to determine the min imal group structure of the resultant spacetime Manifold able to support a fermionic structure. From this fact, the relation between antisymmetric torsion and Dirac structure of the spacetime is determined and the exist ence of an important contribution of the torsion to the giromagnetic factor of the fermions, shown. Also we resume and analyze previous cosmological solutions in this new UFT where, as in our work [Class. Quantum Grav. 22 (2005) 4987–5004] for the non abelian BornInfeld model, the Hosoya and Ogura ansatz is intro duced for the important cases of tratorial, totally antisymmetric and general torsion fields. In the case of spacetimes with torsion the real meaning of the spinframe alignment is find and the question of the minimal coupling is discussed. Keywords: Unified field theories, affine geometries DOI: 10.1134/S106377961305002X 1
CONTENTS 1. Motivation and summary of the results 2. The spacetime manifold and the geometrical action 3. The dynamical equations 3.1. Analysis and reduction of the dynamical equations 3.2. A potential for the torsion 4. Exact solutions in the New UFT theory 4.1. Totally antisymmetric torsion 4.2. “Tratorial” torsion 4.3. General case 4.4. Coexistence of both type of torsion in cosmological spacetimes 5. The underlying Dirac structure of the spacetime manifold 5.1. Field equations and group structure 5.2. Antisymmetric torsion and fermionic structure of the spacetime 1 The article is published in the original.
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5.3. About the equivalence principle (EP) and the antisymmetry of the torsion tensor: a theorem 5.4. The Ginvariance of the action 6. Dirac structure, electromagnetic field and anomalous gyromagnetic factor 7. Spacetime and structural cohomologies 8. Concluding remarks 8.1. On the geometrical structure 8.2. On the energy concept 9. Acknowledgments 10. References
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