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2023, no. 3-4

GENERALIZED KALUZA-KLEIN MODELS WITH GAUSS-BONNET LAGRANGIANS

Kerner R.

The five-dimensional generalization of Einsteins theory of gravitation proposed first by Th. Kaluza (1921) and improved a few years later by O. Klein (1926) has led to the Kaluza-Klein model incorporating electromagnetism and gravitation, and a variant of the Brans-Dicke theory of gravity, containing a scalar field interacting with metric tensor field. However, neither of these models did use the possibilities offered by the enlargement of the Einstein-Hilbert variational principle via including the Gauss-Bonnet invariant, which in 5 dimensions is no more a pure divergence, and modifies substantially the equations of motion of the theory.
After recalling the basics of the Kaluza-Klein model, including the non-abelian case. we give a short review of multi-dimensional cosmological models with scalar fields generated by gauge fields defined on the structural group, including the generalized lagrangian containing the Gauss-Bonnet term \(R_{ABCD}R^{ABCD} - 4R_{AB}R^{AB} + R^2\).
Then we turn our attention back to the 5-dimensional Kaluza-Klein model, without scalar field and neglecting gravity, but with variational principle enriched by the Gauss-Bonnet term, This leads, in the Minkowskian space-time, to an interesting variant of non-linear Electrodynamics. After discussing the modified Maxwell's equations, we show how a toroidal soliton can be constructed, and show that it displays the most essential features of Dirac's electron: electric charge, magnetic moment, and spin. It also predicts the particle-anti particle symmetry.

Keywords: Kaluza-Klein theory, Gauss-Bonnet invariants, Non-linear Electrodynamics, Fibre Bundles, Cosmology in 10 dimensions.

UDC: 537.8, 530.12

PACS: 03.50.-z, 03.50.Kk, 04.50.+h, 04.50.Kd

DOI: 10.17238/issn2226-8812.2023.3-4.166-187

Please cite this article in English as:
Kerner R. Generalized Kaluza-Klein models with Gauss-Bonnet lagrangians. Space, Time and Fundamental Interactions, 2023, no. 3-4, pp. 166-187.