Generalized Kaluza-Klein models with Gauss-Bonnet lagrangians

The five-dimensional generalization of Einstein’s 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 𝑅𝐴𝐵𝐶𝐷𝑅𝐴𝐵𝐶𝐷-4𝑅𝐴𝐵𝑅𝐴𝐵+𝑅2. 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.