Spherically symmetric T- and R-solutions of the equations of the 5-dimensional Kaluza-Klein theory and its generalizations

Using the Hamilton-Jacobi technique, we find spherically symmetric solutions of the 5-dimensional (5D) Kaluza-Klein type theory with the generalized Lagrangian depending on the parameter 𝜖. For = 1, the Lagrangian describes the Kaluza-Klein theory; for = 1/ √ 3 it represents the effective Lagrangian for the low energy limit of superstring theory; finally, for = 0, the Lagrangian describes the Einstein-Maxwell theory with a minimally coupled scalar field. We restrict ourselves to constructing 5D configurations, the geometry of which depends only on the time (T-solution) or radial (R-solution) coordinate. For each case, we go to the configuration space and obtain the metric of this space and the Einstein-Hamilton-Jacobi equation. With the help of this equation, the fields trajectories in the configuration space are found. Further, the evolutionary coordinate is restored, and the metrics and fields are constructed in the coordinate space for the models under consideration. The found T-solution corresponds to the cosmological model of the Kantovsky-Sachs type with the hypercylinder topology, where scalar and electromagnetic fields interacting with each other in a contact manner. On the other hand, with an appropriate choice of the integration constants, they correspond to the inner region of the black hole. It turns out that the set of R-solutions is more meaningful than T-solutions, what leads to the need to construct an appropriate classification of R-solutions. Further, the symmetry of the configuration space of R-models is studied, and a brief analysis of solutions is given.