Analysis of the vacuum solution of the five-dimensional Einstein field equations with negative cosmological constant via variational symmetries

The Kaluza-Klein theory can be reckoned as a classical unified field theory of two of the significant forces of nature gravitation and electromagnetism. This formulation geometrically demonstrates the effects of a gravitational and an electromagnetic field by investigating a five-dimensional space with a metric constructed via the spacetime metric and the four-potential of the electromagnetic field. For the purpose of exploring the influences of dimensionality on the distinct physical variables, inquiring into stationary Kaluza-Klein rotating fluids is of particular significance. In this research, an extensive investigation of the variational symmetries for a specific vacuum solution of the (4+1)-dimensional Einstein field equations with negative cosmological constant is presented. For this purpose, first of all, the variational symmetries of our analyzed model are completely determined and the construction of the Lie algebra of the resulted symmetries is accurately analyzed. It is represented that the Lie algebra of local symmetries interrelated to the system of geodesic equations is non-solvable and not semi-simple and the algebraic organization of the derived quotient Lie algebra is accurately evaluated. Mainly, the adjoint representation group is effectively utilized intended for establishing an optimal system of group invariant solutions; which unequivocally yields a conjugate relation in the set of all one-dimensional symmetry subalgebras. Accordingly, the associated set of invariant solutions can be regarded as the slightest list from that the alternative invariant solutions of one-dimensional subalgebras are thoroughly determined unambiguously by virtue of transformations. Literally, all the corresponding local conservation laws of the resulted variational symmetries are totally calculated. Indeed, the symmetries of the metric of our analyzed space-time lead to the constants of motion for the point particles.