(Pseudo) generalized Kaluza–Klein G-spaces and Einstein equations

Abstract

Introducing the Lie algebroid generalized tangent bundle of a KaluzaKlein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza-Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza-Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza-Klein G-spaces and we develop the Einstein equations in this general framework. IntroductionRecently, Lie algebroids are important issues in physics and mechanics since the extension of Lagrangian and Hamiltonian systems to their entity [11,13,19,27] and catching the poisson structure [20]. Then, Arcuş introduced generalized Lie algebroids as the extension of Lie algebroids and he studied geometrical and physical concepts for these spaces [1,2,3,4]. Indeed a generalized Lie algebroid is a extension of Lie algebroid from one base manifold to a pair of diffeomorphic base manifolds.The Einstein theory of general relativity and Maxwell theory of electromagnetism was independent developed in the world of physicists. In [12] Kaluza proposed to unify these two theories using a five-dimensional manifold. Kaluza's achievement was possible if the components of the pseudo-Riemannian metric on the five-dimensional manifold does not depend on the fifth coordinate (cylinder condition). Then, Klein [14] added the condition of compactification, namely the space is closed by a very small circle in the direction of the fifth dimension. Therefore, the Kaluza-Klein theory emerged to unify of the Einstein theory of general gravity and Maxwell theory of electromagnetism. 1 In [10], Einstein and Bergmann proposed a first generalization of KaluzaKlein theory using a pseudo-Riemannian metric such that its components are periodic in the fifth coordinate. So, the cylinder condition was partially satisfied and, for the firs time, a covariant derivative of a vector bundle over the fivedimensional manifold was introduced.A well known generalization of the five-dimensional Kaluza-Klein theory is the so called the space time matter (STM) theory. Cosmological solutions in which both the cylinder condition and the compactification condition were removed are presented in [9,15,17,18]. An excelent survey on STM theory is presented in the paper of Overduin and Wesson [16].The idea to construct exact solutions with Lie and Clifford algebroid symmetries in modified and extra dimension gravity and matter field theories was elaborated originally in a series of preprints by S. Vacaru [21,22,23]; see further developments and reviews of results on nonh...

Kaluza theory with zero-length extra dimensions

“…Another remarkable approach is based on the vertical and horizontal distributions, and a sub-Riemannian metric defined on the horizontal distribution, and can be applied to both Kaluza-Klein theory, and to space-time-matter theory [31,32] . Very interesting results, based on of connections and generalized Lie algebroids and connections, were obtained in [33] . Other developments are in the direction of superstring theory, which is believed by many to include the Standard Model, see for example the review article [34] .…”