Kinematic Quantities and Raychaudhuri Equations in a 5D5D Universe

Based on some ideas emerged from the classical Kaluza-Klein theory, we present a $5D$ universe as a product bundle over the $4D$ spacetime. This enables us to introduce and study two categories of kinematic quantities (expansions, shear, vorticity) in a $5D$ universe. One category is related to the fourth dimension (time), and the other one comes from the assumption of the existence of the fifth dimension. The Raychaudhuri type equations that we obtain in the paper, lead us to results on the evolution of both the $4D$ expansion and $5D$ expansion in a $5D$ universe.Comment: 27 page

On Higher Dimensional Kaluza-Klein Theories

We present a new method for the study of general higher dimensional Kaluza-Klein theories. Our new approach is based on the Riemannian adapted connection and on a theory of adapted tensor fields in the ambient space. We obtain, in a covariant form, the fully general 4D equations of motion in a (4 + n)D general gauge Kaluza-Klein space. This enables us to classify the geodesics of the (4 + n)D space and to show that the induced motions in the 4D space bring more information than motions from both the 4D general relativity and the 4D Lorentz force equations. Finally, we note that all the previous studies on higher dimensional Kaluza-Klein theories are particular cases of the general case considered in the present paper

Equations of motion with respect to the

We continue our research work started in Bejancu (Eur Phys J C 75:346, 2015), and obtain in a covariant form the equations of motion with respect to the $$(1+1+3)$$ threading of a 5D universe $$(\bar{M}, \bar{g})$$. The natural splitting of the tangent bundle of $$\bar{M}$$ leads to the study of three categories of geodesics: spatial geodesics, temporal geodesics, and vertical geodesics. As an application of the general theory, we introduce and study what we call the 5D Robertson–Walker universe