Abstract
We continue our research work started in "Kinematic Quantities and Raychaudhuri Equations in a $5D$ Universe" (Eur. Phys. J. C, 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 us 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.
Highlights
This paper is a continuation of our previous paper [1] on kinematic quantities and Raychaudhuri equations in a 5D universe
In case T M ⊕ V Mis a Killing vector bundle, we show that spatial geodesics coincide with autoparallel curves of ∇
If T M ⊕ V Mis a Killing vector bundle, we show that the spatial geodesics coincide with the autoparallel curves of ∇
Summary
This paper is a continuation of our previous paper [1] on kinematic quantities and Raychaudhuri equations in a 5D universe. The kinematic quantities together with the spatial tensor fields and the Riemannian spatial connection enable us to obtain, in a covariant form, the equations of motion in (M , g). We apply the general theory to what we call the 5D Robertson– Walker universe, which can be thought of as a disjoint union of 4D Robertson–Walker spacetimes. In this case, the above three categories of geodesics are completely determined
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