Equations of Motion with Respect to the $(1+1+3)$ Threading of a $5D$ Universe

TL;DR

This paper derives covariant equations of motion for a 5D universe with a specific threading, analyzing different geodesic types and introducing a 5D Robertson-Walker model.

Contribution

It extends previous work by formulating covariant equations of motion in a 5D universe with a (1+1+3) threading, including new classifications of geodesics.

Findings

Derived covariant equations of motion for 5D universe

Classified spatial, temporal, and vertical geodesics

Introduced and studied the 5D Robertson-Walker universe

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.