Kinematic Quantities and Raychaudhuri Equations in a $5D$ Universe

TL;DR

This paper extends the classical Raychaudhuri equations to a five-dimensional universe inspired by Kaluza-Klein theory, analyzing the evolution of kinematic quantities in this higher-dimensional framework.

Contribution

It introduces a novel framework for studying kinematic quantities and Raychaudhuri equations in a 5D universe based on a product bundle structure.

Findings

Derived Raychaudhuri equations for 5D universe.

Analyzed evolution of 4D and 5D expansion.

Provided insights into higher-dimensional cosmological dynamics.

Abstract

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.