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

Abstract

We obtain a new and simple splitting of Einstein field equations with respect to the $(1+1+3)$ threading of a $5D$ universe $(\bar{M}, \bar{g})$. The study is based on the spatial tensor fields and on the Riemannian spatial connection, which behave as $3D$ geometric objects. All the equations are expressed with respect to the adapted frame field and the adapted coframe field induced by the $(1+1+3)$ threading of $(\bar{M}, \bar{g})$. In particular, we obtain the splitting of the Einstein field equations in a $5D$ Robertson-Walker universe.