On The splitting of the Einstein field equations with respect to a general $(1+3)$ threading of spacetime

Based on a general $(1+3)$ threading of the spacetime $(M,g)$, we obtain a new and simple splitting of a both the Einstein field equations (EFE) and the conservation laws in $(M,g)$. As an application we obtain the splitting of (EFE) in an almost FLRW universe with energy-momentum tensor of a perfect fluid. In particular, we state the perturbation Friedman equations in an almost FLRW universe.