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

TL;DR

This paper introduces a new, simplified method for splitting Einstein's field equations using a general (1+3) threading of spacetime, with applications to almost FLRW universes and perturbation equations.

Contribution

It presents a novel and straightforward approach to decompose Einstein's equations based on a general (1+3) threading, extending previous methods.

Findings

Derived a new splitting of Einstein's equations and conservation laws.

Applied the method to almost FLRW universes with perfect fluid.

Formulated perturbation Friedmann equations for almost FLRW models.

Abstract

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.