On the relation between the Einstein field equations and the Jacobi-Ricci-Bianchi system

TL;DR

This paper demonstrates that Einstein's field equations can be viewed as integrability conditions for the Jacobi, Ricci, and Bianchi equations within the 1+3 covariant formalism, applicable to both timelike and null congruences.

Contribution

It establishes a novel interpretation of Einstein's equations as integrability conditions within the Jacobi-Ricci-Bianchi system in covariant form.

Findings

Einstein equations linked to Jacobi and Bianchi systems

Applicable to timelike and null congruences

Provides a unified geometric framework

Abstract

The 1+3 covariant equations, embedded in an extended tetrad formalism and describing a space-time with an arbitrary energy-momentum distribution, are reconsidered. It is shown that, provided the 1+3 splitting is performed with respect to a generic timelike congruence with tangent vector u, the Einstein field equations can be regarded as the integrability conditions for the Jacobi and Bianchi equations together with the Ricci equations for u. The same conclusion holds for a generic null congruence in the Newman-Penrose framework.