On the relation between the Einstein field equations and the Jacobi-Ricci-Bianchi system
arXiv:1302.6448 · doi:10.1088/0264-9381/30/14/145010
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.
15 pages; more references added and typos corrected