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

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Abstract
The 1 + 3 covariant equations, embedded in an extended tetrad formalism and describing a spacetime with an arbitrary energy–momentum distribution, are reconsidered. It is shown that, provided the 1 + 3 splitting is performed with respect to a generic time-like congruence with a 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.
Keywords
GENERAL-RELATIVITY, GRAVITY, NEWMAN-PENROSE FORMALISM, UNIVERSES, VARIABLES, COVARIANT, WAVES

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Citation

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Chicago
Van den Bergh, Norbert. 2013. “On the Relation Between the Einstein Field Equations and the Jacobi–Ricci–Bianchi System.” Classical and Quantum Gravity 30 (14).
APA
Van den Bergh, N. (2013). On the relation between the Einstein field equations and the Jacobi–Ricci–Bianchi system. CLASSICAL AND QUANTUM GRAVITY, 30(14).
Vancouver
1.
Van den Bergh N. On the relation between the Einstein field equations and the Jacobi–Ricci–Bianchi system. CLASSICAL AND QUANTUM GRAVITY. Institute of Physics; 2013;30(14).
MLA
Van den Bergh, Norbert. “On the Relation Between the Einstein Field Equations and the Jacobi–Ricci–Bianchi System.” CLASSICAL AND QUANTUM GRAVITY 30.14 (2013): n. pag. Print.
@article{3260562,
  abstract     = {The 1 + 3 covariant equations, embedded in an extended tetrad formalism and describing a spacetime with an arbitrary energy--momentum distribution, are reconsidered. It is shown that, provided the 1 + 3 splitting is performed with respect to a generic time-like congruence with a 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.},
  author       = {Van den Bergh, Norbert},
  issn         = {0264-9381},
  journal      = {CLASSICAL AND QUANTUM GRAVITY},
  keyword      = {GENERAL-RELATIVITY,GRAVITY,NEWMAN-PENROSE FORMALISM,UNIVERSES,VARIABLES,COVARIANT,WAVES},
  language     = {eng},
  number       = {14},
  pages        = {14},
  publisher    = {Institute of Physics},
  title        = {On the relation between the Einstein field equations and the Jacobi--Ricci--Bianchi system},
  url          = {http://dx.doi.org/10.1088/0264-9381/30/14/145010},
  volume       = {30},
  year         = {2013},
}

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