The relationship between the Einstein and Yang-Mills equations.

*(English. Russian original)*Zbl 1183.81106
Russ. Math. 53, No. 9, 62-66 (2009); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 9, 69-74 (2009).

The Yang-Mills equations on a four-dimensional conformally connected manifold represent a system of sixty differential equations in sixty functions on four variables. This system is always consistent, yet, it is very rigid. However, if tensor constraints are added, the Yang-Mills equation system can be simplified because the number of unknown functions becomes smaller. In particular, in a chargeless conformally connected space, the number of unknown functions reduces to twenty. These functions are coefficients of two quadratic forms, one of them can be interpreted as a potential of the gravitational field, and the other one as the matter tensor. Then, a closed system of gravitational equations are obtained, where the metric is related to the energy-momentum tensor via Einstein equations, and the energy-momentum tensor itself satisfies a system of differential equations which depends on the metric.

In this paper, the authors prove that special requirements to Yang-Mills equations on a four dimensional conformally connected manifold allow to reduce them to a system of Einstein equations and additional ones that bind components of the energy-impulse tensor. An algorithm that gives conditions for the embedding of the metric of the gravitational field into a special (uncharged) Yang-Mills conformally connected manifold, is proposed. As an application of this algorithm it is proved that the metric of any Einstein space and the Robertson-Walker metric are embeddable into the specified manifold.

In this paper, the authors prove that special requirements to Yang-Mills equations on a four dimensional conformally connected manifold allow to reduce them to a system of Einstein equations and additional ones that bind components of the energy-impulse tensor. An algorithm that gives conditions for the embedding of the metric of the gravitational field into a special (uncharged) Yang-Mills conformally connected manifold, is proposed. As an application of this algorithm it is proved that the metric of any Einstein space and the Robertson-Walker metric are embeddable into the specified manifold.

Reviewer: Marian Ioan Munteanu (Iaşi)

##### MSC:

81T13 | Yang-Mills and other gauge theories in quantum field theory |

53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |

83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |

83C20 | Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory |

##### Keywords:

Yang-Mills equations; Einstein equations; Robertson-Walker metric; energy-impulse tensor; 4-dimensional conformally connected manifold
PDF
BibTeX
XML
Cite

\textit{L. N. Krivonosov} and \textit{V. A. Luk'yanov}, Russ. Math. 53, No. 9, 62--66 (2009; Zbl 1183.81106); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2009, No. 9, 69--74 (2009)

Full Text:
DOI

##### References:

[1] | É. Cartan, Spaces of Affine, Projective and Conformal Connection (Kazansk. Gos. Univ, Kazan, 1962) [in Russian]. |

[2] | L. D. Landau and E. M. Lifshitz, Field Theory (Nauka, Moscow, 1973) [in Russian]. |

[3] | W. L. Burke, Spacetime, Geometry, Cosmology (University Science Books, Mill Valley, 1980; Mir, Moscow, 1985). |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.