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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.

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
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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).
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