This article is cited in 5 papers

Brief communications

The relationship between the Einstein and Yang–Mills equations

L. N. Krivonosov^a, V. A. Luk'yanov<sup>b

a</sup> Chair of Applied Mathematics, Nizhni Novgorod State Technical University, Nizhni Novgorod, Russia
^b Chair of General Educational and Professional Disciplines, Zavolzh'ye Branch of Nizhni Novgorod Technical State University, Zavolzh'ye, Nizhegorodski region, Russia

Abstract: In this paper we prove that special requirements to Yang–Mills equations on a 4-dimensional conformally connected manifold allow one to reduce them to a system of Einstein equations and additional ones that bind components of the energy-impulse tensor. We propose an algorithm that gives conditions for the embedding of the metric of the gravitational field into a special (uncharged) Yang–Mills conformally connected manifold. As an application of the algorithm, we prove that the metric of any Einstein space and the Robertson–Walker metric are embeddable into the specified manifold.

Keywords: curvature of the connection, Robertson–Walker metric, Hodge operator, energy-impulse tensor, Bianchi identities, Einstein equations, Yang–Mills equations, 4-dimensional conformally connected manifold.

UDC: 515.1

Received: 31.03.2008

 English version: Russian Mathematics (Izvestiya VUZ. Matematika), 2009, 53:9, 62–66

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