A connection between the Einstein and Yang-Mills equations

Abstract

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.