Forces Unification in the Framework of Transitive Lie Algebroids

Abstract

Yang-Mills field equations describe non gravitational forces in the context of principal bundles, and Einstein field equation describes gravity in the context of semi-Riemannian metrics. Transitive Lie algebroids simultaneously contain tangent bundle of its base manifold and some Lie algebra bundle, so in this framework we can describe both gravity and other forces. Fortunately, suitable metrics on transitive algebroids provide us good apparatus to describe gravity and other forces in the same manner. The field equations are derived from an action which is formed naturally by scalar curvature of the metrics on a transitive Lie algebroid. The derived equations contain Einstein and Yang-Mills equations in vacuum, simultaneously.