E. Cartan moment of rotation in Ashtekar’s self-dual representation of gravitation

Abstract

The geometric construction of the E. Cartan moment of rotation associated to the spacetime curvature provides a geometric interpretation of the gravitational field sources and describes geometrically how the sources are ‘‘wired’’ to the field in standard geometrodynamics. The E. Cartan moment of rotation yields an alternate way (as opposed to using variational principles) to obtain Einstein equations. The E. Cartan construction uses, in an essential way, the soldering structure of the frame bundle underlying the geometry of the gravitational field of general relativity. The geometry of Ashtekar’s connection formulation of gravitation is based on a complex-valued self-dual connection that is not defined on the frame bundle of spacetime. It is shown how to transfer the construction of the E. Cartan moment of rotation of Ashtekar’s formulation of the theory of gravity and demonstrate that no spurious equations are produced via this procedure.