E. Cartan moment of rotation in classical and quantum gravity. Final report

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. 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 theory is based on a complex-valued self-dual connection that is defined not on the frame bundle of spacetime but, rather, on its complexification. We show how to transfer the construction of the E. Cartan moment of rotation to Ashtekar`s theory of gravity and demonstrate that no spurious equations are produced via this procedure.