Mtric from Non-Metric Action of Gravity

The action of general relativity proposed by Capovilla, Jacobson and Dell is written in terms of $SO(3)$ gauge fields and gives Ashtekar's constraints for Einstein gravity. However, it does not depend on the space-time metric nor its signature explicitly. We discuss how the space-time metric is introduced from algebraic relations of the constraints and the Hamiltonian by focusing our attention on the signature factor. The system describes both Euclidian and Lorentzian metrics depending on reality assignments of the gauge connections. That is, Euclidian metrics arise from the real gauge fields. On the other hand, self-duality of the gauge fields, which is well known in the Ashtekar's formalism, is also derived in this theory from consistency condition of Lorentzian metric. We also show that the metric so determined is equivalent to that given by Urbantke, which is usually accepted as a definition of the metric for this system.