Abstract
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 spacetime metric nor its signature explicitly. We discuss how the spacetime metric is introduced from algebraic relations of the constraints and the form of the Hamiltonian by focusing our attention on the signature factor. The system describes both Euclidean and Lorentzian metrics depending on reality assignments of the gauge connections. That is, Euclidean metrics arise from the real gauge fields. On the other hand, self-duality of the gauge fields, which is well known in the Ashtekar formalism, is derived in this theory from the consistency condition of the 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.
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