Reality conditions and Ashtekar variables: A different perspective

We give in this paper a modified self-dual action that leads to the SO (3) ADM formalism without having to face the difficult second-class constraints present in other approaches (for example, if the starting point is the Hilbert-Palatini action). We use the new action principle to gain some new insight into the problem of the reality conditions that must be imposed in order to get real formulations from complex general relativity. We derive also a real formulation for Lorentzian general relativity in the Ashtekar phase space by using the modified action presented in the paper.