Lorentzian dynamics in the Ashtekar gravity

We examine the advantages of the SO(3)-ADM (Ashtekar) formulation of general relativity, from the point of following the dynamics of the Lorentzian spacetime in direction of applying this into numerical relativity. We describe our strategy how to treat new constraints and reality conditions, together with a proposal of new variables. We show an example of passing a degenerate point in flat spacetime numerically by posing `reality recovering' conditions on spacetime. We also discuss some available advantages in numerical relativity.