About The Real Conditions Of The Constraint Of The Adm And Ashtekar Gravity In General Relativity Theory

The Research about the ADM Constraint Equations in General Relativity andthe Reality Conditions for the Ashtekar Gravitational TheoryFirst, two different methods are given, by which the wel1-kno~ Amowitt-Deser-Misner (ADM) constraint equations for Einstein fields can be derived and the Lorentzian and Euclidean gravitational theory can be expressed in a unified way. Then, we analyze the reality conditions for Lorentzian and Euclidean gravity in the Ashtekar formulation through a double constant conformal transformation The result demonstrates that in the values of the double conformal factor have to be either real or double purely imaginary. Either Lorentzian or Euclidean gravitational theory is up to the different values of the double conformal factor.