New ways of deriving Arnowitt-Deser-Misner constraint equations in four-dimensional gravity

In this paper, the Arnowitt-Deser-Misner (ADM) constraint equations are naturally derived in two different ways. One method is to construct a one-parametric gravitational action in the Lorentzian spacetime. Hence, the one-parametric ADM constraint equations can be obtained. The other method is to apply the double complex function method to Einstein-Hilbert gravitational fields in Hamiltonian formulation. Therefore the double ADM constraint equations can be obtained, in which the well-known ADM constraint equations are included as a special case.