Wheeler–DeWitt equation for 4D supermetric and ADM with massless scalar field as internal time

The main result of this paper is the 4-dimensional supermetric version of the Wheeler–DeWitt equation, that uses only one time variable for the both roles – as internal time and for the ADM split, as Hamiltonian evolution parameter. We study the ADM split with respect to the scalar massless field serving as internal time. The 4-dimensional hyper-surfaces Σϕ=constΣϕ=const span the 5-dimensional space with the scalar field being the fifth coordinate. As a result we obtain the analog of the Wheeler–DeWitt equation for the 4-dimensional supermetric. We compare the ADM action with the non-compactified Kaluza–Klein action for the same physical space and obtain the equation for the extrinsic curvature and the scalar massless field.