  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 $\Sigma_{\phi = const}$ 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.
