Wheeler-DeWitt Equation for 4D Supermetric and ADM with Massless Scalar Field as Internal Time

TL;DR

This paper derives a 4D supermetric version of the Wheeler-DeWitt equation using a massless scalar field as internal time, unifying the roles of time in quantum gravity and ADM formalism.

Contribution

It introduces a novel formulation of the Wheeler-DeWitt equation in 4D supermetric using a scalar field as internal time, linking ADM split with Kaluza-Klein approaches.

Findings

Derived the 4D supermetric Wheeler-DeWitt equation.

Established the scalar field as internal time in quantum gravity.

Compared ADM and Kaluza-Klein actions for the same physical space.

Abstract

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.