  The classical Einstein-Hilbert (EH) action for general relativity (GR) is
shown to be formally analogous to the classical system with position-dependent
mass (PDM) models. The analogy is developed and used to build the covariant
classical Hamiltonian as well as defining an alternative phase portrait for GR.
The set of associated Hamilton's equations in the phase space is presented as a
first-order system dual to the Einstein field equations. Following the
principles of quantum mechanics, I build a canonical theory for the classical
general. A fully consistent quantum Hamiltonian for GR is constructed based on
adopting a high dimensional phase space. It is observed that the functional
wave equation is timeless. As a direct application, I present an alternative
wave equation for quantum cosmology. In comparison to the standard
Arnowitt-Deser-Misner(ADM) decomposition and quantum gravity proposals, I
extended my analysis beyond the covariant regime when the metric is decomposed
into the $3+1$ dimensional ADM decomposition. I showed that an equal
dimensional phase space can be obtained if one applies ADM decomposed metric.
