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The classical Einstein-Hilbert (EH) action for general relativity (GR) isshown to be formally analogous to the classical system with position-dependentmass (PDM) models. The analogy is developed and used to build the covariantclassical 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 afirst-order system dual to the Einstein field equations. Following theprinciples of quantum mechanics, I build a canonical theory for the classicalgeneral. A fully consistent quantum Hamiltonian for GR is constructed based onadopting a high dimensional phase space. It is observed that the functionalwave equation is timeless. As a direct application, I present an alternativewave equation for quantum cosmology. In comparison to the standardArnowitt-Deser-Misner(ADM) decomposition and quantum gravity proposals, Iextended my analysis beyond the covariant regime when the metric is decomposedinto the $3+1$ dimensional ADM decomposition. I showed that an equaldimensional phase space can be obtained if one applies ADM decomposed metric.

Position-Dependent Mass Quantum systems and ADM formalism