Abstract
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+13+1 dimensional ADM decomposition. I showed that an equal dimensional phase space can be obtained if one applies ADM decomposed metric.
Highlights
Classical mechanics is Galilean invariant, i.e, time parameter t and position coordinate q(t) are explicitly functions of each other
Since quantum mechanics is Galilean invariant there is no simple way to build a locally Lorentz invariant theory with single particle interpretation
If we adopt any of these proposals, it seems that time problem remains unsolved both in the quantum gravity and cosmology [4]
Summary
Classical mechanics is Galilean invariant, i.e, time parameter t and position coordinate q(t) are explicitly functions of each other. In GR as a classical dynamical system (but with second order derivative of the position ), if we make an analogous Riemannian metric gab with the coordinate q(t) and if we use the spacetime derivatives of the metric ∂c gab (which is proportional to the Christoffel symbols Γdab ), instead of the velocity , i.e, the time variation of the coordinate q, and by adopting a symmetric connection , we can rewrite EH Lagrangian in terms of the metric, first and second derivatives of it It looks like a classical system in the form of L(q, q, q) (see TABLE I).
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