Comparative description of the evolving universe in classical and quantum geometrodynamics

TL;DR

This paper compares classical and quantum descriptions of the universe's evolution, deriving a non-linear Hamilton-Jacobi equation in quantum geometrodynamics and analyzing semiclassical wave functions under different matter-energy dominances.

Contribution

It introduces a new quantum gravitational source in the geometrodynamics framework and derives semiclassical wave functions for various matter-energy scenarios.

Findings

Quantum geometrodynamics reduces to a non-linear Hamilton-Jacobi equation.

Semiclassical wave functions are obtained for different matter-energy dominances.

Transition probability between radiation and dust-dominated states is calculated.

Abstract

The description of the universe evolving in time according to general relativity is given in comparison with the quantum description of the same universe in terms of semiclassical wave functions. The spacetime geometry is determined by the Robertson-Walker metric. It is shown that the main equation of the quantum geometrodynamics is reduced to the non-linear Hamilton-Jacobi equation. Its non-linearity is caused by a new source of the gravitational field, which has a purely quantum dynamical nature, and is additional to ordinary matter sources. In the semiclassical approximation, the non-linear equation of motion is linearized and reduces to the Friedmann equation with the additional quantum source of gravity (or anti-gravity) in the form of the stiff Zel'dovich matter. The semiclassical wave functions of the universe, in which different types of matter-energies dominate, are obtained. As examples, the cases of the domination of radiation, barotropic fluid, or new quantum matter-energy are discussed. The probability of the transition from the quantum state, where radiation dominates into the state, in which barotropic fluid in the form of dust is dominant, is calculated. This probability has the same order of magnitude as the matter density contrast in the era of matter-radiation equality.