On the Schr\"{o}dinger equation for the free gravitational field

TL;DR

This paper derives a Schr"{o}dinger equation for the free gravitational field using Hamiltonian methods, revealing that harmonic gravitational waves cannot be observed as actual motions and exploring the internal structure of solitary waves.

Contribution

It introduces a new form of the Hamiltonian for General Relativity to derive a Schr"{o}dinger equation for free gravitational fields, providing insights into gravitational wave motions.

Findings

Harmonic gravitational waves cannot be observed as actual motions.

The Schr"{o}dinger equation for the gravitational field is formulated.

Internal structure of solitary gravitational waves is analyzed.

Abstract

The Hamiltonian of the metric General Relativity derived in our earlier study (Gravitation, {\bf 17}, 314 - 323 (2011)) is analyzed by the methods of Matrix Quantum Mechanics. This Hamiltonian is a quadratic function of the momenta $\pi^{mn}$ conjugate to the spatial components $g_{mn}$ of the metric tensor $g_{\alpha\beta}$. The Hamiltonian is reduced to the new form which is more convenient to derive the Schr\"{o}dinger equation for the free gravitational field. In turn, this Schr\"{o}dinger equation is used to study possible motions of the free gravitational field(s). In particular, it is shown that harmonic oscillations of the free gravitational field, or harmonic gravitational waves, cannot be observed as an actual motion of this field. We also investigate the internal structure of the solitary gravitational wave.