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The Hamiltonian of the metric General Relativity derived in our earlier study(Gravitation, {\bf 17}, 314 - 323 (2011)) is analyzed by the methods of MatrixQuantum 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 moreconvenient to derive the Schr\"{o}dinger equation for the free gravitationalfield. In turn, this Schr\"{o}dinger equation is used to study possible motionsof the free gravitational field(s). In particular, it is shown that harmonicoscillations of the free gravitational field, or harmonic gravitational waves,cannot be observed as an actual motion of this field. We also investigate theinternal structure of the solitary gravitational wave.

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