Quantum mechanical equivalence of the metrics of a centrally symmetric gravitational field

TL;DR

This paper demonstrates that quantum mechanical descriptions of particles in various centrally symmetric gravitational metrics are equivalent, showing that physical results do not depend on the specific metric coordinate system used.

Contribution

It proves the quantum mechanical equivalence of different metrics describing a static, centrally symmetric gravitational field, including Schwarzschild, Eddington-Finkelstein, Painlevé-Gullstrand, Lemaître-Finkelstein, and Kruskal-Szekeres.

Findings

Existence of degenerate stationary bound states of fermions with zero energy in all considered metrics.

Quantum results are independent of the choice of metric coordinate system.

Wave functions are restricted to the domain outside the event horizon.

Abstract

We analyze the quantum mechanical equivalence of the metrics of a centrally symmetric uncharged gravitational field. We consider the static Schwarzschild metric in spherical and isotropic coordinates, the stationary Eddington-Finkelstein and Painlev\'e-Gullstrand metrics, and nonstationary Lema\^itre-Finkelstein and Kruskal-Szekeres metrics. When the real radial functions of the Dirac equation and of the second-order equation in the Schwarzschild field are used, the domain of wave functions is restricted to the range $r>r_{0}$, where $r_{0}$ is the radius of the event horizon. A corresponding constraint also exists in other coordinates for all considered metrics. For the considered metrics, the second-order equations admit the existence of degenerate stationary bound states of fermions with zero energy. As a result, we prove that physically meaningful results for a quantum mechanical description of a particle interaction with a gravitational filed are independent of the choice of a solution for the centrally symmetric static gravitational field used.