  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.
