Analysis of the quantum-mechanical equivalence of the metrics of a centrally symmetric uncharged gravitational field

Abstract

In the paper we analyze the quantum-mechanical equivalence of the metrics of a centrally symmetric uncharged gravitational field. We consider the Schwarzschild metrics in the spherical, isotropic and harmonic coordinates, and the Eddington-Finkelstein, Painleve-Gullstrand, Lemaitre-Finkelstein, Kruskal metrics. The scope of the analysis includes domains of the wave functions of Dirac's equation, hermiticity of Hamiltonians, and the possibility of existence of stationary bound states of spin-half particles. The constraint on the domain of the wave functions of the Hamiltonian in a Schwarzschild field in spherical coordinates (r > r_{0}) resulting from the fulfillment of Hilbert's condition g_{00} > 0 also holds in other coordinates for all the metrics considered. The self-adjoint Hamiltonians for the Schwarzschild metrics in the spherical, isotropic and harmonic coordinates and also for the Eddington-Finkelstein and Painleve-Gullstrand metrics are Hermitian, and for them the existence of stationary bound states of spin-half particles is possible. The self-adjoint Hamiltonians for non-stationary Lemaitre-Finkelstein and Kruskal metrics have the explicit dependence on the temporal coordinates and stationary bound states of spin-half particles cannot be defined for these Hamiltonians. The results of this study can be useful when addressing the issues related to the evolution of the universe and interaction of collapsars with surrounding matter.