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

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 Painlevé-Gullstrand metrics, and nonstationary Lemaître-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.

40 pages

References in corpus (2)

• Uniqueness and Self-Conjugacy of Dirac Hamiltonians in arbitrary Gravitational Fields
• Solution of the problem of uniqueness and hermiticity of hamiltonians for Dirac particles in gravitational fields

Cited by in corpus (3)

• Quantum mechanics of stationary states of particles in a space-time of classical black holes
• Quantum mechanics of stationary states of particles in external singular spherically and axially symmetric gravitational and electromagnetic fields
• Quantum probing of singularities at event horizons of black holes