Quantum-mechanical nonequivalence of metrics of centrally symmetric uncharged gravitational field

Abstract

Quantum-mechanical analysis shows that the metrics of a centrally symmetric uncharged gravitational field, which are exact solutions of the general relativity equations, are physically non-equivalent. The classical Schwarzschield metric and the Schwarzschild metrics in isotropic and harmonic coordinates provide for the existence of stationary bound states of Dirac particles with a real energy spectrum. The Hilbert condition g_{00}>0 is responsible for zero values of the wave functions under the "event horizon" that leads to the absence of Hawking radiation. For the Eddington-Finkelstein and Painleve-Gullstrand metrics, stationary bound states of spin-half particles cannot exist because Dirac Hamiltonians are non-Hermitian. For these metrics, the condition g_{00}>0 also leads to the absence of Hawking evaporation. For the Finkelstein-Lemaitre and Kruskal metrics, Dirac Hamiltonians are explicitly time-dependent, and stationary bound states of spin-half particles cannot exist for them. The Hilbert condition for these metrics does not place any constraints on the domains of the wave functions. Hawking evaporation of black holes is possible in this case. The results can lead to revisiting some concepts of the standard cosmological model related to the evolution of the universe and interaction of collapsars with surrounding matter.