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

TL;DR

This paper investigates the quantum-mechanical properties of different metrics of a centrally symmetric uncharged gravitational field, revealing non-equivalence and implications for black hole radiation and cosmology.

Contribution

It demonstrates that various classical solutions of general relativity exhibit distinct quantum behaviors, challenging the assumption of metric equivalence at the quantum level.

Findings

Schwarzschild metric allows stationary bound states of Dirac particles.

Eddington-Finkelstein and Painleve-Gullstrand metrics do not support stationary bound states.

Hawking radiation is absent in metrics where the Hilbert condition g_{00}>0 is violated.

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.