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Theoretical and Mathematical Physics

, Volume 198, Issue 3, pp 425–454 | Cite as

Quantum Mechanical Equivalence of the Metrics of a Centrally Symmetric Gravitational Field

  • M. V. Gorbatenko
  • V. P. NeznamovEmail author
Article

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, stationary Eddington–Finkelstein and Painlevé–Gullstrand metrics, and nonstationary Lemaˆıtre–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 > r0, where r0 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 field are independent of the choice of a solution for the centrally symmetric static gravitational field used.

Keywords

coordinate transformation Dirac Hamiltonian second-order equation for fermions effective potential degenerate bound state 

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Russian Federal Nuclear CenterAll-Russian Scientific Research Institute of Experimental PhysicsSarov, Nizhny Novgorod OblastRussia
  2. 2.National Research Nuclear University MEPhIMoscowRussia

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