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Quantum mechanical equivalence of the metrics of a centrally symmetric gravitational field
General Physics · Submitted 16 Apr 2019
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 Painlev\'e-Gullstrand metrics, and nonstationary Lema\^itre-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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We analyze the quantum mechanical equivalence of the metrics of a centrally symmetric uncharged gravitational field. | contribution 0.98 | 0.93 | 0.06 | asserted 0.51 |
| s1 | We consider the static Schwarzschild metric in spherical and isotropic coordinates, the stationary Eddington-Finkelstein and Painlev\'e-Gullstrand metrics, and nonstationary Lema\^itre-Finkelstein and Kruskal-Szekeres metrics. | method 0.54 | 0.53 | 0.02 | asserted 0.77 |
| s2 | 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. | result 0.95 | 0.79 | 0.04 | asserted 1.00 |
| s3 | A corresponding constraint also exists in other coordinates for all considered metrics. | result 1.00 | 0.76 | 0.04 | asserted 1.00 |
| s4 | For the considered metrics, the second-order equations admit the existence of degenerate stationary bound states of fermions with zero energy. | result 1.00 | 0.86 | 0.09 | asserted 0.87 |
| s5 | 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. | result 0.67 | 0.94 | 0.94 | asserted 1.00 |
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The paper
Original 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 Painlev\'e-Gullstrand metrics, and nonstationary Lema\^itre-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.
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How the literature uses it
2 arXiv papers cite this in 13 sentences
Papers whose main result relates to this one's
States a conflicting result 1 paper
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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.
Quantum-mechanical nonequivalence of metrics of centrally symmetric uncharged gravitational field · 0.99
Papers citing it · 8 sentences without a confident label
Uses its method or tool 1 paper
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The nonzero tetrads in the Schwinger gauge of $H̃_α^μ$ equal [equation] In compliance with ([ref]), the matrixes of $γ̃^α$ are as follows: [equation] The bispinor connections of $Φ̃_α$ are as follows: [equation] Taking into account ([ref]) - ([ref]), the Dirac Hamiltonian of ([ref]) is given by [equation] Hamiltonian ([ref]) is pseudo-Hermitian with the weight operator $ρ__P =f_S^-1/2$ [ref] , [ref] , [ref] .
Quantum probing of singularities at event horizons of black holes · The Dirac equation · 0.88
Cites it as support 1 paper
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In this case, the mode of a particle ”fall” on the event horizons is preserved (see, in Ref. [ref] solutions ([ref]) and ([ref]), Hamiltonian ([ref]) and the system of equations for radial functions ([ref])).
Quantum probing of singularities at event horizons of black holes · Static metrics and the Schwarzschild metric with ”tortoise” coordinate · 0.99
Cites it as background 1 paper
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These results are presented in many textbooks, monographs and papers (see, for instance, Refs. [ref] , [ref] as well as our papers Refs. [ref] , [ref] ).
Quantum probing of singularities at event horizons of black holes · Introduction · 0.98
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