On the Dirac operator for a test electron in a Reissner–Weyl–Nordström black hole spacetime


The present paper studies the Dirac Hamiltonian of a test electron with a domain of bi-spinor wave functions supported on the static region inside the Cauchy horizon of the subextremal RWN black hole spacetime, respectively inside the event horizon of the extremal RWN black hole spacetime. It is found that this Dirac Hamiltonian is not essentially self-adjoint, yet has infinitely many self-adjoint extensions. Including a sufficiently large anomalous magnetic moment interaction in the Dirac Hamiltonian restores essential self-adjointness; the empirical value of the electron’s anomalous magnetic moment is large enough. In the subextremal case the spectrum of the self-adjoint Dirac operator with anomalous magnetic moment is purely absolutely continuous and consists of the whole real line; in particular, there are no eigenvalues. The same is true for the spectrum of any self-adjoint extension of the Dirac operator without anomalous magnetic moment interaction, in the subextremal black hole context. In the extremal black hole sector the point spectrum, if non-empty, consists of a single eigenvalue, which is identified.

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  1. 1.

    One may be tempted to consider this result as a vindication for the widespread opinion that “naked singularities are considered unphysical” (cf. [16], p.562). However, this opinion propagates an unfortunate myth. It is based on a misunderstanding of Penrose’s weak cosmic censorship hypothesis, which surmises that gravitational collapse of cosmic matter does not form a naked singularity. In its strict sense the surmise is wrong, as shown first by Christodoulou [7, 8] for spherically symmetric collapse of scalar matter, and most recently by Rodnianski and Shlapentokh-Rothman [27] for collapsing gravitational waves without symmetry assumption; yet it is expected that these scenarios are not generic (this was confirmed for the spherically symmetric scalar case, also by Christodoulou [9]), and that generically (or: typically) a gravitational collapse of cosmic matter will not form a naked singularity. However, the point nuclei used in quantum-mechanical models of hydrogenic ions of the kind created in our laboratories are not assumed to have formed through gravitational collapse of charged matter in cosmic proportions. In short, the weak cosmic censorship hypothesis, even if generically true, is entirely irrelevant to the problem of general-relativistic hydrogenic ions.

  2. 2.

    The distinguished self-adjoint extention is defined by allowing \(Z\in {\mathbb {C}}\) and demanding analyticity in Z. The real threshold values then become \(Z =\sqrt{3}/2{\alpha ^{}_{{{\text {S}}}}}\) instead of \(Z=118\), and \(Z=1/{\alpha ^{}_{{{\text {S}}}}}\) instead of \(Z=137\). Here, \({\alpha ^{}_{{{\text {S}}}}}:= {e^2}/{\hbar c} \approx 1/137.036\) is Sommerfeld’s fine structure constant.

  3. 3.

    Here, “\(f(x) \sim g(x)\) as \(x\rightarrow x_*\)” means \(\exists C>0\) such that \(f(x)/g(x) \rightarrow C\) as \(x\rightarrow x_*\), where \(x_*=0\) or \(\infty \).


  1. 1.

    Balasubramanian, M.K.: Spectrum of the Dirac Hamiltonian for hydrogenic atoms on spacetimes with mild singularities, arXiv:2009.11986 [math-ph]; see also: Scalar fields and spin-half fields on mildly singular spacetimes, Ph.D. thesis, Rutgers Univ. (2015)

  2. 2.

    Behncke, H.: The Dirac equation with an anomalous magnetic moment. Math. Z. 174, 213–225 (1980)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Belgiorno, F.: Massive Dirac fields in naked and in black hole Reissner-Nordström manifolds. Phys. Rev. D 58, 084017 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  4. 4.

    Belgiorno, F., Martellini, M., Baldicchi, M.: Naked Reissner-Nordström singularities and the anomalous magnetic moment of the electron field. Phys. Rev. D 62, 084014 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  5. 5.

    Born, M.: Modified field equations with a finite radius of the electron. Nature 132, 282 (1933)

    ADS  Article  Google Scholar 

  6. 6.

    Brill, D.R., Cohen, J.M.: Cartan frames and the general relativistic Dirac equation. J. Math. Phys. 7, 238–243 (1966)

    ADS  MathSciNet  Article  Google Scholar 

  7. 7.

    Christodoulou, D.: Violation of cosmic censorship in the gravitational collapse of a dust cloud. Commun. Math. Phys. 93, 171–195 (1984)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Christodoulou, D.: Examples of naked singularity formation in the gravitational collapse of a scalar field. Ann. Math. 140, 607–653 (1994)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Christodoulou, D.: The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math. 149, 183–217 (1999)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Cohen, J.M., Powers, R.T.: The general-relativistic hydrogen atom. Commun. Math. Phys. 86, 96–86 (1982)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Droz, S., Israel, W., Morsink, S.M.: Black holes: the inside story. Phys. World 9, 34–37 (1996)

    ADS  Article  Google Scholar 

  12. 12.

    Einstein, A.: Letter to Arnold Sommerfeld, Dec. 9 (1915)

  13. 13.

    Esteban, M.J., Loss, M.: Self-adjointness of Dirac operators via Hardy–Dirac inequalities. J. Math. Phys. 48, 112107(8) (2007)

    ADS  MathSciNet  Article  Google Scholar 

  14. 14.

    Finster, F., Smoller, J., Yau, S.T.: Non-existence of time-periodic solutions of the Dirac equation in a Reissner–Nordström black hole background. J. Math. Phys. 41, 2173–2194 (2000)

    ADS  MathSciNet  Article  Google Scholar 

  15. 15.

    Gesztezy, F., Simon, B., Thaller, B.: On the self-adjointness of Dirac operators with anomalous magnetic moment. Proc. AMS 94, 115–118 (1985)

    Article  Google Scholar 

  16. 16.

    Greiner, W., Müller, B., Rafelski, J.: Quantum Electrodynamics of Strong Fields. Springer, Berlin (1985)

    Google Scholar 

  17. 17.

    Hawking, S., Ellis, G.F.R.: The Large-scale Structure of Spacetime. Cambridge University Press, Cambridge (1973)

    Google Scholar 

  18. 18.

    Hinton, D.B., Mingarelli, A.B., Shaw, J.K.: Dirac systems with discrete spectra. Can. J. Math. XXXIX XXXIX, 100–122 (1987)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Hartle, J., Hawking, S.: Solutions of the Einstein-Maxwell equations with many black holes. Commun. Math. Phys. 26, 87–101 (1972)

    ADS  MathSciNet  Article  Google Scholar 

  20. 20.

    Hoffmann, B.: Gravitational and electromagnetic mass in the Born-Infeld electrodynamics. Phys. Rev. 47, 877–880 (1935)

    ADS  Article  Google Scholar 

  21. 21.

    Kalf, H., Schmincke, U.-W., Walter, J., Wüst, R.: On the spectral theory of Schrödinger and Dirac operators with strongly singular potentials, Lect. Notes Math. 448:182–226 Springer, Berlin (1975)

  22. 22.

    Kato, T.: Perturbation Theory for Linear Operators. Springer, New York (1966)

    Google Scholar 

  23. 23.

    Kiessling, M.K.-H., Tahvildar-Zadeh, A.S., Toprak, E.: On general-relativistic hydrogen and hydrogenic ions. J. Math. Phys. 61, 092303 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  24. 24.

    Narnhofer, H.: Quantum theory for \(1/r^2\) potentials. Acta Phys. Aust. 40, 306–322 (1974)

    Google Scholar 

  25. 25.

    Nordström, G.: On the energy of the gravitational field in Einstein’s theory. Proc. Konigl. Ned. Akad. Wet. 20, 1238–1245 (1918)

    ADS  Google Scholar 

  26. 26.

    Reissner, H.: Über die Eigengravitation des elektrischen Feldes nach der Einsteinschen Theorie. Ann. Phys. 59, 106–120 (1916)

    Article  Google Scholar 

  27. 27.

    Rodnianski, I., Shlapentokh-Rothman, Y.: Naked singularities for the Einstein vacuum equations: The exterior solution, arXiv:1912.08478 [math.AP] (2019)

  28. 28.

    Schrödinger, E.: Diracsches Elektron im Schwerefeld I, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl. 1932, pp.436–460; Verlag Akad. Wiss. (1932)

  29. 29.

    Tahvildar-Zadeh, A.S.: On the static spacetime of a single point charge. Rev. Math. Phys. 23, 309–346 (2011)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Thaller, B.: The Dirac Equation. Springer, Berlin (1992)

    Google Scholar 

  31. 31.

    Thaller, B.: The Dirac operator, chpt. 2, pp. 23–106, in: Relativistic electronic structure theory. Part 1. Fundamentals. Ed. Peter Schwerdtfeger. Theor. Comp. Chem. 11:1–926 (2002)

  32. 32.

    Weidmann, J.: Absolut stetiges Spektrum bei Sturm–Liouville–Operatoren und Dirac–Systemen. Math. Z. 180, 423–427 (1982)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Weidmann, J.: Spectral Theory of Ordinary Differential Operators. Springer, Berlin (1987)

    Google Scholar 

  34. 34.

    Weyl, H.: Zur Gravitationstheorie. Ann. Phys. 54, 117–145 (1917)

    Article  Google Scholar 

  35. 35.

    Weyl, H.: Gravitation and the electron. Proc. Nat. Acad. Sci. USA 15, 323–334 (1929)

    ADS  Article  Google Scholar 

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Kiessling, M.KH., Tahvildar-Zadeh, A.S. & Toprak, E. On the Dirac operator for a test electron in a Reissner–Weyl–Nordström black hole spacetime. Gen Relativ Gravit 53, 15 (2021). https://doi.org/10.1007/s10714-021-02789-0

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  • Reissner–Weyl–Nordström spacetime
  • Charged black holes
  • Dirac equation
  • Mathematical relativity