Abstract
A stationary solution of the Dirac equation in the metric of a Reissner-Nordström black hole has been found. Only one stationary regular state outside the black hole event horizon and only one stationary regular state below the Cauchy horizon are shown to exist. The normalization integral of the wave functions diverges on both horizons if the black hole is non-extremal. This means that the solution found can be only the asymptotic limit of a nonstationary solution. In contrast, in the case of an extremal black hole, the normalization integral is finite and the stationary regular solution is physically self-consistent. The existence of quantum levels below the Cauchy horizon can affect the final stage of Hawking black hole evaporation and opens up the fundamental possibility of investigating the internal structure of black holes using quantum tunneling between external and internal states.
Similar content being viewed by others
References
V. Fock, Z. Phys. 57, 261 (1929); V. Fock and D. Iwanenko, Z. Phys. 54, 798 (1929).
S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (Wiley, New York, 1972; Platon, Volgograd, 2000).
D. R. Brill and J. A. Wheeler, Rev. Mod. Phys. 29, 465 (1957).
B. Mukhopadhyay, Classical Quantum Gravity 17, 2017 (2000).
V. A. Berezin, Preprint: Neutrino Forces and the Schwarzschild Metric (Institute for Nuclear Research, Academy of Sciences of the Soviet Union, Moscow, 1975).
A. Burinskii, Gravitation Cosmol. 14, 109 (2008).
N. Deruelle and R. Ruffini, Phys. Lett. B 52, 437 (1974).
I. M. Ternov, V. R. Khalilov, G. A. Chizhov, and A. B. Gaina, Sov. Phys. J. 21(9), 1200 (1978).
L. A. Kofman, Phys. Lett. A 87, 281 (1982).
M. Soffel, B. Muller, and W. Greiner, J. Phys. A: Math. Gen. 10, 551 (1977).
I. M. Ternov, A. B. Gaina, and G. A. Chizhov, Sov. Phys. J. 23(8), 695 (1980).
D. V. Gal’tsov, G. V. Pomerantseva, and G. A. Chizhov, Sov. Phys. J. 26(8), 743 (1983).
M. V. Gorbatenko and V. P. Neznamov, arXiv:1205.4348 [gr-qc].
M. A. Vronsky, M. V. Gorbatenko, N. S. Kolesnikov, V. P. Neznamov, E. Yu. Popov, and I. I. Safronov, arXiv:1301.7595 [gr-qc].
V. Dzhunushaliev, arXiv:1202.5100 [gr-qc].
S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, Oxford, 1983; Mir, Moscow, 1986), Part 1, Chap. 5.
V. I. Dokuchaev, Classical Quantum Gravity 28, 235015 (2011).
J. Bičák, Z. Stuchlík, and V. Balek, Bull. Astron. Inst. Czech. 40, 65 (1989); J. Bičák, Z. Stuchlík, and V. Balek, Bull. Astron. Inst. Czech. 40, 133 (1989).
E. Hackmann, C. Lämmerzahl, V. Kagramanova, and J. Kunz, Phys. Rev. D: Part. Fields 81, 044020 (2010).
S. Grunau and V. Kagramanova, Phys. Rev. D: Part. Fields 83, 044009 (2011).
M. Olivares, J. Saavedra, C. Leiva, and J. R. Villanueva, arXiv:1101.0748 [gr-qc].
B. J. Carr, J. H. Gilbert, and J. E. Lidsey, Phys. Rev. D: Part. Fields 50, 4853 (1994).
A. D. Dolgov, P. D. Naselsky, and I. D. Novikov, arXiv:astro-ph/0009407.
B. J. Carr, Lect. Notes Phys. 631, 301 (2003); B. J. Carr, arXiv:astro-ph/0310838.
M. A. Markov, Sov. Phys. JETP 24(3), 584 (1966).
G. E. Volovik, JETP Lett. 69(9), 705 (1999).
Y. Miao, Z. Xue, and S. Zhang, Europhys. Lett. 96, 10008 (2011).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © V.I. Dokuchaev, Yu.N. Eroshenko, 2013, published in Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2013, Vol. 144, No. 1, pp. 85–91.
Rights and permissions
About this article
Cite this article
Dokuchaev, V.I., Eroshenko, Y.N. Stationary solutions of the Dirac equation in the gravitational field of a charged black hole. J. Exp. Theor. Phys. 117, 72–77 (2013). https://doi.org/10.1134/S1063776113080049
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063776113080049