Quantum mechanics of stationary states of particles in a space–time of classical black holes

M. V. Gorbatenko   ; V. P. Neznamov

Teoretičeskaâ i matematičeskaâ fizika, Tome 205 (2020) no. 2, pp. 284-323

Résumé

We consider interactions of scalar particles, photons, and fermions in Schwarzschild, Reissner–Nordström, Kerr, and Kerr–Newman gravitational and electromagnetic fields with a zero and nonzero cosmological constant. We also consider interactions of scalar particles, photons, and fermions with nonextremal rotating charged black holes in a minimal five-dimensional gauge supergravity. We analyze the behavior of effective potentials in second-order relativistic Schrödinger-type equations. In all cases, we establish the existence of the regime of particles “falling” on event horizons. An alternative can be collapsars with fermions in stationary bound states without a regime of particles “falling.”

Détail

Keywords: quantum mechanical hypothesis of cosmic censorship, Schrödinger-type equation, effective potential, Schwarzschild, Reissner–Nordström, and Kerr–Newman black holes with zero and nonzero cosmological constant, anti-de Sitter black hole in five-dimensional supergravity.
Mots-clés : scalar particle, photon, fermion, Kerr

Bibliographie

[1] V. P. Neznamov, “Uravneniya vtorogo poryadka dlya fermionov v prostranstve-vremeni Shvartsshilda, Raissnera–Nordstrema, Kerra i Kerra–Nyumena”, TMF, 197:3 (2018), 493 | DOI | DOI

[2] L. D. Landau, E. M. Lifshits, Teoreticheskaya fizika, v. III, Kvantovaya mekhanika. Nerelyativistskaya teoriya, Fizmatlit, M., 1963 | MR | MR

[3] A. M. Perelomov, V. S. Popov, ““Padenie na tsentr” v kvantovoi mekhanike”, TMF, 4:1 (1970), 48–65 | DOI

[4] V. P. Neznamov, I. I. Safronov, “Statsionarnye resheniya uravneniya vtorogo poryadka dlya tochechnykh fermionov v gravitatsionnom pole Shvartsshilda”, ZhETF, 154:4(10) (2018), 761–773, arXiv: 1809.08940 | DOI

[5] V. P. Neznamov, I. I. Safronov, V. E. Shemarulin, “Statsionarnye resheniya uravneniya vtorogo poryadka dlya fermionov v prostranstve-vremeni Raissnera–Nordstrema”, ZhETF, 154:4(10) (2018), 802–825, arXiv: 1810.01960 | DOI

[6] V. P. Neznamov, I. I. Safronov, V. E. Shemarulin, “Statsionarnye resheniya uravneniya vtorogo poryadka dlya fermionov v prostranstve-vremeni Kerra–Nyumena”, ZhETF, 155:1 (2019), 69–95, arXiv: 1904.05791 | DOI

[7] N. Deruelle, R. Ruffini, “Quantum and classical relativistic energy states in stationary geometries”, Phys. Lett. B, 52:4 (1974), 437–441 | DOI

[8] T. Damour, N. Deruelle, R. Ruffini, “On quantum resonances in stationary geometries”, Lett. Nuovo Cimento, 15:2 (1976), 257–262

[9] I. M. Ternov, V. R. Khalilov, G. A. Chizhov, A. B. Gaina, “Finitnoe dvizhenie massivnykh chastits v polyakh Kerra i Shvartsshilda”, Izv. vuzov. Fizika, 1978, no. 9, 109–114 | DOI

[10] A. B. Gaina, G. A. Chizhov, “Radialnoe dvizhenie v pole Shvartsshilda”, Izv. vuzov. Fizika, 1980, no. 4, 120–121

[11] I. M. Ternov, A. B. Gaina, G. A. Chizhov, “Finitnoe dvizhenie elektronov v pole mikroskopicheski- malykh chernykh dyr”, Izv. vuzov. Fizika, 1980, no. 8, 56–62 | DOI

[12] D. V. Galtsov, G. V. Pomerantseva, G. A. Chizhov, “Zapolnenie elektronami kvazisvyazannykh sostoyanii v pole Shvartsshilda”, Izv. vuzov. Fizika, 26:8 (1983), 75–77 | DOI

[13] I. M. Ternov, A. B. Gaina, “Ob energeticheskom spektre uravneniya Diraka v polyakh Shvartsshilda i Kerra”, Izv. vuzov. Fizika, 31:2 (1988), 86–92 | DOI | MR

[14] A. B. Gaina, O. B. Zaslavskii, “On quasilevels in the gravitational field of a black hole”, Class. Quantum Grav., 9:3 (1992), 667–676 | DOI | MR

[15] A. B. Gaina, N. I. Ionescu-Pallas, “The fine and hyperfine structure of fermionic levels in gravitational fields”, Rom. J. Phys., 38 (1993), 729–730

[16] A. Lasenby, C. Doran, J. Pritchard, A. Caceres, S. Dolan, “Bound states and decay times of fermions in a Schwarzschild black hole background”, Phys. Rev. D, 72:10 (2005), 105014, 14 pp. | DOI

[17] S. Dolan, D. Dempsey, “Bound states of the Dirac equation on Kerr spacetime”, Class. Quantum Grav., 32:18 (2015), 184001, 31 pp. | DOI | MR

[18] D. Batic, M. Nowakowski, K. Morgan, “The Problem of embedded eigenvalues for the Dirac equation in the Schwarzschild black hole metric”, Universe, 2:4 (2016), 31, 14 pp., arXiv: 1701.03889 | DOI

[19] F. Finster, J. Smoller, S.-T. Yau, “Non-existence of time-periodic solutions of the Dirac equation in a Reissner–Nordström black hole background”, J. Math. Phys., 41:4 (2000), 2173–2194 | DOI | MR

[20] F. Finster, N. Kamran, J. Smoller, S.-T. Yau, “Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry”, Comm. Pure Appl. Math., 53:7 (2000), 902–929 | 3.0.CO;2-4 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[21] F. Finster, N. Kamran, J. Smoller, S.-T. Yau, “Erratum: Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry”, Comm. Pure Appl. Math., 53:9 (2000), 1201 | 3.0.CO;2-T class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[22] G. T. Horowitz, D. Marolf, “Quantum probes of spacetime singularities”, Phys. Rev. D, 52:10 (1995), 5670–5675, arXiv: gr-qc/9504028 | DOI | MR

[23] G. Bete, E. Solpiter, Kvantovaya mekhanika atomov s odnim i dvumya elektronami, Fizmatgiz, M., 1960

[24] R. Penrose, “Gravitational collapse: the role of general relativity”, Riv. Nuovo Cimento, 1 (1969), 252–276

[25] K. Meetz, “Singular potentials in nonrelativistic quantum mechanics”, Nuovo Cimento, 34:3 (1964), 690–708 | DOI

[26] H. Behncke, “Some remarks on singular attractive potentials”, Nuovo Cimento A, 55:4 (1968), 780–785 | DOI

[27] A. S. Wightman, “Introduction to some aspects of the relativistic dynamics of quantized fields”, High Energy Electromagnetic Interactions and Field Theory (Cargèse, France, September 1964), ed. M. Levy, eds. B. d' Espagnat, M. Lévy, Gordon and Breach, New York, 1967, 171–192

[28] V. P. Neznamov, I. I. Safronov, V. E. Shemarulin, “Statsionarnye resheniya uravneniya vtorogo poryadka dlya fermionov v prostranstve-vremeni Kerra–Nyumena”, ZhETF, 155:1 (2019), 69–95

[29] I. Ya. Pomeranchuk, Ya. A. Smorodinskii, “Ob urovnyakh energii sistem s $Z > 137$”, J. Phys. USSR, 9:2 (1945), 97

[30] W. Pieper, W. Griener, “Interior electron shells in superheavy nuclei”, Z. Phys., 218:4 (1969), 327–340 | DOI

[31] Ya. B. Zeldovich, V. S. Popov, “Elektronnaya struktura sverkhtyazhelykh atomov”, UFN, 105:11 (1971), 403–440 | DOI

[32] W. Greiner, B. Müller, J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, Berlin–Heidelberg, 1985 | DOI

[33] D. Andrae, “Finite nuclear charge density distributions in electronic structure calculations for atoms and molecules”, Phys. Rep., 336:6 (2000), 413–525 | DOI

[34] R. N. Boyer, R. W. Lindquist, “Maximal analytic extension of the Kerr metric”, J. Math. Phys., 8:2 (1967), 265–281 | DOI | MR

[35] V. B. Bezerra, H. S. Vieira, A. A. Costa, “The Klein–Gordon equation in the spacetime of a charged and rotating black hole”, Class. Quantum Grav., 31:4 (2014), 045003, 11 pp., arXiv: 1312.4823v1 | DOI | MR

[36] A. S. Eddington, “A comparison of Whitehead's and Einstein's formulæ”, Nature, 113:2832 (1924), 192 | DOI

[37] D. Finkelstein, “Past-future asymmetry of the gravitational field of a point particle”, Phys. Rev., 110:4 (1958), 965–967 | DOI

[38] P. Painlevé, “La mécanique classique et la théorie de la relativité”, C. R. Acad. Sci. (Paris), 173 (1921), 677–680 | Zbl

[39] A. Gullstrand, Allgemeine Lösung des Statischen Einkörperproblems in der Einsteinschen Gravitationstheorie, Arkiv. Mat. Astron. Fys., 16, no. 8, Almqvist and Wiksell, Stockholm, 1922

[40] G. Lemaítre, “L'univers en expansion”, Ann. Soc. Sci. Bruxelles A, 53 (1933), 51–85

[41] M. D. Kruskal, “Maximal extension of Schwarzschild metric”, Phys. Rev., 119:5 (1960), 1743–1745 | DOI

[42] G. Szekeres, “On the singularities of a Riemannian manifold”, Publ. Math. Debrecen, 7 (1960), 285–301 | MR

[43] M. V. Gorbatenko, V. P. Neznamov, “Kvantovo-mekhanicheskaya ekvivalentnost metrik tsentralno-simmetrichnogo gravitatsionnogo polya”, TMF, 198:3 (2019), 489–522 | DOI | DOI

[44] R. P. Kerr, “Gravitational field of a spinning mass as an example of algebraically special metrics”, Phys. Rev. Lett., 11:5 (1963), 237–238 | DOI | MR

[45] S. A. Teukolsky, “Rotating black holes: separable wave equations for gravitational and electromagnetic perturbations”, Phys. Rev. Lett., 29:16 (1972), 1114–1118 ; “Perturbations of a rotating black hole. I. Fundamental equations for gravitational, electromagnetic, and neutrino-field perturbations”, Astrophys. J., 185 (1973), 635–648 | DOI | DOI

[46] W. Kinnersley, “Type $D$ vacuum metrics”, J. Math. Phys., 10:7 (1969), 1195–1205 | DOI | MR

[47] E. Newman, R. Penrose, “An approach to gravitational radiation by a method of spin coefficients”, J. Math. Phys., 3:3 (1962), 566–578 | DOI | MR

[48] O. Lunin, “Maxwell's equations in the Myers–Perry geometry”, JHEP, 2017:12 (2017), 138, 86 pp., arXiv: 1708.06766 | DOI

[49] S. Chandrasekhar, “The solution of Dirac's equation in Kerr geometry”, Proc. Roy. Soc. London Ser. A, 349:1659 (1976), 571–575 | DOI | MR

[50] D. N. Page, “Dirac equation around a charged, rotating black hole”, Phys. Rev. D, 14:6 (1976), 1509–1510 | DOI

[51] Z. Stuchlík, G. Bao, E. Østgaard, S. Hledík, “Kerr–Newman–de Sitter black holes with a restricted repulsive barrier of equatorial photon motion”, Phys. Rev. D., 58:8 (1998), 084003, 8 pp. | DOI | MR

[52] J. B. Griffiths, J. Podolský, Exact Spacetimes in Einstein's General Relativity, Cambridge Monographs in Mathematical Physics, Cambridge Univ. Press, Cambridge, 2009 | MR

[53] B. Carter, “Global structure of the Kerr family of gravitational fields”, Phys. Rev., 174:5 (1968), 1559–1571 | DOI

[54] Z. Stuchlík, S. Hledík, “Equatorial photon motion in the Kerr–Newman spacetimes with a non-zero cosmological constant”, Class. Quantum Grav., 17:21 (2000), 4541–4576 | DOI | MR

[55] G. V. Kraniotis, “The Klein–Gornon–Fock equation in the curved spacetime of the Kerr–Newman (anti) de Sitter black hole”, Class. Quantum Grav., 33:22 (2016), 225011, 50 pp., arXiv: 1602.04830 | DOI | MR

[56] G. V. Kraniotis, “The massive Dirac equation in the Kerr–Newman–de Sitter and Kerr–Newman black hole spacetimes”, J. Phys. Commun., 3:3 (2019), 035026, 41 pp., arXiv: 1801.03157 | DOI

[57] F. Finster, N. Kamran, J. Smoller, S.-T. Yau, “Nonexistence of time-periodic solutions of the Dirac equation in an axisymmetric black hole geometry”, Commun. Pure Appl. Math., 53:7 (2000), 902–909 | 3.0.CO;2-4 class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI

[58] S. Q. Wu, “Separability of massive field equations for spin-0 and spin-1/2 charged particles in the general nonextremal rotating charged black hole spacetimes in minimal five-dimensional gauged supergravity”, Phys. Rev. D, 80:8 (2009), 084009, 16 pp., arXiv: 0906.2049 | DOI

[59] M. Heusler, Black Holes Uniqueness Theorems, Cambridge Univ. Press, Cambridge, 1996 | MR

[60] H. Quevedo, “Mass quadrupole as a source of naked singularities”, Internat. J. Modern Phys., 20:10 (2011), 1779–1787, arXiv: 1012.4030 | DOI

[61] S. Toktarbay, H. Quevedo, “A stationary $q$-metric”, Grav. Cosmol., 20:4 (2014), 252–254, arXiv: 1510.04155 | DOI | MR

[62] V. P. Neznamov, V. E. Shemarulin, “Motion of spin-half particles in the axially symmetric field of naked singularities of the static $q$-metric”, Grav. Cosmol., 23:2 (2017), 149–161, arXiv: 1706.05229 | DOI | MR

[63] K. Schwarzschild, “Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie”, Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften, Reimer, Berlin, 1916, 424–434

[64] R. Tolmen, Otnositelnost, termodinamika i kosmologiya, Nauka, M., 1974

[65] Dzh. Sing, Obschaya teoriya otnositelnosti, IL, M., 1963

[66] M. V. Gorbatenko, “Reshetochnye dirakovskie matritsy i formalizm standartnoi modeli”, VANT. Ser.: Teoreticheskaya i prikladnaya fizika, 1 (2018), 19–30