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М. V . Gorbatenko , V . P . Neznamov

Quantum mechanics of stationary states of particles in external singular fields of black holes with event horizons of zero thickness. М. V . Gorbatenko , V . P . Neznamov. International Bogolubov Conference Problems of Theoretical and Mathematical Physics September 9-13, 2019,

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М. V . Gorbatenko , V . P . Neznamov

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  1. Quantum mechanics of stationary states of particlesin external singular fields of black holes with event horizons of zero thickness М.V.Gorbatenko, V.P.Neznamov International Bogolubov Conference Problems of Theoretical and Mathematical Physics September 9-13, 2019, Moscow-Dubna, Russia

  2. Plan of talk Introduction Quantum mechanical hypothesis of cosmological censorship Hydrogen-like atom in the strong Coulomb field Effective potentials of the relativistic Schrödinger-type equations in the geometries of classical solutions of the general relativity theorywith zero and non-zero cosmological constant for: - bosons (S=0) - photons (S=1) - fermions (S=1/2) Asymptotics of effective potentials in neighborhoods of event horizons Incompatibility of quantum theory with the classical conception of black holes with event horizons of zero thickness 7. Discussion and conclusion 8. Myths of classical solutions of the general relativity theory 2

  3. Introduction What we do “Z>137 catastrophe” Mode: «fall» to the center “Cure”: taking into account the finite-size nuclei “Z>137 catastrophe” Mode: «fall» to the center “Cure”: taking into account the finite-size nuclei classical solutions of general relativity theory curved space-time flat space-time classical mechanics, field theory effects of vacuum polarization as Z>137 etc. Black holes Mode: «fall» to the event horizon “Cure”: no quantum mechanics quantum mechanics atomic physics Lamb shift anomalous magnetic moment of electron quantum field theory quantum field theory 3

  4. Introduction The report considers the interaction of scalar particles, photons and fermions with the gravitational and electromagnetic Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman fields. The behavior of effective potentials in the relativistic Schrödinger-type second-order equations is analyzed. It was found that the quantum theory is incompatible with the hypothesis of the existence of classical black holes with event horizons of zero thickness that were predicted based on solutions of the general relativity with zero and non-zero cosmological constant . The alternative may be presented by compound systems, i.e., collapsars with fermions in stationary bound states. 4

  5. Introduction The following papers are the base of report 1. Stationary solutions of second-order equations for point fermions in the Schwarzschild gravitational field // V.P.Neznamov and I.I.Safronov,J. Exp. Theor. Phys. (2018) 127: 647. 2. Stationary solutions of second-order equations for fermions in Reissner–Nordström space-time // V.P.Neznamov, I.I.Safronov, and V.E.Shemarulin, J. Exp. Theor. Phys. (2018) 127: 684. 3. Stationary solutions of the second-order equation for fermions in Kerr–Newman space-time // V.P.Neznamov, I.I.Safronov, and V.E.Shemarulin, J. Exp. Theor. Phys. (2019) 128:64. 4. Second-order equations for fermions on Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman space-times // V.P.Neznamov,Theor. Math. Phys. (2018) 197: 1823. 5. Quantum mechanical equivalence of the metric of a centrally symmetric gravitational field // М.V.Gorbatenko and V.P.Neznamov, Theor. Math. Phys. (2019)198:425. 6. Stationary solutions of the second-order equation for fermions in the external Coulomb field // V.P.Neznamov and I.I.Safronov,J. Exp. Theor. Phys. (2019) 155: 792. 5

  6. Introduction The following papers are ready for publication Quantum mechanics of particle stationary states in external singular spherically and axially symmetric gravitational fields // M.V.Gorbatenko and V.P.Neznamov. 2. Quantum mechanics of particle stationary states in external singular spherically and axially symmetric gravitational fields with non-zero cosmological constant // M.V.Gorbatenko and V.P.Neznamov. 3. Quantum mechanics of stationary states of particles interacting with non-extreme rotating charged black holes in minimal five-dimensional gauged supergravity // M.V.Gorbatenko and V.P.Neznamov. 6

  7. Introduction • We also used the results of separation of variables: • 1. in Klein-Gordon equation • for Kerr-Newman metric - V.B.Bezerra, H.S.Viera and André A.Costa, Class. Quantum Grav. 31, 045003 (2014); • for Kerr-Newman-(A)dS metric - G.V.Kraniotis, Class. Quantum Grav. 33, 225011 (2016); • for five-dimensional geometry of Kerr-Newman-AdS - S.Q.Wu, Phys. Rev. D 80 (2009) 084009, arxiv:0906.2049v4 [hep-th]; • 2. in Maxwell equations • for Kerr, Kerr-(A)dS metrics, five-dimensional geometry of Myers-Perry - O.Lunin, J. High Energ. Phys. (2017) 2017:138, arxiv: 1708.06766v2 [hep-th]; • 3. in Dirac equation • for Kerr-Newman metric - S.Chandrasekhar, Proc. Roy. Soc. London ser. A 349, 571 (1976); • D.Page, Phys. Rev. D 14, 1509 (1976); F.Finster, N.Kamran, J.Smoller and S.-T. Yau, • Comm. Pure Appl. Math. 53, 1201 (2000); • for Kerr-Newman-(A)dS metric- C.V.Kraniotis, J. Phys. Commun. 3, 035026 (2019), arxiv: 1801.03157; • for five-dimensional Kerr-Newman-AdS metric - S.Q.Wu, Phys. Rev. D 80 (2009) 084009, arxiv: 0906.2049v4 [hep-th]. 7

  8. Introduction Hereafter, we use dimensionless variables for the second-order equations for particles with energy , mass and electrical charge in the spice-time of considered metrics is the Compton wave length of a particle; is the Planck mass; is the electromagnetic constant of the fine structure; are the gravitational and electromagnetic coupling constants; are the dimensionless constants characterizing the source of the electromagnetic field with the charge and the ratio of an angular momentum to the mass in the Kerr and Kerr-Newman metrics. 8

  9. Quantum mechanical hypothesis of cosmological censorship For classical physics, the hypothesis of cosmic censorship proposed by Penrose forbids the existence of singularities not covered by event horizons in nature. «Nature has an aversion for naked singularity» R.Penrose «Природа питает отвращение к голой сингулярности» Р.Пенроуз 9

  10. Quantum mechanical hypothesis of cosmological censorship In the quantum mechanics, G.T.Horowitz and D.Marolf(Phys. Rev. D 52, 5670 (1995)) actually proposed a quantum mechanical hypothesis of cosmic censorship. They write in the Introduction: “We will say that a system is nonsingular, when the evolution of any state is uniquely defined for all time. If this is not the case, then there is some loss of predictability and we will say that the system is singular” and incompatible with the quantum theory. Similarly to Penrose, we should add that such singular systems cannot exist in nature. 10

  11. Quantum mechanical hypothesis of cosmological censorship For the radial second-order equations reduced to the relativistic Schrödinger-type equations with the effective potentials , the behavior of these potentials in the neighborhood of event horizons is essential. For all the considered metrics, the behavior of effective potentials in the neighborhood of the event horizons has often the form of an infinitely deep potential well 11

  12. Quantum mechanical hypothesis of cosmological censorship If , a so-called “fall” of a particle to the event horizon occurs. In this case the system is singular. The behavior of the radial function from the Schrödinger-type equation has the following form As , the functionsof stationary states of discrete and continuous spectrum have an unlimited number of zeros,discrete levels of energy “ dive” into the area of negative continuum.The functions do not have the defined values as . Also, the system will be singular, if the exponent of the denominator in the expression for the effective potential higher than two. In this case where are arbitrary phases, is the exponent in the effective potential. 12

  13. Quantum mechanical hypothesis of cosmological censorship In the Hamiltonian formulation, the mode of a ''particle fall'' to the event horizons means that the Hamiltonian has nonzero deficiency indexes. For elimination of this mode, it is necessary to choose the additional boundary conditions on the event horizons. The self-conjugate extension of the Hermitian operator is determined by this choice. 13

  14. Hydrogen-like atom in the strong Coulomb field In the quantum mechanics there is an example confirming the quantum mechanical hypothesis of cosmic censorship. For hydrogen-like atoms, the Sommerfeld formula for the fine structure of the energy levels has the form As the expression for energy becomes imaginary («the catastrophe»). Let us consider solutions of the relativistic Schrödinger-type equation with the effective potential for fermions in the Coulomb field. The asymptotics of the effective potential ashas the form 14

  15. Hydrogen-like atom in the strong Coulomb field In the asymptotics,dependently on it is possible to single out three characteristic areas. For example, we consider these areas for the bound states . In the first area asthere exists the positive barrier with the following potential well.As the potential barrier disappears; as for asthe potential well remains. In the second area the coefficient is , which permits the existence of the fermion stationary bound states. In the third area as there is the potential well with , which is indicative of the implementation of the regime “fall” to the center. In Fig.1 for the dependencies as are shown. There for comparison there are shown the dependence of the Coulomb potential . 15

  16. Hydrogen-like atom in the strong Coulomb field a) b) Fig. 1. The dependenciesand. 16

  17. Hydrogen-like atom in the strong Coulomb field In the third area with the system “a fermion in the Coulomb field” is singular, and incompatible with the quantum theory. For elimination of mode “fall” to the center, it was offered to take into account the finite sizes of nuclei (I.Ya.Pomeranchuk and A.Smorodinsky (1945). W.Paper and W.Griener (1969)).As a result, at the characteristic lengths of the nuclei sizes it is cut either the Coulomb potential, or the effective one (see Fig.1 b).Presently, there are ~ 30 such cutting methods (D. Andrae, Physics Reports 336, 413 (2000)). The system “an electron in the Coulomb field of the finite-size atomic nucleus'' is nonsingular. 17

  18. Hydrogen-like atom in the strong Coulomb field a) b) Fig. 1. The dependenciesand. 18

  19. Effective potentials For a closed system of “a particle in the external force field”, quantum mechanics allows existence of stationary states with definite real energies of particle. The stationary states involve both the states of discrete spectrum (bound states) and states of continuous spectrum (scattering states). In this case, the wave function of the particle is written as where is a particle energy. Here and hereafter, we use the system of units . We will explore in closed systems the existence possibility of stationary states at interaction of scalar particles , photon , fermions with the Schwarzschild, Reissner-Nordström, Kerr and Kerr-Newman black holes. We consider the four-dimensional geometries with zero and non-zero cosmological constant and also AdSblack holegeometry in five-dimensional gauged supergravity. 19

  20. Effective potentials Scheme of our analysis I. Scalar uncharged particles 1. The Klein-Gordon equation : 2. Separation of variables : (for the spherically symmetric Schwarzschild, Reissner-Nordströmmetrics, are the spherical harmonics) (for the axially symmetric Kerr, Kerr-Newman metrics, are the oblate spheroidal harmonic functions, where ) 3. Second-order equation for radial functions : 20

  21. Effective potentials 4. Reduction to form of the Schrödinger equation with the effective potential : In the second equation the summand is singled out and at the same time added to the third equation. On the one hand, it is done, in order to impart to equation to form of the Schrödinger-type equation, and, on the other hand, ensure the classical asymptotics of the effective potential as . 5. Study of behavior of effective potential in the neighborhood of the event horizons. 21

  22. Effective potentials II. Photons The Maxwell equations for the spherically symmetric Schwarzschild and Reissner-Nordstrom metrics were written in three-dimensional form (in three-dimensional space with metric . Separation of the variables was performed via expansion of electric and magnetic fields in terms of the vector harmonics of the electric, magnetic and longitudinal types. For the axially symmetric Kerr, Kerr-Newman metrics, the variables separation procedure of O.Lunin was used. As a result, the effective potentials of the Schrodinger-type relativistic equations were obtained. III. Fermion with charge For metrics under consideration, the effective potentials of the Schrodinger-type equation were obtained in our papers (V.P.Neznamov, I.I.Safronov, V.E.Shemarulin, J. Exp. Theor. Phys.127(2018), 128(2019) ).These papers also contain the solutions for the stationary bound states with energies . 22

  23. Asymptotics of effective potentials in neighborhoods of event horizons fermion fermion fermion Schwarzschild field photon scalar particle fermion Reissner-Nordström field scalar particle photon 23

  24. Asymptotics of effective potentials in neighborhoods of event horizons scalar particle fermion fermion Kerr, Kerr-Newman fields photon 24

  25. Asymptotics of effective potentials in neighborhoods of event horizons Non-zero cosmological constant We have proved that Kerr-Newman-(A)dS, Kerr-(A)dS, Reissner-Nordström-(A)dS andSchwarzschild-(A)dSblack holes have the same nature of the divergence of effective potentials near the event horizons as in case of . For example, for the most general Kerr-Newman-(A)dS field, for the de Sitter solutionat scalar particle fermion photon We also proved that the Kerr-Newman-AdSblack hole in the minimal five-dimensional gauged supergravityhas the same nature of the divergence of effective potentials near the event horizons as in the four-dimensional case. AdS black hole CFT 25

  26. Asymptotics of effective potentials in neighborhoods of event horizons Coordinate transformations of the Schwarzschild metric For the scalar particles, our analysis can be supplemented with the Eddington-Finkelstein and the Painlevé-Gullstrand metrics, for which the leading singularity in the neighborhood of event horizon remains at the real axis and it is similar to those for the Schwarzschild metrics. For fermions, for the Schwarzschild space-time in the isotropic coordinates, as well as the Eddington-Finkelstein, Painlevé-Gullstrand, Lemaitre-Finkelstein and Kruskal-Szekeres metrics, it is proved that in case of the stationary bound state the leading singularity in the event horizon neighborhood is preserved the same as in the initial Schwarzschild metrics. 26

  27. Incompatibility of quantum theory with the classical conception of black holes with event horizons of zero thickness The analysis shows that for in the considered gravitational fields the existence of the stationary particle states is impossible. Systems “a particle in the gravitational and electromagnetic fields” are singularand incompatible with the quantum theory. 27

  28. Incompatibility of quantum theory with the classical conception of black holes with event horizons of zero thickness Existence of the stationary bound states with does not change the previous conclusion because to attain the values within the range of ,it is necessary to change a particular particle initial energy up to this level.However, quantum mechanical stationary states with the energy do not exist in the considered gravitational and electromagnetic fields. For all the considered metrics and particles with different spins it is characteristic the universal nature of the divergence of the effective potentials near to the event horizons. Uncovered singularities doesn’t allow using the quantum theory in full measure that leads to the necessity of changing the initial formulation of the physical problem. 28

  29. Discussion and conclusion • Is it possible to cure the concerned GR solutions from the standpoint of quantum mechanics? The answer to this question is negative.Actually, the uniqueness theorem for the black holes affirms that the most general asymptotically flat solution of the GR equations is the Kerr metrics with the monopole mass and the angular momentum (M. Heusler, Cambrige, UK (1996)). Any deviation from the spherically symmetric mass distribution leads to the event horizon disappearance and appearance of series of naked singularities instead (see the static and stationary q-metrics). 29

  30. Discussion and conclusion 1.1 Deformed distribution of the collapsar mass: prolate or oblate along the axis the Schwarzschild metric: The static q-metric (Quevedo, Int. J.Mod. Phys. D (2011)) is theprolate mass distribution is theoblate mass distribution 30

  31. Discussion and conclusion As a result, we have “naked singularity” instead of a black hole as for any . One can build a stationary q-metric(Quevedoat al., arxiv: 1510.04155v1). The similar conclusion: a black hole transforms to naked singularity on the surface covering the Schwarzschild event horizon. 1.2 Nonpoint collapsar and linkage. If, similarly to the Coulomb potentiasl as , we perform the linkage of the GR external vacuum solutions with the internal solution options, preserving the continuity of the metric tensor and its first derivatives, then the linkage radius will larger than the radius of the external event horizon. This is not a black hole!!! As a result, in the mentioned above cases we deviate from the classical black holes with the event horizons. 31

  32. Discussion and conclusion 2. Is it possible to use the concerned classical GR solutions from the standpoint of quantum mechanics? Is it possible to blur the event horizons in a natural way? Our answer to these questions is positive and we propose to supplement the gravitational collapse mechanism. Let us, at the last collapse stage the gravitational field captures the half-spin particles, which after the formation of the event horizons will get into the stationary bound states with both under the inner and above external event horizons. For the next particles interacting with such compound systems, the self-consistent gravitational and electromagnetic fields will be determined both by mass and charge of collapsar, and by the masses and charges of the fermions in the stationary bound states with in the neighborhood of the event horizons. 32

  33. Discussion and conclusion Obviously, such system may be nonsingular. To provide a strong proof, there are required exact calculations of the self-consistent gravitational and electromagnetic fields, as well as the proof that the stationary states of the quantum-mechanical probe particles exist in these fields. The discussed compound systems may be considered as the dark matter carriers. On the other hand, these systems may be the building blocks for joining new particles and formation of the macroscopic objects in the long run. 33

  34. Discussion and conclusion CONCLUSION The quantum theory is incompatible with the existence of the Schwarzschild, Reissner-Nordström, Kerr, Kerr-Newman black holes with event horizons of zero thickness that were predicted based on the GR solutions with zero and non-zero cosmological constant . 34

  35. Myths of classical solutions of the general relativity theory unstable system electron + proton stable atomic systems 1 + ergoregionin Kerr, Kerr-Newman metricsfor rotating system ergoregion does not appear in the quantum Klein-Gordon-Fock, Maxwell, Dirac equations and in effective potentials 2 - ring singularity in Kerr, Kerr-Newman metrics ring singularity does not appear in quantum equations and in effective potentials 3 - there is not confirmation for Reissner-Nordström, Kerr and Kerr-Newman metrics in fermion Schrödinger-type equations with effective potentials + hypothesis of Penrose’s cosmic censorship 4 - quantum theory is incompatible with the conception of black holes with event horizons of zero thickness black holes with event horizons of zero thickness 5 - phenomenology: firewalls, generalized principle of uncertainty etc. - 6 + - compound systems: collapsars with fermions in stationary bound states - 35

  36. Thank you for your attention!

  37. References 1.D. Batic, M. Nowakowski and K. Morgan, Universe, 2, 31 (2016). 2.F. Finster, J. Smoller, S.-T. Yau, J. Math. Phys., 41, 2173 (2000). 3.F. Finster, N. Kamran, J. Smoller, and S.-T. Yau, Comm. Pure Appl. Math. 53, 902 (2000). 4.F. Finster, N. Kamran, J. Smoller, and S.-T. Yau, Comm. Pure Appl. Math. 53, 1201 (2000). 5.H. Schmid. Mathematische Nachrichten, 274-275 (1), 117 (2004); arxiv: math-ph/0207039v2. 6.C. L. Pekeris and K. Frankowski. Proc. Natl. Acad. Sci. USA, 83, 1978 (1986). 7.F. Belgiorno and M. Martellini, Phys. Lett. B 453, 17 (1999). 8.M. Winklmeier and O. Yamada, J. Math. Phys. 47, 102503 (17) (2006). 9.V. P. Neznamov, I.I.Safronov, J. Exp. Theor. Phys. 127, 647 (2018), arxiv: 1809.08940 [gr-qc, hep-th]. 10.V. P. Neznamov, I.I.Safronov, V. E. Shemarulin, J. Exp. Phys. 127, 684 (2018), arxiv: 1810.01960 [gr-qc, hep-th]. 11.V. P. Neznamov, I.I.Safronov, V. E. Shemarulin, J. Exp. Phys. 128, 64 (2019), arxiv: 1904.05791 [gr-qc, hep-th]. 12.G. T. Horowitz and D. Marolf, Phys. Rev. D 52, 5670 (1995). 13.R. N. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967). 14.R. Penrose, Rivista del Nuovo Cimento, Serie I, 1, Numero Speciale: 252 (1969). 15.D. M. Zipoy, J. Math. Phys. 7, 1137 (1966). 37

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  39. References 34.V. B. Berestetsky, Е. М. Lifshitz, L. P. Ptayevsky, Quantum electrodynamics. PhizMatLit, Moscow (2006). 35.R. P. Kerr, Phys. Rev. Lett. 11, 237 (1963). 36.S. A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972); S. A. Teukolsky, Astrophys. J. 185, 635 (1973). 37.W. Kinnersley, J. Math. Phys. 10, 1195 (1969). 38.E. Newman and R. Penrose, J. Math. Phys. 3, 566 (1962). 39.O. Lunin, J. High Energ. Phys. (2017) 2017:138, arxiv: 1708.06766v2 [hep-th]. 40.S. Chandrasekhar, Proc. Roy. Soc. London Ser. A 349, 571 (1976). 41.D. Page, Phys. Rev. D 14, 1509 (1976). 42.M. Heusler. Black holes uniqueness theorems. Cambridge University Press, Cambridge, UK (1996). 43.H. Quevedo, Int. J. Mod. Phys. 20, 1779 (2011). 44.S. Toktarbay and H. Quevedo. Grav. Cosmol. 20, 252-254 (2014); arxiv: 1510.04155v1. 45.V. P. Neznamov and V. E. Shemarulin. Grav. Cosmol. 23, 149 (2017). 46.K. Schwarzschild. Sitz. Preuss. Akad. Wiss. S. 424 (1916). 47.R. C. Tolmen. Relativity, thermodynamics and cosmology. Oxford at the Clarendon Press (1969). 48.J. L. Synge, Relativity: The General Theory. North-Holland Furlishing Company, Amsterdam (1960). 49.M.V. Gorbatenko, VANT, Series Theoretical and Applied Physics, 2018, i. 3. 39

  40. Effective potentials For the stationary states, the spherically symmetry of the RN metric allows searching for solutions of and in the form of the multipole expansion by the states with the angular momentum and its projection on the axis . For example, for the electric field components Here are the vector harmonics of the electric , magnetic and longitudinaltype. 40

  41. Effective potentials For the electric and magnetic fields there exist two independent solutions, which have a certain parity and angular momentum. Similarly to the flat space-time, the solution of the equations may be called “the magnetic J-pole radiation”. The second solution may be referred to as “the electric J-pole radiation”.Below, we consider the behavior the radial functions in these expressions. (1) (2) 41

  42. Effective potentials In the Maxwell equations, we can perform the separation of the variables and obtain the solutions (1) – (2). The radial functions connected by the ratios 3. The second-order equation forhas the form Here 42

  43. Effective potentials 4. After the transitions to the Schrödinger-type equation, we can obtain the effective potential 5. Study of behavior of effective potential as,and in the neighborhood of the event horizons. 43

  44. Effective potentials 2.2.Photon in the Kerr and Kerr-Newman fields Lunin (J. High Energ. Phys. 2017:138) introduced a new connection of the ansatz for of the master equation with the components of electromagnetic field potential. As a result, it was performed the separation of the variables with the real equation for the radial function. The separation of the variables according to Lunin’s terminology was performed for “the electric polarization” and “the magnetic polarization” (earlier Tukolskydidn’t perform the separation of the variables for “the magnetic polarization”). Aiming at better understanding, let us write out the Lunin’sfinal formulas. 44

  45. Effective potentials 2.2.1Electric polarization Here are the components of the Kinnersley tetrad. 45

  46. Effective potentials 2.2.2Magnetic polarization Here 46

  47. Effective potentials 2.2.3Effective potentials From the equations for the radial functions , we can obtain Schrödinger type equations for the functions with the effective potentials and . The asymptotics of the potentials are the same at and . Further, we perform the analysis of . 47

  48. Effective potentials III. Fermions • The Dirac equation 2. Separation of variables(Chandrasekhar; Page; Finster, Kamran, Smoller, Yau) 3. Equations for radial functions Where are determined from the form of Hamiltonian for concrete metrics. 48

  49. Effective potentials 4. The Schrödinger type equation with effective potential 5. The sum of and the mentioned above expressions leads to the expression for the effective potential . 49

  50. Effective potentials 5.1 The Kerr-Newman field 50

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