The existence of degenerate stationary bound states with square-integrable radial wavefunctions was proven when second-order equations are used with the effective potential of the Reissner–Nordström ...(RN) field with two event horizons for charged and uncharged fermions. The fermions in such states are localized near event horizons within the ranges from zero to several fractions of Compton wavelength of fermions versus the values of gravitational and electromagnetic coupling constants and the values of angular and orbital momenta
j
,
l
. In case of extreme RN fields, the absence of stationary bound states of fermions with the energies of
E
<
mc
2
is shown for solutions of the second-order equation for any value of gravitational and electromagnetic coupling constants. The existence of a discrete energy spectrum is shown for the naked RN singularity, due to the solution of the second-order equation at definite values of physical parameters. The discrete spectrum exists for both charged and uncharged fermions. The naked RN singularity in quantum mechanics with the second-order equation for half-spin particles poses no threat to cosmic censorship, since it is covered with an infinitely large potential barrier. Electrically neutral systems of atomic type (RN collapsars with the definite number of fermions in degenerate bound states) are proposed to consider as particles of dark matter.
Stable and reliable numerical integration of second-order radial equations in the fields of classical black holes can be performed by using the Prüfer transformation and by passing to the use of ...phase functions, which allows uniquely selecting the solutions with physical asymptotics in numerical calculations.
Quantum Black Holes in the Expanding Universe Neznamov, V. P.; Safronov, I. I.; Shemarulin, V. E.
Doklady. a journal of the Russian Academy of Sciences. Physics,
07/2023, Letnik:
68, Številka:
7
Journal Article
Recenzirano
In the expanding Universe, the evolution of quantum black holes with modified Schwarzschild and Reissner–Nordström geometries is considered. All quantum black holes at definite moments of the ...cosmological time lose event horizons and are transformed to gravitating objects with masses
M
.
The self-conjugate Dirac Hamiltonian is obtained in the Kerr–Newman field. A transition is implemented to a Schrödinger-type relativistic equation. For the case where the angular and radial variables ...are not separated, the method of obtaining effective potentials is generalized. Effective potentials have isolated singularities on the event horizons as well as at certain parameters of the Kerr–Newman field and of the fermion in the neighborhoods of some values of the radial coordinate. For the extreme Kerr–Newman field, the impossibility of existence of stationary bound states of half-spin particles is proved.
Quantum-mechanical motion of a spin-half particle is examined in the axially symmetric fields of static naked singularities formed by a mass distribution with a quadrupole moment (
q
-metric). The ...analysis is performed by means of the method of effective potentials of the Dirac equation generalized to the case where radial and angular variables are not separated. If −1 <
q
<
q
lim, |
q
lim| ≪ 1, where
q
is the quadrupolemoment in proper units, the naked singularities do not exclude the existence of stationary bound states of Dirac particles for a prolate mass distribution in the
q
-metric along the axial axis. For an oblate mass distribution, the naked singularities of the
q
-metric are separated from a Dirac particle by infinitely large repulsive barriers followed by a potential well which deepens while moving apart from the equator (from
θ
=
θ
min
or
θ
=
π
−
θ
min
) toward the poles. The poles make an exception, and at 0 <
q
<
q
*, there are some points
θ
i
for particle states with
j
≥ 3/2.
The natural Hamiltonian systems (systems with separable Hamiltonians) are considered. The variety of explicit three-stage symplectic schemes is described. A classification of the third-order accurate ...schemes is given. All fourth-order schemes are found (there are seven of them). It is proved that there are no fifth-order schemes. The schemes with improved properties, such as invertibility and optimality with respect to the phase error, are listed. Numerical results that demonstrate the properties of these schemes are presented, and their comparative analysis with respect to the accuracy–efficiency criterion is given. The disbalance of total energy is used as the accuracy criterion.
After transition from the Dirac equation to the Schrödinger-type relativistic equation with effective potentials of the Schwarzschild and Reissner–Nordström (RN) fields, the existence of the ...stationary state of fermions with real square integrable radial wave functions is proved. The fermions are localized near the event horizons within the range from zero to several fractions or a few units of the Compton wavelength of a fermion as a function of both the gravitational and electromagnetic coupling constants and the angular and orbital momenta
. Electrically neutral atomic-type systems (Schwarzschild and RN collapsars with fermions in bound states) are proposed as particles of dark matter.
We propose a quantum model of spinning black holes with the integrable ring singularities. For the charged Kerr-Newman quantum metric, the complete regularization takes place at fixing of the maximal ...(cut-off) energy of gravitons, \(k_{UV}^{reg} = \hbar c/R_{S}^{reg}\).The domains of existence of one, two and several event horizons \(r_{q}\) are shown depending on the parameters of modified Kerr and Kerr-Newman metrics.
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