Stationary Solutions of the Second-Order Equation for Fermions in Kerr–Newman Space-Time

Abstract

When using the quantum mechanical second-order equation with the effective potential of the Kerr–Newman (KN) field for fermions, results were obtained that qualitatively differ from the results obtained when using the Dirac equation. In the presence of two event horizons, existence of degenerate stationary bound states was proved for charged and uncharged fermions with square integrable wavefunctions vanishing on event horizons. The fermions in such states are localized near the event horizons with the maxima of probability densities away from the event horizons by fractions of the Compton wavelength of fermions versus the values of coupling constants, the values of angular and orbital momenta j, l, and the value of the azimuthal quantum number m_φ. In the case of extreme KN fields, absence of stationary bound states of fermions was shown for any values of coupling constants. Existence of discrete energy spectra was shown for charged and uncharged fermions in the field of KN naked singularity at definite values of physical parameters. The KN naked singularity poses no threat to cosmic censorship because of the regular behavior of the effective potentials of the KN field in quantum mechanics with the second-order equation.

APPENDIX

1.1 Effective Potentials of Gravitational and Electromagnetic Fields in Self-Conjugate Second-Order Equations

1. Kerr–Newman field:

In compliance with (68)–(71), we can obtain

$$\begin{gathered} \frac{3}{8}\frac{1}{{B_{{KN}}^{2}}}{{\left( {\frac{{d{{B}_{{KN}}}}}{{d\rho }}} \right)}^{2}} = \frac{3}{8}\left\{ {{{{\frac{{{{f}_{{KN}}}}}{{{{\omega }_{{KN}}} + \sqrt {{{f}_{{KN}}}} }}}}_{{_{{}}}}}} \right. \\ \times \left[ { - \frac{1}{{f_{{KN}}^{2}}}f_{{KN}}^{'}({{\omega }_{{KN}}} + \sqrt {{{f}_{{KN}}}} )} \right. \\ {{\left. {\left. { + \frac{1}{{{{f}_{{KN}}}}}\left( {\omega _{{KN}}^{'} + \frac{{f_{{KN}}^{'}}}{{2\sqrt {{{f}_{{KN}}}} }}} \right)} \right]} \right\}}^{2}}, \\ \end{gathered} $$

((A1))

$$\begin{gathered} - \frac{1}{4}\frac{1}{{{{B}_{{KN}}}}}\frac{{{{d}^{2}}{{B}_{{KN}}}}}{{d{{\rho }^{2}}}} = - \frac{1}{4}\frac{{{{f}_{{KN}}}}}{{{{\omega }_{{KN}}} + \sqrt {{{f}_{{KN}}}} }} \\ \times \left[ {\frac{2}{{f_{{KN}}^{3}}}{{{(f_{{KN}}^{'})}}^{2}}({{\omega }_{{KN}}} + \sqrt {{{f}_{{KN}}}} ) - \frac{1}{{f_{{KN}}^{2}}}} \right. \\ \, \times f_{{KN}}^{{''}}({{\omega }_{{KN}}} + \sqrt {{{f}_{{KN}}}} ) - \frac{{2f_{{KN}}^{'}}}{{f_{{KN}}^{2}}}\left( {\omega _{{KN}}^{'} + \frac{{f_{{KN}}^{'}}}{{2\sqrt {{{f}_{{KN}}}} }}} \right) \\ \left. { + \frac{1}{{{{f}_{{KN}}}}}\left( {\omega _{{KN}}^{{''}} + \frac{{f_{{KN}}^{{''}}}}{{2\sqrt {{{f}_{{KN}}}} }} - \frac{{{{{(f_{{KN}}^{'})}}^{2}}}}{{4f_{{KN}}^{{3/2}}}}} \right)} \right], \\ \end{gathered} $$

((A2))

$$\frac{1}{4}\frac{d}{{d\rho }}(A - D) = \frac{\lambda }{2}\left[ {\frac{1}{2}\frac{{f_{{KN}}^{'}}}{{\rho f_{{KN}}^{{3/2}}}} + \frac{1}{{{{\rho }^{2}}f_{{KN}}^{{1/2}}}}} \right],$$

((A3))

$$\begin{gathered} - \frac{1}{4}\frac{{A - D}}{B}\frac{{dB}}{{d\rho }} = \frac{\lambda }{{2\rho f_{{KN}}^{{1/2}}}}\left( { - \frac{{f_{{KN}}^{'}}}{{{{f}_{{KN}}}}}} \right. \\ \left. { + \frac{1}{{{{\omega }_{{KN}}} + \sqrt {{{f}_{{KN}}}} }}\left( {\omega _{{KN}}^{'} + \frac{{f_{{KN}}^{'}}}{{2\sqrt {{{f}_{{KN}}}} }}} \right)} \right), \\ \end{gathered} $$

((A4))

$$\frac{1}{8}{{(A - D)}^{2}} = \frac{{{{\lambda }^{2}}}}{{2{{f}_{{KN}}}{{\rho }^{2}}}},$$

((A5))

$$\frac{1}{2}BC = - \frac{1}{{2f_{{KN}}^{2}}}(\omega _{{KN}}^{2} - {{f}_{{KN}}}).$$

((A6))

In (A1)–(A6),

$${{f}_{{KN}}} = 1 - \frac{{2\alpha }}{\rho } + \frac{{\alpha _{a}^{2} + \alpha _{Q}^{2}}}{{{{\rho }^{2}}}},$$

$$f_{{KN}}^{'} \equiv \frac{{d{{f}_{{KN}}}}}{{d\rho }} = \frac{{2\alpha }}{{{{\rho }^{2}}}} - \frac{{2(\alpha _{a}^{2} + \alpha _{Q}^{2})}}{{{{\rho }^{3}}}},$$

$$f_{{KN}}^{{''}} \equiv \frac{{{{d}^{2}}{{f}_{{KN}}}}}{{d{{\rho }^{2}}}} = - \frac{{4\alpha }}{{{{\rho }^{3}}}} + \frac{{6(\alpha _{a}^{2} + \alpha _{Q}^{2})}}{{{{\rho }^{4}}}},$$

$${{\omega }_{{KN}}} = \varepsilon \left( {1 + \frac{{\alpha _{a}^{2}}}{{{{\rho }^{2}}}}} \right) - \frac{{{{\alpha }_{a}}{{m}_{\varphi }}}}{{{{\rho }^{2}}}} - \frac{{{{\alpha }_{{em}}}}}{\rho },$$

$$\omega _{{KN}}^{'} \equiv \frac{{d{{\omega }_{{KN}}}}}{{d\rho }} = - \frac{{2\varepsilon {{\alpha }_{a}}}}{{{{\rho }^{3}}}} + \frac{{2{{\alpha }_{a}}{{m}_{\varphi }}}}{{{{\rho }^{3}}}} + \frac{{{{\alpha }_{{em}}}}}{{{{\rho }^{2}}}},$$

$$\omega _{{KN}}^{{''}} \equiv \frac{{{{d}^{2}}{{\omega }_{{KN}}}}}{{d{{\rho }^{2}}}} = \frac{{6\varepsilon \alpha _{a}^{2}}}{{{{\rho }^{4}}}} - \frac{{6{{\alpha }_{a}}{{m}_{\varphi }}}}{{{{\rho }^{4}}}} - \frac{{2{{\alpha }_{{em}}}}}{{{{\rho }^{3}}}}.$$

The sum of expressions ${{E}_{{{\text{Schr}}}}} = \frac{1}{2}({{\varepsilon }^{2}} - 1)$ and (A1)–(A6) leads to the expression for effective potential $U_{{{\text{eff}}}}^{F}$ (79). For the rest of the electromagnetic and gravitational fields examined in the paper, the structure of the expressions for the effective potentials does not change. Only the expressions for $f,f ',f '',\omega ,\omega ',\omega ''$ change.

2. Kerr field $({{\alpha }_{Q}} = 0,{{\alpha }_{{em}}} = 0)$:

$${{f}_{K}} = 1 - \frac{{2\alpha }}{\rho } + \frac{{\alpha _{a}^{2}}}{{{{\rho }^{2}}}},\quad f_{K}^{'} = \frac{{2\alpha }}{{{{\rho }^{2}}}} - \frac{{2\alpha _{a}^{2}}}{{{{\rho }^{3}}}},$$

$$f_{K}^{{''}} = - \frac{{4\alpha }}{{{{\rho }^{3}}}} + \frac{{6\alpha _{a}^{2}}}{{{{\rho }^{4}}}},\quad {{\omega }_{K}} = \varepsilon \left( {1 + \frac{{\alpha _{a}^{2}}}{{{{\rho }^{2}}}}} \right) - \frac{{{{\alpha }_{a}}{{m}_{\varphi }}}}{{{{\rho }^{2}}}},$$

$$\omega _{K}^{'} = - \frac{{2\varepsilon {{\alpha }_{a}}}}{{{{\rho }^{3}}}} + \frac{{2{{\alpha }_{a}}{{m}_{\varphi }}}}{{{{\rho }^{3}}}},\quad \omega _{K}^{{''}} = \frac{{6\varepsilon \alpha _{a}^{2}}}{{{{\rho }^{4}}}} - \frac{{6{{\alpha }_{a}}{{m}_{\varphi }}}}{{{{\rho }^{4}}}}.$$

3. Reissner–Nordström field (α_a = 0):

$${{f}_{{RN}}} = 1 - \frac{{2\alpha }}{\rho } + \frac{{\alpha _{Q}^{2}}}{{{{\rho }^{2}}}},\quad f_{{RN}}^{'} = \frac{{2\alpha }}{{{{\rho }^{2}}}} - \frac{{2\alpha _{Q}^{2}}}{{{{\rho }^{3}}}},$$

$$f_{{RN}}^{{''}} = - \frac{{4\alpha }}{{{{\rho }^{3}}}} + \frac{{6\alpha _{Q}^{2}}}{{{{\rho }^{4}}}},\quad {{\omega }_{{RN}}} = \varepsilon - \frac{{{{\alpha }_{{em}}}}}{\rho },$$

$$\omega _{{RN}}^{'} = \frac{{{{\alpha }_{{em}}}}}{{{{\rho }^{2}}}},\quad \omega _{{RN}}^{{''}} = - \frac{{2{{\alpha }_{{em}}}}}{{{{\rho }^{3}}}},\quad \lambda = \kappa .$$

4. Schwarzschild field $\left( {{{\alpha }_{Q}} = 0,{{\alpha }_{a}} = 0,{{\alpha }_{{em}}} = 0} \right)$:

$${{f}_{S}} = 1 - \frac{{2\alpha }}{p},\quad f_{S}^{'} = \frac{{2\alpha }}{{{{\rho }^{2}}}},\quad f_{S}^{{''}} = - \frac{{4\alpha }}{{{{\rho }^{3}}}},$$

$${{\omega }_{S}} = \varepsilon ,\quad \omega _{S}^{'} = \omega _{S}^{{''}} = 0,\quad \lambda = \kappa .$$

5. Coulomb field (the plane space-time, α = 0, ${{\alpha }_{Q}} = 0,{{\alpha }_{a}} = 0$):

$${{f}_{C}} = 1,\quad f_{C}^{'} = f_{C}^{{''}} = 0,\quad {{\omega }_{C}} = \varepsilon - \frac{{{{\alpha }_{{em}}}}}{\rho },$$

$$\omega _{C}^{'} = \frac{{{{\alpha }_{{em}}}}}{{{{\rho }^{2}}}},\quad \omega _{C}^{{''}} = - \frac{{2{{\alpha }_{{em}}}}}{{{{\rho }^{3}}}},\quad \lambda = \kappa .$$