Stationary Solutions of Second-Order Equations for Fermions in Reissner–Nordström Space-Time

Abstract

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 < mc² 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.

Appendices

APPENDIX A

1.1 SELF-CONJUGATE SECOND-ORDER EQUATIONS FOR SPINOR WAVEFUNCTIONS OF FERMIONS IN SCHWARZSCHILD AND REISSNER–NORDSTRÖM FIELDS

Let us introduce dimensionless variables and denotations in Hamiltonian (9)

$${{H}_{\eta }} = {{H}_{1}} + V(\rho ),$$

((A.1))

where V(ρ) = α_em/ρ. Taking into account (10) and (A.1), equation (5) has the form of

$$\left( {\varepsilon - V(\rho ) - {{H}_{1}}} \right){{\Psi }_{\eta }}\left( {\rho ,\theta ,\varphi } \right) = 0.$$

((A.2))

Let us multiply equation (A.2) on the left by operator $\left( {\varepsilon - V\left( \rho \right) + {{H}_{1}}} \right)$. Then,

$$\left( {\varepsilon - V(\rho ) + {{H}_{1}}} \right)\left( {\varepsilon - V(\rho ) - {{H}_{1}}} \right){{\Psi }_{\eta }}\left( {\rho ,\theta ,\varphi } \right) = 0.$$

((A.3))

$$\begin{gathered} \left\{ {{{{\left( {\varepsilon - V} \right)}}^{2}} - {{f}_{{R - N}}} + \left( {{{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)} \right. \\ \times \,\,\left( {{{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right) + \frac{{{{f}_{{R - N}}}}}{{{{\rho }^{2}}}} \\ \times \,\,\left[ {\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right)\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right)} \right. \\ + \,\,\frac{1}{{{{{\sin }}^{2}}\theta }}\frac{{{{\partial }^{2}}}}{{\partial {{\varphi }^{2}}}} + i{{\Sigma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{3} }}}\frac{\partial }{{\partial \theta }}\left( {\frac{1}{{\sin \theta }}} \right)\frac{\partial }{{\partial \varphi }} \\ \end{gathered} $$

$$\begin{gathered} \, + i{{\gamma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} }}}{{\gamma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{3} }}}{{f}_{{R - N}}}\frac{{dV}}{{d\rho }} - i{{\gamma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{0} }}}{{\gamma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{3} }}}{{f}_{{R - N}}}\frac{d}{{d\rho }}(\sqrt {{{f}_{{R - N}}}} ) \\ \, + {{f}_{{R - N}}}\frac{d}{{d\rho }}\left( {\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }} \right) \\ \end{gathered} $$

((A.4))

$$\begin{gathered} \left. { \times \,\,\left[ {i{{\Sigma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{2} }}}\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right) - i{{\Sigma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} }}}\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \varphi }}} \right]} \right\} \\ \times \,\,{{\Psi }_{\eta }}(\rho ,\theta ,\varphi ) = 0. \\ \end{gathered} $$

As earlier, in (A.4), the equivalent substitution of matrices (12) was performed,

$${{\Sigma }^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{k} }}} = \left( {\begin{array}{*{20}{c}} {{{\sigma }^{k}}}&0 \\ 0&{{{\sigma }^{k}}} \end{array}} \right).$$

The Dirac equations for upper and lower bispinor components

$${{\Psi }_{\eta }}\left( {\rho ,\theta ,\varphi ,t} \right) = \left( {\begin{array}{*{20}{c}} {U\left( {\rho ,\theta ,\varphi } \right)} \\ {W\left( {\rho ,\theta ,\varphi } \right)} \end{array}} \right){{e}^{{ - i\varepsilon t}}}$$

((A.5))

have the form of

$$\begin{gathered} (\varepsilon - V - \sqrt {{{f}_{{R - N}}}} )U \\ = \left( { - i{{\sigma }^{3}}\left( {{{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)} \right. \\ \, - i{{\sigma }^{1}}\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right)\left. { - \,i{{\sigma }^{2}}\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \varphi }}} \right)W, \\ \end{gathered} $$

$$\begin{gathered} (\varepsilon - V + \sqrt {{{f}_{{R - N}}}} )W = \left( { - i{{\sigma }^{3}}\left( {{{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)} \right. \\ \, - i{{\sigma }^{1}}\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right) \\ \end{gathered} $$

((A.6))

$$\,\left. { - i{{\sigma }^{2}}\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \varphi }}} \right)U.$$

As a result, equation (A.4), taking into account (A.6), can be written for one of the spinors $U\left( {\rho ,\theta ,\varphi } \right)$ or $W\left( {\rho ,\theta ,\varphi } \right)$. For the spinor $U\left( {\rho ,\theta ,\varphi } \right)$, equation (A.4) has the form of

$$\left\{ {{{{\left( {\varepsilon - V} \right)}}^{2}} - {{f}_{{R - N}}} + {{{\left( {{{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)}}^{2}}} \right.$$

$$\begin{gathered} + \frac{{{{f}_{{R - N}}}}}{{{{\rho }^{2}}}}\left[ {{{{\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right)}}^{2}} + \frac{1}{{{{{\sin }}^{2}}\theta }}\frac{{{{\partial }^{2}}}}{{\partial {{\varphi }^{2}}}}} \right. \\ \left. {\, + i{{\sigma }^{3}}\frac{\partial }{{\partial \theta }}\left( {\frac{1}{{\sin \theta }}} \right)\frac{\partial }{{\partial \varphi }}} \right] + {{f}_{{R - N}}}\frac{d}{{d\rho }}\left( {\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }} \right) \\ \, \times \left[ {i{{\sigma }^{2}}\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right) - i{{\sigma }^{1}}\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \varphi }}} \right] \\ \end{gathered} $$

$$\begin{gathered} + \left( {{{f}_{{R - N}}}\frac{d}{{d\rho }}(\sqrt {{{f}_{{R - N}}}} ) - {{f}_{{R - N}}}\frac{{dV}}{{d\rho }}} \right) \\ \times \frac{1}{{\varepsilon - V + \sqrt {{{f}_{{R - N}}}} }}\left[ { - {{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} - \frac{1}{\rho } + \frac{\alpha }{{{{\rho }^{2}}}}} \right. \\ \end{gathered} $$

$$\begin{gathered} - \,i{{\sigma }^{2}}\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right) \\ \,\left. {\left. { + i{{\sigma }^{1}}\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }\frac{1}{{\sin \theta }}\frac{\partial }{{\partial \varphi }}} \right]} \right\}U\left( {\rho ,\theta ,\varphi } \right) = 0. \\ \end{gathered} $$

((A.7))

Then, the variables can be separated. It follows from representation (10) that

$$U\left( {r,\theta ,\varphi } \right) = F\left( \rho \right)\xi \left( \theta \right){{e}^{{i{{m}_{\varphi }}\varphi }}}.$$

((A.8))

Using Brill–Wheeler equation (11) and its squared representation [16]

$$\begin{gathered} \left[ {{{{\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right)}}^{2}} + \frac{1}{{{{{\sin }}^{2}}\theta }}\frac{{{{\partial }^{2}}}}{{\partial {{\varphi }^{2}}}}} \right. \\ \left. {\, + i{{\sigma }^{3}}\frac{\partial }{{\partial \theta }}\left( {\frac{1}{{\sin \theta }}} \right)\frac{\partial }{{\partial \varphi }}} \right]\xi (\theta ){{e}^{{i{{m}_{\varphi }}\varphi }}} = - {{\kappa }^{2}}\xi (\theta ){{e}^{{i{{m}_{\varphi }}\varphi }}}, \\ \end{gathered} $$

((A.9))

we can obtain the second-order equation for the radial function F(ρ)

$$\begin{gathered} \left\{ {{{{\left( {\varepsilon - V} \right)}}^{2}} - {{f}_{{R - N}}} + {{{\left( {{{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)}}^{2}}} \right. \\ - \frac{{{{f}_{{R - N}}}{{\kappa }^{2}}}}{{{{\rho }^{2}}}} + {{f}_{{R - N}}}\kappa \frac{d}{{d\rho }}\left( {\sqrt {{{f}_{{R - N}}}} \frac{1}{\rho }} \right) \\ \end{gathered} $$

((A.10))

$$ - \left( {{{f}_{{R - N}}}\frac{d}{{d\rho }}(\sqrt {{{f}_{{R - N}}}} )\, - \,{{f}_{{R - N}}}\frac{{dV}}{{d\rho }}} \right)\frac{1}{{\varepsilon \, - \,V\, + \,\sqrt {{{f}_{{R - N}}}} }}\frac{{\kappa \sqrt {{{f}_{{R - N}}}} }}{\rho }$$

$$\begin{gathered} - \left( {{{f}_{{R - N}}}\frac{d}{{d\rho }}(\sqrt {{{f}_{{R - N}}}} ) - {{f}_{{R - N}}}\frac{{dV}}{{d\rho }}} \right) \\ \times \left. {\frac{1}{{\varepsilon - V + \sqrt {{{f}_{{R - N}}}} }}\left( {{{f}_{{R - N}}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)} \right\}F(\rho ) = 0. \\ \end{gathered} $$

The third and the last summands in equation (A.10) are not self-conjugate. Let us perform nonunitary similarity transformation for self-conjugacy of (A.10).

$$F\left( \rho \right) = g_{F}^{{ - 1}}\left( \rho \right){{\psi }_{F}}\left( \rho \right).$$

((A.11))

If we denote in equation (15) that

$$A\left( \rho \right) = - \frac{1}{{{{f}_{{R - N}}}}}\left( {\frac{{1 + \kappa \sqrt {{{f}_{{R - N}}}} }}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right),$$

((A.12))

$$B\left( \rho \right) = \frac{1}{{{{f}_{{R - N}}}}}\left( {\varepsilon - \frac{{{{\alpha }_{{{\text{em}}}}}}}{\rho } + \sqrt {{{f}_{{R - N}}}} } \right),$$

((A.13))

$$C\left( \rho \right) = - \frac{1}{{{{f}_{{R - N}}}}}\left( {\varepsilon - \frac{{{{\alpha }_{{{\text{em}}}}}}}{\rho } + \sqrt {{{f}_{{R - N}}}} } \right),$$

((A.14))

$$D\left( \rho \right) = - \frac{1}{{{{f}_{{R - N}}}}}\left( {\frac{{1 - \kappa \sqrt {{{f}_{{R - N}}}} }}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)$$

((A.15))

and besides introduce the denotations of

$${{A}_{F}}\left( \rho \right) = - \frac{1}{B}\frac{{dB}}{{d\rho }} - A - D,$$

((A.16))

$${{A}_{G}}\left( \rho \right) = - \frac{1}{C}\frac{{dC}}{{d\rho }} - A - D,$$

((A.17))

then, the sought transformation is

$${{g}_{F}}(\rho ) = \exp \left( {\frac{1}{2}\int {{{A}_{F}}(\rho ')d\rho '} } \right).$$

((A.18))

As the result, we write equation (A.10) as

$$\hat {M}F(\rho ) = 0,$$

then, the transformed self-conjugate equation has the view of

$${{g}_{F}}\hat {M}g_{F}^{{ - 1}}{{\psi }_{F}}(\rho ) = 0$$

((A.19))

Equation (A.19) can be written in the form of the Schrödinger-type second-order equation with the effective potential $U_{{{\text{eff}}}}^{F}(\rho )$

$$\frac{{{{d}^{2}}{{\psi }_{F}}}}{{d{{\rho }^{2}}}} + 2({{E}_{{{\text{Schr}}}}} - U_{{{\text{eff}}}}^{F}){{\psi }_{F}} = 0,$$

((A.20))

where

$${{E}_{{{\text{Schr}}}}} = \frac{1}{2}({{\varepsilon }^{2}} - 1),$$

((A.21))

$$\begin{gathered} U_{{{\text{eff}}}}^{F} = - \frac{1}{4}\frac{1}{B}\frac{{{{d}^{2}}B}}{{d{{\rho }^{2}}}} + \frac{3}{8}{{\left( {\frac{1}{B}\frac{{dB}}{{d\rho }}} \right)}^{2}} \\ - \frac{1}{4}\left( {A - D} \right)\frac{1}{B}\frac{{dB}}{{d\rho }} + \frac{1}{4}\frac{d}{{d\rho }}\left( {A - D} \right) \\ + \frac{1}{8}{{\left( {A - D} \right)}^{2}} + \frac{1}{2}BC + {{E}_{{{\text{Schr}}}}}. \\ \end{gathered} $$

((A.22))

The summand E_Schr (A.21) in equation (A.20) is separated and simultaneously added to (A.22). This is done, on the one hand, for equation (A.20) to take the form of Schrödinger-type equation, on the other hand, to ensure the classical asymptotics of the effective potential at $\rho \to \infty$.

For the lower spinor $W(\rho ,\theta ,\varphi )$ with radial function G(ρ), the appropriate formulas have the view

$$G\left( \rho \right) = g_{G}^{{ - 1}}{{\psi }_{G}}\left( \rho \right),$$

((A.23))

$${{g}_{G}}(\rho ) = \exp \left( {\frac{1}{2}\int {{{A}_{G}}(\rho ')d\rho '} } \right),$$

((A.24))

$$\frac{{{{d}^{2}}{{\psi }_{G}}}}{{d{{\rho }^{2}}}} + 2({{E}_{{{\text{Schr}}}}} - U_{{{\text{eff}}}}^{G}){{\psi }_{G}} = 0,$$

((A.25))

$$\begin{gathered} U_{{{\text{eff}}}}^{G} = - \frac{1}{4}\frac{1}{C}\frac{{{{d}^{2}}C}}{{d{{\rho }^{2}}}} + \frac{3}{8}{{\left( {\frac{1}{C}\frac{{dC}}{{d\rho }}} \right)}^{2}} \\ + \frac{1}{4}\frac{{\left( {A - D} \right)}}{C}\frac{{dC}}{{d\rho }} - \frac{1}{4}\frac{d}{{d\rho }}\left( {A - D} \right) \\ + \frac{1}{8}{{\left( {A - D} \right)}^{2}} + \frac{1}{2}BC + {{E}_{{{\text{Schr}}}}}. \\ \end{gathered} $$

((A.26))

APPENDIX B

1.1 EFFECTIVE POTENTIAL OF THE RN FIELD IN SCHRÖDINGER-TYPE EQUATION

According to (A.12)–(A.15), (A.22), we can obtain

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

((B.1))

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

$$ - \frac{1}{{f_{{R - N}}^{2}}}f_{{R - N}}^{{''}}(\omega + \sqrt {{{f}_{{R - N}}}} )$$

((B.2))

$$\begin{gathered} - \frac{2}{{f_{{R - N}}^{2}}}f_{{R - N}}^{'}\left( {\omega ' + \frac{{f_{{R - N}}^{'}}}{{2\sqrt {{{f}_{{R - N}}}} }}} \right) \\ \left. { + \frac{1}{{{{f}_{{R - N}}}}}\left( {\omega '' + \frac{{f_{{R - N}}^{{''}}}}{{2\sqrt {{{f}_{{R - N}}}} }} - \frac{{{{{(f_{{R - N}}^{'})}}^{2}}}}{{4f_{{R - N}}^{{3/2}}}}} \right)} \right], \\ \end{gathered} $$

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

((B.3))

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

((B.4))

$$\frac{1}{8}{{\left( {A - D} \right)}^{2}} = \frac{{{{\kappa }^{2}}}}{{2{{f}_{{R - N}}}{{\rho }^{2}}}},$$

((B.5))

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

((B.6))

In (B.1)–(B.6),

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

$$f_{{R - N}}^{'} \equiv \frac{{d{{f}_{{R - N}}}}}{{d\rho }} = \frac{{2\alpha }}{{{{\rho }^{2}}}} - \frac{{2\alpha _{Q}^{2}}}{{{{\rho }^{3}}}},$$

$$f_{{R - N}}^{{''}} \equiv \frac{{df_{{R - N}}^{2}}}{{d{{\rho }^{2}}}} = - \frac{{4\alpha }}{{{{\rho }^{3}}}} + \frac{{6\alpha _{Q}^{2}}}{{{{\rho }^{4}}}},$$

$$\omega = \varepsilon - \frac{{{{\alpha }_{{{\text{em}}}}}}}{\rho },\quad \omega ' \equiv \frac{{d\omega }}{{d\rho }} = \frac{{{{\alpha }_{{{\text{em}}}}}}}{{{{\rho }^{2}}}},$$

$$\omega '' \equiv \frac{{{{d}^{2}}\omega }}{{d{{\rho }^{2}}}} = - \frac{{2{{\alpha }_{{{\text{em}}}}}}}{{{{\rho }^{3}}}}.$$

The sum of expressions E_Schr and (B.1)–(B.6) leads to the expression for the effective potential $U_{{{\text{eff}}}}^{F}$ (A.22).