Stationary Solutions of Second-Order Equations for Point Fermions in the Schwarzschild Gravitational Field

Abstract

When using a second-order Schrödinger-type equation with the effective potential of the Schwarzschild field, the existence of a stationary state of half-spin particles with energy E = 0 is proved. For each of the values of quantum numbers j, l, the physically meaningful energy E = 0 (the binding energy is ${{E}_{b}} = m{{c}^{2}}$) is implemented at the value of the gravitational coupling constant $\alpha \geqslant {{\alpha }_{{\min }}}$. The particles with E = 0 are, with the overwhelming probability, at some distance from the event horizon within the range from zero to several fractions of the Compton wavelength of a fermion depending on value of the gravitational coupling constants and the values of j, l. In this paper, similar solutions of the second-order equation are announced for bound states of fermions in the Reissner–Nordström, Kerr, Kerr–Newman fields. Atomic-type systems (the point sources of the Schwarzschild gravitational field) with fermions in bound states are proposed as particles of dark matter.

Appendices

APPENDIX A

1.1 SELF-CONJUGATE SECOND-ORDER EQUATION FOR HALF-SPIN PARTICLES IN THE GRAVITATIONAL SCHWARZSCHILD FIELD

Dirac Eq. (6) with Hamiltonian (5) for the stationary states of ${{\psi }_{\eta }}(t,\rho ,\theta ,\varphi )$ = ${{e}^{{ - i\varepsilon t}}}{{\psi }_{\eta }}(\rho ,\theta ,\varphi )$ can be written as

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

((А.1))

Let us multiply equality on the left (A.1) by the operator $(\varepsilon + {{H}_{\eta }})$:

$$(\varepsilon + {{H}_{\eta }})(\varepsilon - {{H}_{\eta }}){{\psi }_{\eta }}(\rho ,\theta ,\varphi ) = 0.$$

((А.2))

Taking into account (5), we obtain

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

((А.3))

In (А.3), as well as earlier in item 2.1, there was an equivalent substitution of matrices (9);

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

Dirac equations for upper and lower components of the bispinor

$${{\psi }_{\eta }}(\rho ,\theta ,\varphi ,t) = \left( \begin{gathered} U(\rho ,\theta ,\varphi ) \\ W(\rho ,\theta ,\varphi ) \\ \end{gathered} \right){{e}^{{ - i\varepsilon t}}}$$

((А.4))

have the form

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

((А.5))

As a result, taking into account (А.5), Eq. (А.3) can be written for one of the spinors $U(\rho ,\theta ,\varphi )$ or $W(\rho ,\theta ,\varphi )$. For the spinor $U(\rho ,\theta ,\varphi )$, Eq. (А.3) has the view

$$\begin{gathered} \left\{ {{{\varepsilon }^{2}} - {{f}_{S}} + {{{\left( {{{f}_{S}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)}}^{2}}} \right. \\ \, + \frac{{{{f}_{S}}}}{{{{\rho }^{2}}}}\left[ {{{{\left( {\frac{\partial }{{\partial \theta }} + \frac{1}{2}{\text{cot}}\theta } \right)}}^{2}} + \frac{1}{{{{{\sin }}^{2}}\theta }}\frac{{{{\partial }^{2}}}}{{\partial {{\varphi }^{2}}}}} \right. \\ \end{gathered} $$

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

((А.6))

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

Then, the variables can be separated. From representation (7) it follows that

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

((А.7))

Using Brill–Wheeler Eq. (8) and its squared representation [41]

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

((А.8))

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

$$\begin{gathered} \left\{ {{{\varepsilon }^{2}} - {{f}_{S}} + {{{\left( {{{f}_{S}}\frac{\partial }{{\partial \rho }} + \frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)}}^{2}} - \frac{{{{f}_{S}}{{\kappa }^{2}}}}{{{{\rho }^{2}}}}} \right. \\ \, + {{f}_{S}}\kappa \frac{d}{{d\rho }}\left( {\sqrt {{{f}_{S}}} \frac{1}{\rho }} \right) - {{f}_{S}}\frac{d}{{d\rho }}(\sqrt {{{f}_{S}}} )\frac{1}{{\varepsilon + \sqrt {{{f}_{S}}} }}\frac{{\kappa \sqrt {{{f}_{S}}} }}{\rho } \\ \left. {\, - {{f}_{S}}\frac{d}{{d\rho }}(\sqrt {{{f}_{S}}} )\frac{1}{{\varepsilon \, + \sqrt {{{f}_{S}}} }}\left( {{{f}_{S}}\frac{\partial }{{\partial \rho }}\, + \,\frac{1}{\rho } - \frac{\alpha }{{{{\rho }^{2}}}}} \right)} \right\}F(\rho )\, = \,0. \\ \end{gathered} $$

((A.9))

In Eq. (А.9), the third and the last summands are not self-conjugate. For self-conjugacy (А.9), let us perform nonunitary similarity transformation

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

((А.10))

If in Eq. (12), we denote

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

((А.11))

$$B(\rho ) = \frac{1}{{{{f}_{S}}}}(\varepsilon + \sqrt {{{f}_{S}}} ),$$

((А.12))

$$C(\rho ) = - \frac{1}{{{{f}_{S}}}}(\varepsilon - \sqrt {{{f}_{S}}} ),$$

((А.13))

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

((А.14))

and, besides, introduce denotations of

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

((А.15))

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

((А.16))

then, the desired transformation is

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

((А.17))

As a result, if we write Eq. (А.9) as

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

then the transformed self-conjugate equation has the form

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

((А.18))

Equation (А.18) can be written as the Schrödinger-type second-order equation with the effective potential $U_{{{\text{eff}}}}^{F}(\rho )$

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

((А.19))

where

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

((А.20))

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

((А.21))

In Eq. (А.19), summand ${{E}_{{{\text{Schr}}}}}$ (А.20) is selected and simultaneously added to (А.21). On the one hand, it is done for Eq. (А.19) to have the form of the Schrödinger-type equation and, on the other hand, to ensure classical asymptotics of the effective potential at $\rho \to \infty$.

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

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

((А.22))

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

((А.23))

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

((А.24))

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

((А.25))

APPENDIX B

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

In compliance with (А.11)–(А.14), (А.21), we can obtain

$$\begin{gathered} \frac{3}{8}\frac{1}{{{{B}^{2}}}}{{\left( {\frac{{dB}}{{d\rho }}} \right)}^{2}} \\ = \frac{3}{8}{{\left( {\frac{{2\alpha }}{{\rho (\rho - 2\alpha )}} - \frac{\alpha }{{\varepsilon \rho {{{(\rho (\rho - 2\alpha ))}}^{{1/2}}} + \rho (\rho - 2\alpha )}}} \right)}^{2}}, \\ \end{gathered} $$

((B.1))

$$\begin{gathered} - \frac{1}{4}\frac{1}{B}\frac{{{{d}^{2}}B}}{{d{{\rho }^{2}}}} = - \frac{{2{{\alpha }^{2}}}}{{{{\rho }^{2}}{{{(\rho - 2\alpha )}}^{2}}}} - \frac{\alpha }{{{{\rho }^{2}}(\rho - 2\alpha )}} \\ + \frac{5}{4}\frac{{{{\alpha }^{2}}}}{{\varepsilon {{\rho }^{{5/2}}}{{{(\rho - 2\alpha )}}^{{3/2}}} + {{\rho }^{2}}{{{(\rho - 2\alpha )}}^{2}}}} \\ + \frac{\alpha }{{2[\varepsilon {{\rho }^{{5/2}}}{{{(\rho - 2\alpha )}}^{{1/2}}} + {{\rho }^{2}}(\rho - 2\alpha )]}}, \\ \end{gathered} $$

((B.2))

$$\frac{1}{4}\frac{d}{{d\rho }}\left( {A - D} \right) = \frac{{\kappa (\rho - \alpha )}}{{2{{\rho }^{{3/2}}}{{{(\rho - 2\alpha )}}^{{3/2}}}}},$$

((B.3))

$$\begin{gathered} - \frac{1}{4}\frac{{\left( {A - D} \right)}}{B}\frac{{dB}}{{d\rho }} = - \frac{{\alpha \kappa }}{{{{\rho }^{{3/2}}}{{{(\rho - 2\alpha )}}^{{3/2}}}}} \\ + \frac{{\alpha \kappa }}{{2[\varepsilon {{\rho }^{2}}(\rho - 2\alpha ) + {{\rho }^{{3/2}}}{{{(\rho - 2\alpha )}}^{{3/2}}}]}}, \\ \end{gathered} $$

((B.4))

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

((B.5))

$$\frac{1}{2}BC = - \frac{1}{2}\frac{{{{\rho }^{2}}{{\varepsilon }^{2}}}}{{{{{\left( {\rho - 2\alpha } \right)}}^{2}}}} + \frac{1}{2}\frac{\rho }{{\rho - 2\alpha }}.$$

((B.6))

The sum of expressions of ${{E}_{{{\text{Schr}}}}}$ and (B.1)–(B.6), leads to the desired expression for the effective potential $U_{{{\text{eff}}}}^{F}$.

The asymptotics is

$$U_{{{\text{eff}}}}^{F}(\varepsilon = 0){{{\text{|}}}_{{\rho \to 2\alpha }}} = - \frac{3}{{32{{{(\rho - 2\alpha )}}^{2}}}}.$$

((B.7))