Thin and Thick Disks around Black Holes and Wormholes

Abstract

The paper explores the distribution of matter in thick disks around black holes and wormholes both numerically and analytically. Kerr and Lame metrics are considered, and exact analytical solutions are derived. The influence of toroidal magnetic fields on the structure of the thick disk is taken into account. Images of thin disks are constructed depending on the values of metric parameters.

Appendices

APPENDIX A

The metric coefficients of the Lame metric are

$$\begin{gathered} {{g}_{{tt}}} = {{\omega }^{2}}{{e}^{{2\psi }}} - {{e}^{{2\nu }}} = - \left( {1 - \frac{{2M(r)r}}{\Sigma }} \right),\quad {{g}_{{rr}}} = {{e}^{{2\lambda }}} = \frac{\Sigma }{\Delta }, \\ {{g}_{{\theta \theta }}} = {{e}^{{2\mu }}} = \Sigma , \\ {{g}_{{\phi \phi }}} = {{e}^{{2\psi }}} = \frac{{{{{\sin }}^{2}}\theta }}{\Sigma }\left( {{{{\left( {{{r}^{2}} + {{a}^{2}}} \right)}}^{2}} - \Delta {{a}^{2}}{{{\sin }}^{2}}\theta } \right), \\ \end{gathered} $$

$$\begin{gathered} {{g}_{{t\phi }}} = - \omega {{e}^{{2\psi }}} = - \frac{{2M(r)ra{{{\sin }}^{2}}\theta }}{\Sigma }{\kern 1pt} , \\ {{g}^{{tt}}} = - {{e}^{{ - 2\nu }}} = - \frac{{{{{\left( {{{r}^{2}} + {{a}^{2}}} \right)}}^{2}} - \Delta {{a}^{2}}{{{\sin }}^{2}}\theta }}{{\Sigma \Delta }}, \\ \end{gathered} $$

$$\begin{gathered} {{g}^{{rr}}} = {{e}^{{ - 2\lambda }}} = \frac{\Delta }{\Sigma }{\kern 1pt} ,\quad {{g}^{{\theta \theta }}} = {{e}^{{ - 2\mu }}} = \frac{1}{\Sigma }{\kern 1pt} , \\ {{g}^{{\phi \phi }}} = {{e}^{{ - 2\psi }}} - {{\omega }^{2}}{{e}^{{ - 2\nu }}} = \frac{{\Delta - {{a}^{2}}{{{\sin }}^{2}}\theta }}{{\Delta \Sigma {{{\sin }}^{2}}\theta }}{\kern 1pt} , \\ {{g}^{{t\phi }}} = - \omega {{e}^{{ - 2\nu }}} = - \frac{{2M(r)ra}}{{\Sigma \Delta }}. \\ \end{gathered} $$

The determinant of the metric is

$$g = - {{e}^{{2\lambda + 2\mu + 2\psi + 2\nu }}} = - {{\Sigma }^{2}}\mathop {\sin }\nolimits^2 \theta {\kern 1pt} .$$

The Christoffel symbols are

$$\begin{gathered} \Gamma _{{tt}}^{t} = \Gamma _{{t\phi }}^{t} = \Gamma _{{rr}}^{t} = \Gamma _{{r\theta }}^{t} = \Gamma _{{\theta \theta }}^{t} = \Gamma _{{\phi \phi }}^{t} = \Gamma _{{t\theta }}^{\theta } = 0, \\ \Gamma _{{tr}}^{r} = \Gamma _{{t\theta }}^{r} = \Gamma _{{r\phi }}^{r} = \Gamma _{{\theta \phi }}^{r} = \Gamma _{{tr}}^{\theta } = \Gamma _{{r\phi }}^{\theta } = \Gamma _{{\theta \phi }}^{\theta } = 0, \\ \Gamma _{{tt}}^{\phi } = \Gamma _{{t\phi }}^{\phi } = \Gamma _{{rr}}^{\phi } = \Gamma _{{r\theta }}^{\phi } = \Gamma _{{\theta \theta }}^{\phi } = \Gamma _{{\phi \phi }}^{\phi } = 0, \\ \end{gathered} $$

$$\begin{gathered} \Gamma _{{tr}}^{t} = - \frac{{\left( {p + rM{\kern 1pt} '{\text{/}}\Sigma } \right)B}}{\Delta },\quad \Gamma _{{t\theta }}^{\phi } = \frac{{2qa\cos \theta }}{{\Sigma \sin \theta }}, \\ \Gamma _{{r\phi }}^{t} = \frac{{a{{{\sin }}^{2}}\theta }}{\Delta }\left( {B(p + rM{\kern 1pt} '{\kern 1pt} {\text{/}}\Sigma ) + 2qr} \right), \\ \Gamma _{{\theta \phi }}^{t} = - \frac{{q{{a}^{3}}}}{\Sigma }{{\sin }^{2}}\theta \sin 2\theta ,\quad \Gamma _{{\theta \theta }}^{r} = - \frac{{r\Delta }}{\Sigma }, \\ \end{gathered} $$

$$\begin{gathered} \Gamma _{{tt}}^{r} = - \frac{\Delta }{\Sigma }(p + rM{\kern 1pt} '{\text{/}}\Sigma ),\quad \Gamma _{{r\theta }}^{\theta } = \frac{r}{\Sigma }, \\ \Gamma _{{rr}}^{r} = \frac{r}{\Sigma } + \frac{{M - r + rM{\kern 1pt} '}}{\Delta },\quad \Gamma _{{r\theta }}^{r} = - \frac{{{{a}^{2}}\sin 2\theta }}{{2\Sigma }}, \\ \Gamma _{{\theta \theta }}^{\theta } = - \frac{{{{a}^{2}}\sin 2\theta }}{{2\Sigma }},\quad \Gamma _{{rr}}^{\theta } = \frac{{{{a}^{2}}\sin 2\theta }}{{2\Sigma \Delta }}, \\ \end{gathered} $$

$$\begin{gathered} \Gamma _{{\phi \phi }}^{r} = - \frac{{\Delta {{{\sin }}^{2}}\theta }}{\Sigma }\left( {r + \left( {p + rM{\kern 1pt} '{\text{/}}\Sigma } \right){{a}^{2}}{{{\sin }}^{2}}\theta } \right), \\ \Gamma _{{t\theta }}^{t} = \frac{{q{{a}^{2}}\sin 2\theta }}{\Sigma }, \\ \Gamma _{{tt}}^{\theta } = \frac{{q{{a}^{2}}\sin 2\theta }}{{{{\Sigma }^{2}}}},\quad \Gamma _{{t\phi }}^{r} = \frac{{a\Delta {{{\sin }}^{2}}\theta }}{\Sigma }\left( {p + rM{\kern 1pt} '{\kern 1pt} {\text{/}}\Sigma } \right), \\ \end{gathered} $$

$$\begin{gathered} \Gamma _{{t\phi }}^{\theta } = - \frac{{qaB\sin 2\theta }}{{{{\Sigma }^{2}}}},\quad \Gamma _{{tr}}^{\phi } = - \frac{a}{\Delta }\left( {p + rM{\kern 1pt} '{\text{/}}\Sigma } \right), \\ \Gamma _{{\phi \phi }}^{\theta } = - \frac{{\sin 2\theta }}{{2\Sigma }}\left[ {B - 2{{a}^{2}}q{{{\sin }}^{2}}\theta \left( {2 + \frac{{{{a}^{2}}{{{\sin }}^{2}}\theta }}{\Sigma }} \right)} \right], \\ \end{gathered} $$

$$\begin{gathered} \Gamma _{{r\phi }}^{\phi } = \frac{r}{\Delta }(1 + 2q) + \frac{{{{a}^{2}}{{{\sin }}^{2}}\theta }}{\Delta }\left( {p + rM{\kern 1pt} '{\text{/}}\Sigma } \right), \\ \Gamma _{{\theta \phi }}^{\phi } = \frac{{\cos \theta }}{{\Delta \sin \theta }}\left[ {(1 + 2q)\left( {B - 2q{{a}^{2}}{{{\sin }}^{2}}\theta } \right)\mathop {}\limits_{\mathop {}\limits_{_{{}}} } } \right. \\ \left. { - \;\frac{{2q{{a}^{2}}B{{{\sin }}^{2}}\theta }}{\Sigma }} \right], \\ \end{gathered} $$

where notations $B = {{r}^{2}} + {{a}^{2}}$, $q = - {\kern 1pt} \frac{{rM(r)}}{\Sigma }$, p = $M(r)\frac{{\Sigma - 2{{r}^{2}}}}{{{{\Sigma }^{2}}}}$ are introduced. Notation $M{\kern 1pt} '$ represents the derivative of parameter $M$ with respect to radius $r$: $M{\kern 1pt} ' = \frac{{6{{m}^{2}}{{b}^{2}}r\left| r \right|}}{{{{{\left( {2m{{b}^{2}} + {{{\left| r \right|}}^{3}}} \right)}}^{2}}}}$.

APPENDIX B

This section presents the solution obtained in [13] for the Lame metric. According to the findings of [13], the angular momentum is determined by the expression $l = {{u}_{\phi }}{{u}^{t}}$. Using this expression, Eq. (10) can be rewritten as

$$\frac{1}{{p + \rho }}{\kern 1pt} \frac{{\partial P}}{{\partial a}} = - {{\nu }_{{,a}}} - l{{\omega }_{{,a}}} + \frac{{ - 1 + \sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} }}{2}{{\chi }_{{,a}}},$$

(21)

where the notation $\chi = \psi - \nu$ was introduced [13]. The above expression (21) can be integrated explicitly. In the end, we obtain

$$\begin{gathered} \ln (h) = - \nu - l\omega + \frac{1}{2}\ln \left( {1 + \sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} } \right) \\ \, - \frac{1}{2}{\kern 1pt} \sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} - \ln ({{h}_{{in}}}). \\ \end{gathered} $$

This solution can be further expanded by substituting metric coefficients

$$\begin{gathered} \ln (h) = \frac{1}{2}\ln \left( {\frac{{1 + \sqrt {1 + \frac{{4{{l}^{2}}{{\Sigma }^{2}}\Delta }}{{{{A}^{2}}{{{\sin }}^{2}}\theta }}} }}{{\frac{{\Sigma \Delta }}{A}}}} \right) \\ - \;\frac{1}{2}\sqrt {1 + \frac{{4{{l}^{2}}{{\Sigma }^{2}}\Delta }}{{{{A}^{2}}{{{\sin }}^{2}}\theta }}} - \frac{{2arM(r)l}}{A} - \ln ({{h}_{{in}}}). \\ \end{gathered} $$

(22)

This expression coincides with Eq. (3.6) in [13] when the magnetic charge is zero $b = 0$. The $\ln ({{h}_{{in}}})$ value is the integration constant and is determined by initial conditions. In [13], the initial conditions were set as follows. At the inner radius of the disk in the plane perpendicular to the axis of rotation of the black hole (in this case, a black hole or a wormhole), the enthalpy is zero. From this condition, it follows that

$$\begin{gathered} \ln ({{h}_{{in}}}) = \frac{1}{2}\ln \left( {\frac{{1 + \sqrt {1 + \frac{{4{{l}^{2}}\Sigma _{{in}}^{2}{{\Delta }_{{in}}}}}{{A_{{in}}^{2}}}} }}{{\frac{{{{\Sigma }_{{in}}}{{\Delta }_{{in}}}}}{{{{A}_{{in}}}}}}}} \right) \\ \, - \frac{1}{2}\sqrt {1 + \frac{{4{\kern 1pt} {{l}^{2}}\Sigma _{{in}}^{2}{{\Delta }_{{in}}}}}{{A_{{in}}^{2}}}} - \frac{{2a{{r}_{{in}}}M({{r}_{{in}}})l}}{A}. \\ \end{gathered} $$

(23)

The notation “$in$” means that the radial coordinate is taken at the inner radius of the disk $r = {{r}_{{in}}}$, in the plane perpendicular to the axis of rotation $\theta = \pi {\text{/}}2$.

The angular momentum $l$ can be determined from the condition that in the equatorial plane at the point of maximum density (pressure), the derivative of the enthalpy with respect to the radial coordinate is zero. This condition takes the form

$${{[\ln (h({{r}_{{\max }}},\pi {\text{/}}2))]}_{{,r}}} = 0.$$

This expression in expanded form is

$$ - \frac{1}{2}\left( {\nu {\kern 1pt} '\; + \psi {\kern 1pt} '} \right) - l\omega {\kern 1pt} '\; \pm \frac{{\sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} }}{2}\chi {\kern 1pt} ' = 0,$$

where the prime denotes differentiation with respect to the radial coordinate. From this equation, it follows that the angular momentum is determined by the formula

$$l = \frac{{\omega {\kern 1pt} '\left( {\nu {\kern 1pt} ' + \psi {\kern 1pt} '} \right) + \chi {\kern 1pt} '\sqrt {\omega _{{}}^{{'2}} + 4\nu {\kern 1pt} '\psi {\kern 1pt} '{{e}^{{ - 2\chi }}}} }}{{2\left( {\chi _{{}}^{{'2}}{{e}^{{ - 2\chi }}} - \omega _{{}}^{{'2}}} \right)}}.$$

(24)

This value is taken at the point of maximum density of the disk $r = {{r}_{{\max }}}$ in the plane perpendicular to the plane of rotation of the black hole (wormhole) $\theta = \pi {\text{/}}2$. Thus, by specifying two parameters ${{r}_{{in}}}$ and ${{r}_{{\max }}}$, the solution is uniquely determined. Figure 4 shows the contours of the enthalpy boundary $\ln h = 0$; 0.01 for the Kerr metric with a rotation parameter $a = 1$ (solid curve) and for the Lame metric with rotation parameters $a = 1$ and $b = 0.9$ (dashed curve). These parameters were chosen in such a way that the Lame metric corresponds to a wormhole. The inner radius was set equal to ${{r}_{{in}}} = 6$, and the radius of maximum density was ${{r}_{{\max }}} = 12.9$. The solid curve for the Kerr metric coincides with a similar curve shown in [13], Fig. 2.

Fig. 4.

Contours of enthalpy for the case $\ln h = 0$ и 0.01 for Kerr and Lame metrics. The solid curve corresponds to the Kerr metric with parameter $a = 1$, and the dashed curve corresponds to the Lame metric with parameters $a = 1$ and $b = 0.9$. The inner radius was set to ${{r}_{{in}}} = 6$, and the radius of maximum density ${{r}_{{\max }}} = 12.9$.

From Fig. 4, it can be seen that for the selected parameters, the qualitative distribution of ideal fluid in the Kerr and Lame metrics does not differ significantly from each other. This is because a significant difference will be noticeable on the scales of the event horizon $r \approx 2m$ or the throat radius $r \approx b$.

APPENDIX C

In this section, we will write out the solution obtained in [14] for the Lame metric. According to [14], the angular momentum is determined by the expression ${{u}_{\phi }} = - l{{u}_{t}}$. Using this expression, Eq. (10) can be rewritten as

$$\frac{1}{{p + \rho }}{\kern 1pt} \frac{{\partial P}}{{\partial a}} = - {{\nu }_{{,a}}} + \frac{1}{2}{{\left( {\ln \left( {{{{\left( {1 - \omega l} \right)}}^{2}} - {{l}^{2}}{{e}^{{2\nu - 2\psi }}}} \right)} \right)}_{{,a}}}.$$

The above expression is easily integrated explicitly. As a result, we obtain a solution of the form

$$\ln (h) = - \nu + \frac{1}{2}\ln \left( {{{{(1 - \omega l)}}^{2}} - {{l}^{2}}{{e}^{{2\nu - 2\psi }}}} \right) - \ln ({{h}_{{in}}}),$$

where, as before, the $\ln ({{h}_{{in}}})$ value is the integration constant. Substituting the values of metric coefficients into this solution, we rewrite it as

$$\begin{gathered} \ln (h) = \frac{1}{2}\ln \left( {{{{\left( {1 - l\frac{{2arM}}{A}} \right)}}^{2}} - {{l}^{2}}\frac{{\Delta {{\Sigma }^{2}}}}{{{{A}^{2}}{{{\sin }}^{2}}\theta }}} \right){\kern 1pt} \\ \, - \frac{1}{2}\ln \left( {\frac{{\Delta \Sigma }}{A}} \right) - \ln ({{h}_{{in}}}){\kern 1pt} . \\ \end{gathered} $$

(25)

The integration constant $\ln ({{h}_{{in}}})$ can be determined in a similar way as in the previous case. As a result, we obtain

$$\begin{gathered} \ln ({{h}_{{in}}}) = \frac{1}{2}\ln \left( {{{{\left( {1 - l{\kern 1pt} \frac{{2a{{r}_{{in}}}M({{r}_{{in}}})}}{{{{A}_{{in}}}}}} \right)}}^{2}} - {{l}^{2}}{\kern 1pt} \frac{{{{\Delta }_{{in}}}\Sigma _{{in}}^{2}}}{{A_{{in}}^{2}}}} \right) \\ - \frac{1}{2}\ln \left( {\frac{{{{\Delta }_{{in}}}{{\Sigma }_{{in}}}}}{{{{A}_{{in}}}}}} \right). \\ \end{gathered} $$

Similarly, from the condition of maximum density (pressure), we can obtain the value of angular momentum $l$. Simple calculations lead to an expression of the form

$$l = \frac{{\omega {\kern 1pt} ' - 2\omega \nu {\kern 1pt} '\; + \sqrt {\omega _{{}}^{{'2}} + 4\nu {\kern 1pt} '{\kern 1pt} \psi {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}}} }}{{2\left( {\omega \omega {\kern 1pt} '\; + \psi {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}} - {{\omega }^{2}}\nu {\kern 1pt} '} \right)}},$$

where the $l$ value is determined at the point of maximum density $r = {{r}_{{\max }}}$ in the plane perpendicular to the rotation of the black hole (wormhole) $\theta = \pi {\text{/}}2$.

Figure 5 shows the contours of the boundary of $\ln h = 0$; 0.01 for the Kerr metric with a rotation parameter $a = 1$ (solid curve) and for the Lame metric with rotation parameters $a = 1$ and $b = 0.9$ (dashed curve). The Lame metric with the chosen parameters corresponds to a wormhole. The inner radius was set equal to ${{r}_{{in}}} = 6$, and the radius of maximum density was ${{r}_{{\max }}} = 11$.

Fig. 5.

Contours of enthalpy $\ln h = 0$ and 0.01 for a thick disk in Kerr and Lame metrics. The solid curve corresponds to the Kerr metric with parameter $a = 1$, and the dashed curve corresponds to the Lame metric with parameters $a = 1$ and $b = 0.9$. The Lame metric with the chosen parameters corresponds to the wormhole. The inner radius was set to ${{r}_{{in}}} = 6$, and the radius of maximum density ${{r}_{{\max }}} = 11$.

APPENDIX D

In this section, we generalize the solution obtained in [13] to the case of the presence of a toroidal magnetic field in the Lame metric. In this solution, the angular momentum is a constant and is determined by the expression $l = {{u}_{\phi }}{{u}^{t}}$. Using parameter $\beta$ and expressing the four-velocity through the angular momentum $l$, Eq. (16) can be rewritten as

$$\begin{gathered} \frac{\beta }{{1 + \beta }}{\kern 1pt} \frac{1}{{P + \rho }}\frac{{\partial P}}{{\partial a}} + \frac{1}{{2(1 + \beta ){{b}^{2}}}}\frac{{\partial {{b}^{2}}}}{{\partial a}} \\ \, = - {{\nu }_{{,a}}} - l{{\omega }_{{,a}}} + 0.5{{\chi }_{{,a}}}\left( { - 1 + \sqrt {1 + 4{\kern 1pt} {{l}^{2}}{{e}^{{ - 2\chi }}}} } \right) \\ \, - \frac{{{{\psi }_{{,a}}} - l{{\omega }_{{,a}}} + 0.5{{\chi }_{{,a}}}\left( { - 1 + \sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} } \right)}}{{1 + \beta }}. \\ \end{gathered} $$

(26)

This equation can be easily integrated analytically. After integration, we obtain a solution of the form

$$\frac{\beta }{{1 + \beta }}\frac{\gamma }{{\gamma - 1}}\ln \left( {1 + k{{\rho }^{{\gamma - 1}}}} \right) + \frac{1}{{2(1 + \beta )}}\ln {{b}^{2}}$$

$$\begin{gathered} = - \nu - \omega l + 0.5\ln \left( {1 + \sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} } \right) \\ \, - 0.5\sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} \\ - \;\frac{{\psi - l\omega + 0.5\ln \left( {1 + \sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} } \right)}}{{1 + \beta }} \\ \end{gathered} $$

(27)

$$ - \;\frac{{0.5\sqrt {1 + 4{{l}^{2}}{{e}^{{ - 2\chi }}}} }}{{1 + \beta }} - \ln {{h}_{{in}}},$$

where ${{h}_{{in}}}$ is the integration constant and is determined at the inner radius of the disk ($r = {{r}_{{in}}}$) in a plane perpendicular to the axis of rotation of the black hole or wormhole ($\theta = \pi {\text{/}}2$), similar to Appendix B.

The angular momentum $l$ is determined from the extremum condition of the right-hand side of expression (27) in the equatorial plane of the disk $\theta = \pi {\text{/}}2$. By setting the right-hand side of the expression (26) to zero and assuming $a = r$, after some simple transformations, we obtain the expression for the angular momentum $l$ at the extremum point ${{r}_{{\max }}}$:

$$\begin{gathered} l = \frac{1}{{2\left( {\chi _{{}}^{{'2}}{{e}^{{ - 2\chi }}} - \omega _{{}}^{{'2}}} \right)}}\left( {\omega {\kern 1pt} '\left( {\psi {\kern 1pt} '\; + \nu {\kern 1pt} '} \right)\frac{{2 + \beta }}{\beta }} \right. \\ \left. { + \;\chi {\kern 1pt} '\sqrt {{{{\left( {\psi {\kern 1pt} '\; + \nu {\kern 1pt} '} \right)}}^{2}}{{e}^{{ - 2\chi }}}\frac{{{{{(2 + \beta )}}^{2}}}}{{{{\beta }^{2}}}} + \omega _{{}}^{{'2}} - \chi _{{}}^{{'2}}{{e}^{{ - 2\chi }}}} } \right). \\ \end{gathered} $$

(28)

In the limiting case when parameter $\beta \to \infty$, this expression reduces to the expression (24) without the toroidal magnetic field.

Figure 6 shows the contours of the right-hand side of the expression (27) for the case when the left-hand side of the expression (27) is zero for the same initial parameters ${{r}_{{in}}} = 6$ and ${{r}_{{\max }}} = 12.9$, as in Fig. 4. The solid curve, which corresponds to parameter $\beta = \infty$, coincides with the dashed curve in Fig. 4 for $\ln h = 0$.

Fig. 6.

Contours of the right-hand side of the expression (27) for the case where the left-hand side of the expression (27) equals zero for the Lame metric with parameters $a = 1$, $b = 0.9$. Curves are shown for parameter $\beta = \infty$ (solid line), 1000 (dashed line), 100 (dash–dot line), and 20 (dotted line).

From Fig. 6, it can be seen that even for large values of $\beta$ deviations from the solution without a toroidal field are quite significant. This is because, at large but finite parameter $\beta$, parameter $l$ changes, and small changes in parameter $l$ lead to large changes in the right-hand side of the solution (27).

APPENDIX E

In this section, we generalize the solution obtained in [14] to the case of the presence of a toroidal magnetic field in the Lame metric. In this solution, the angular momentum is determined by the expression ${{u}_{\phi }} = - l{{u}_{t}}$. Using the condition stated above, Eq. (16) is rewritten as:

$$\begin{gathered} \frac{1}{{P + \rho + {{b}^{2}}}}{\kern 1pt} \frac{\partial }{{\partial a}}\left( {P + \frac{1}{2}{{b}^{2}}} \right) = - {{\nu }_{{,a}}} \\ + \;\frac{{{{l}^{2}}({{\psi }_{{,a}}} - {{\nu }_{{,a}}}){{e}^{{2\nu - 2\psi }}} - l{{\omega }_{{,a}}}(1 - \omega l)}}{{{{{(1 - l\omega )}}^{2}} - {{l}^{2}}{{e}^{{2\nu - 2\psi }}}}} - \frac{{{{b}^{2}}}}{{P + \rho + {{b}^{2}}}} \\ \times \;\left[ {{{\psi }_{{,a}}} + \frac{{{{l}^{2}}({{\psi }_{{,a}}} - {{\nu }_{{,a}}}){{e}^{{2\nu - 2\psi }}} - l{{\omega }_{{,a}}}(1 - l\omega )}}{{{{{(1 - \omega l)}}^{2}} - {{l}^{2}}{{e}^{{2\nu - 2\psi }}}}}} \right]. \\ \end{gathered} $$

(29)

This equation is integrated analytically. As a result, we obtain

$$\begin{gathered} \frac{\beta }{{1 + \beta }}\frac{\gamma }{{\gamma - 1}}\ln \left( {1 + k{{\rho }^{{\gamma - 1}}}} \right) + \frac{1}{{2(1 + \beta )}}\ln {{b}^{2}} \\ = - \nu + 0.5\left( {\ln \left( {{{{(1 - \omega l)}}^{2}} - {{l}^{2}}{{e}^{{2\nu - 2\psi }}}} \right)} \right) \\ - \;\frac{1}{{1 + \beta }}\left( {\psi + 0.5\left( {\ln \left( {{{{(1 - \omega l)}}^{2}} - {{l}^{2}}{{e}^{{2\nu - 2\psi }}}} \right)} \right)} \right) - \ln {{h}_{{in}}}, \\ \end{gathered} $$

(30)

where $\ln {{h}_{{in}}}$ is the integration constant, which is determined at the inner radius of the disk in a plane perpendicular to the axis of rotation of the black hole or wormhole ($\theta = \pi {\text{/}}2$)similar to Appendix C.

The magnitude of the angular momentum $l$ is determined from the extremum condition of the right-hand side of expression (30) in the plane perpendicular to the axis of rotation of the black hole or wormhole. By setting the right-hand side of the expression (29) to zero and assuming $a = r$, after some simple transformations, we obtain a quadratic equation for the angular momentum $l$ at the extremum point:

$$\begin{gathered} {{l}^{2}}\left[ {\psi {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}} - \nu {\kern 1pt} '{{\omega }^{2}} + \frac{{\beta \omega \omega {\kern 1pt} '\; - \psi {\kern 1pt} '{\kern 1pt} {{\omega }^{2}} + \nu {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}}}}{{1 + \beta }}} \right] \\ + \;l\left[ {2\omega \nu {\kern 1pt} '\; + \frac{{2\omega \psi {\kern 1pt} '\; - \beta \omega {\kern 1pt} '}}{{1 + \beta }}} \right] - \nu {\kern 1pt} '\; - \frac{{\psi {\kern 1pt} '}}{{1 + \beta }} = 0. \\ \end{gathered} $$

Solving this equation yields

$$\begin{gathered} l = \left( { - 2\omega \nu {\kern 1pt} '\; + \frac{{\beta \omega {\kern 1pt} '\; - 2\omega \psi {\kern 1pt} '}}{{1 + \beta }}} \right. \\ \left. { + \;\sqrt {4\nu {\kern 1pt} '{\kern 1pt} \psi {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}} + 4{{e}^{{ - 2\chi }}}\frac{{\psi _{{}}^{{'2}} + \nu _{{}}^{{'2}}}}{{1 + \beta }} + \frac{{{{\beta }^{2}}\omega _{{}}^{{'2}} + 4\psi {\kern 1pt} '{\kern 1pt} \nu {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}}}}{{{{{(1 + \beta )}}^{2}}}}} } \right) \\ \times \;{{\left( {2\left( {\psi {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}} - \nu {\kern 1pt} '{{\omega }^{2}} - \frac{{\psi {\kern 1pt} '{\kern 1pt} {{\omega }^{2}} - \nu {\kern 1pt} '{\kern 1pt} {{e}^{{ - 2\chi }}} - \beta \omega \omega {\kern 1pt} '}}{{1 + \beta }}} \right)} \right)}^{{ - 1}}}. \\ \end{gathered} $$

Figure 7 shows the contours of the right-hand side of the expression (30) for the case when the left-hand side of the expression (30) is zero for the same initial parameters ${{r}_{{in}}} = 6$ and ${{r}_{{\max }}} = 11$, as in Fig. 5. The solid curve, which corresponds to parameter $\beta = \infty$, coincides with the dashed curve in Fig. 5 for $\ln h = 0$. Even for large values of $\beta$ the differences between this solution and the solution without a magnetic field become significant.

Fig. 7.

Contours of the right-hand side of the expression (30) for the case where the left-hand side of the expression (30) equals zero for the Lame metric with parameters $a = 1$, $b = 0.9$. Curves are shown for parameter $\beta = \infty$ (solid line), 1000 (dashed line), 100 (dash–dot line), and 40 (dotted line).

APPENDIX F

In this section, we provide comparisons of wormhole images depending on the observer’s distance. The following model parameters were set: the rotation parameter $a = 0.9$, the magnetic charge $b = 0.3$, and the observer’s inclination angle to the axis of rotation $i = 45^\circ$. The resolution of the images was chosen to be $500 \times 500$ points. The observer was located at a distance ${{r}_{o}} = 50$ and ${{r}_{0}} = 250$. For the observer located at a distance of ${{r}_{o}} = 50$, the image is shown in the second column, in the second row from the top in Fig. 3. Figure 8 shows the difference between the images. If we do not consider the photon rings, where it is difficult to precisely match the rings due to the small resolution and slight shifts, it can be seen from the image of the disk that there is a small shift related to the different positions of the observer. The magnitude of the shift and its sign depend on the metric parameters and the angle of inclination of the observer’s line of sight to the axis of rotation. In general, the qualitative pattern does not depend on the observer’s distance to the wormhole, but for more accurate numerical calculations, it is necessary to place the observer as far away as possible from the central source.

Fig. 8.

The difference between the images obtained for observers located at distances ${{r}_{o}} = 50$ and 250. Model parameters: $a = 0.9$, $b = 0.3$, $i = 45^\circ$.