Construction of Black Hole Shadows: An Analytical Theory

Abstract

Analytical methods for the construction of black hole shadows in the Kerr–Newman metric and possibilities for their observation through very long baseline interferometry (VLBI) are considered.

Appendices

APPENDIX A

The metric coefficients of the Kerr–Newman metric are [16]

$${{g}_{{tt}}} = - \left( {1 - \frac{{2Mr - {{Q}^{2}}}}{\Sigma }} \right),\quad {{g}_{{rr}}} = \frac{\Sigma }{\Delta },\quad {{g}_{{\theta \theta }}} = \Sigma ,$$

$${{g}_{{\phi \phi }}} = \frac{{{{{\sin }}^{2}}\theta }}{\Sigma }({{({{r}^{2}} + {{a}^{2}})}^{2}} - \Delta {{a}^{2}}{{\sin }^{2}}\theta ),$$

$${{g}_{{t\phi }}} = - \frac{{(2Mr - {{Q}^{2}})a{{{\sin }}^{2}}\theta }}{\Sigma },\quad {{g}^{{\theta \theta }}} = \frac{1}{\Sigma },$$

$${{g}^{{tt}}} = - \frac{{{{{({{r}^{2}} + {{a}^{2}})}}^{2}} - \Delta {{a}^{2}}{{{\sin }}^{2}}\theta }}{{\Sigma \Delta }},\quad {{g}^{{rr}}} = \frac{\Delta }{\Sigma },$$

$${{g}^{{\phi \phi }}} = \frac{{\Delta - {{a}^{2}}{{{\sin }}^{2}}\theta }}{{\Delta \Sigma {{{\sin }}^{2}}\theta }},\quad {{g}^{{t\phi }}} = - \frac{{(2Mr - {{Q}^{2}})a}}{{\Sigma \Delta }}.$$

The determinant of the metric is [16]

$$g = - {{\Sigma }^{2}}{{\sin }^{2}}\theta .$$

The Christoffel symbols are [16]

$$\Gamma _{{tr}}^{t} = - \frac{{pb}}{\Delta },\quad \Gamma _{{r\phi }}^{t} = \frac{{a{{{\sin }}^{2}}\theta }}{\Delta }(pb + 2qr),$$

$$\Gamma _{{\theta \phi }}^{t} = - \frac{{q{{a}^{3}}}}{\Sigma }{{\sin }^{2}}\theta \sin 2\theta ,\quad \Gamma _{{tt}}^{r} = - \frac{{\Delta p}}{\Sigma },$$

$$\Gamma _{{rr}}^{r} = \frac{r}{\Sigma } + \frac{{M - r}}{\Delta },\quad \Gamma _{{r\theta }}^{r} = - \frac{{{{a}^{2}}\sin 2\theta }}{{2\Sigma }},$$

$$\Gamma _{{\theta \theta }}^{\theta } = - \frac{{{{a}^{2}}\sin 2\theta }}{{2\Sigma }},\quad \Gamma _{{rr}}^{\theta } = \frac{{{{a}^{2}}\sin 2\theta }}{{2\Sigma \Delta }},$$

$$\Gamma _{{\theta \theta }}^{r} = - \frac{{r\Delta }}{\Sigma },\quad \Gamma _{{r\theta }}^{\theta } = \frac{r}{\Sigma },$$

$$\Gamma _{{\phi \phi }}^{r} = - \frac{{\Delta {{{\sin }}^{2}}\theta }}{\Sigma }(r + p{{a}^{2}}{{\sin }^{2}}\theta ),\quad \Gamma _{{t\theta }}^{t} = \frac{{q{{a}^{2}}\sin 2\theta }}{\Sigma },$$

$$\Gamma _{{tt}}^{\theta } = \frac{{q{{a}^{2}}\sin 2\theta }}{{{{\Sigma }^{2}}}},\quad \Gamma _{{t\phi }}^{r} = \frac{{a\Delta p{{{\sin }}^{2}}\theta }}{\Sigma },$$

$$\Gamma _{{t\phi }}^{\theta } = - \frac{{qab\sin 2\theta }}{{{{\Sigma }^{2}}}},\quad \Gamma _{{t\theta }}^{\phi } = \frac{{2qa\cos \theta }}{{\Sigma \sin \theta }},$$

$$\Gamma _{{\phi \phi }}^{\theta } = - \frac{{\sin 2\theta }}{{2\Sigma }}\left[ {b - 2{{a}^{2}}q{{{\sin }}^{2}}\theta \left( {2 + \frac{{{{a}^{2}}{{{\sin }}^{2}}\theta }}{\Sigma }} \right)} \right],$$

$$\Gamma _{{r\phi }}^{\phi } = \frac{r}{\Delta }(1 + 2q)\,\, + \,\,{{a}^{2}}p\frac{{{{{\sin }}^{2}}\theta }}{\Delta },$$

$$\Gamma _{{\theta \phi }}^{\phi } = \frac{{\cos \theta }}{{\Delta \sin \theta }}\left[ {(1 + 2q)(b - 2q{{a}^{2}}{\text{si}}{{{\text{n}}}^{2}}\theta ) - \frac{{2q{{a}^{2}}b{\text{si}}{{{\text{n}}}^{2}}\theta }}{\Sigma }} \right],$$

$$\Gamma _{{tr}}^{\phi } = - a\frac{{r{{Q}^{2}} + M(\Sigma - 2{{r}^{2}})}}{{\Delta {{\Sigma }^{2}}}},$$

where

$$\begin{gathered} b = {{r}^{2}} + {{a}^{2}},\quad p = \frac{{r{{Q}^{2}} + M(\Sigma - 2{{r}^{2}})}}{{{{\Sigma }^{2}}}}, \\ q = \frac{{{{Q}^{2}} - 2Mr}}{{2\Sigma }}. \\ \end{gathered} $$

APPENDIX B

The expression for the radial potential is a quartic algebraic equation, which is convenient to represent as [17]

$$R = (r - {{r}_{a}})(r - {{r}_{b}})(r - {{r}_{c}})(r - {{r}_{d}}),$$

where the roots of the equation are written in decreasing order, r_a > r_b > r_c > r_d. The solution of this equation can be written in an explicit form [17] as

$${{r}_{a}} = z + \sqrt { - \frac{\mathcal{A}}{2} - {{z}^{2}} - \frac{\mathcal{B}}{{4z}}} ,$$

$${{r}_{b}} = z - \sqrt { - \frac{\mathcal{A}}{2} - {{z}^{2}} - \frac{\mathcal{B}}{{4z}}} ,$$

$${{r}_{c}} = - z + \sqrt { - \frac{\mathcal{A}}{2} - {{z}^{2}} + \frac{\mathcal{B}}{{4z}}} ,$$

$${{r}_{d}} = - z - \sqrt { - \frac{\mathcal{A}}{2} - {{z}^{2}} + \frac{\mathcal{B}}{{4z}}} ,$$

where

$$\mathcal{A} = {{a}^{2}} - \eta - {{\lambda }^{2}},$$

$$\mathcal{B} = 2M(\eta + {{(\lambda - a)}^{2}}),$$

$$\mathcal{C} = - {{a}^{2}}\eta - {{Q}^{2}}(\eta + {{(a - \lambda )}^{2}}),$$

$$\mathcal{P} = - \frac{{{{A}^{2}}}}{{12}} - \mathcal{C},$$

$$\mathcal{Q} = - \frac{\mathcal{A}}{3}\left[ {{{{\left( {\frac{\mathcal{A}}{6}} \right)}}^{2}} - \mathcal{C}} \right] - \frac{{{{\mathcal{B}}^{2}}}}{8},$$

$${{\omega }_{ \pm }} = \sqrt[3]{{ - \frac{\mathcal{Q}}{2} \pm \sqrt {{{{\left( {\frac{\mathcal{P}}{3}} \right)}}^{3}} + {{{\left( {\frac{\mathcal{Q}}{2}} \right)}}^{2}}} }},$$

$$z = \sqrt {\frac{{{{\omega }_{ + }} + {{\omega }_{ - }}}}{2} - \frac{\mathcal{A}}{6}} > 0.$$

APPENDIX C

The incomplete elliptic integrals of the first, second, and third types are defined as follows:

$$F(\phi ,k) = \int\limits_0^\phi {\frac{{d\theta }}{{\sqrt {1 - k{{{\sin }}^{2}}\theta } }},} $$

$$E(\phi ,k) = \int\limits_0^\phi {\sqrt {1 - k{{{\sin }}^{2}}\theta } d\theta ,} $$

$$\Pi ({{\alpha }^{2}},\phi ,k) = \int\limits_0^\phi {\frac{1}{{1 - {{\alpha }^{2}}{{{\sin }}^{2}}\theta }}} \frac{{d\theta }}{{\sqrt {1 - k{{{\sin }}^{2}}\theta } }}.$$

The derivative of the elliptic integral of the second type E with respect to the parameter k is specified as

$$E{\kern 1pt} '(\phi ,k) = \frac{{\partial E(\phi ,k)}}{{\partial k}} = \frac{{E(\phi ,k) - F(\phi ,k)}}{{2k}}.$$

The complete elliptic integrals are defined by the condition ϕ = π/2 and are designated as

$$K(k) = F(\pi {\text{/}}2,k),\quad E(k) = E(\pi {\text{/}}2,k),$$

$$\Pi (n,k) = \Pi (n,\pi {\text{/}}2,k).$$

Let us introduce the notation

$$\nu = \arcsin \sqrt {\frac{{({{r}_{b}} - {{r}_{d}})(r - {{r}_{a}})}}{{({{r}_{a}} - {{r}_{d}})(r - {{r}_{b}})}}} ,\quad {{\alpha }^{2}} = \frac{{{{r}_{a}} - {{r}_{d}}}}{{{{r}_{b}} - {{r}_{d}}}},$$

$$g = \frac{2}{{\sqrt {({{r}_{a}} - {{r}_{c}})({{r}_{b}} - {{r}_{d}})} }},\quad k = \frac{{({{r}_{b}} - {{r}_{c}})({{r}_{a}} - {{r}_{d}})}}{{({{r}_{a}} - {{r}_{c}})({{r}_{b}} - {{r}_{d}})}}.$$

To calculate the integrals I, we used the following tabulated integrals for the case where r > r_a. This integral is calculated in [21]:

$$\int\limits_{{{r}_{a}}}^r {\frac{{dx}}{{\sqrt {(x - {{r}_{a}})(x - {{r}_{b}})(x - {{r}_{c}})(x - {{r}_{d}})} }} = gF(\nu ,k).} $$

This integral is calculated in [21]:

$$\begin{gathered} \int\limits_{{{r}_{a}}}^r {\frac{{xdx}}{{\sqrt {(x - {{r}_{a}})(x - {{r}_{b}})(x - {{r}_{c}})(x - {{r}_{d}})} }}} \\ = g[({{r}_{a}} - {{r}_{b}})\Pi ({{\alpha }^{2}},\nu ,k) + {{r}_{b}}F(\nu ,k)]. \\ \end{gathered} $$

This integral is calculated in [21]:

$$\begin{gathered} \int\limits_{{{r}_{a}}}^r {\frac{{dx}}{{(p - x)\sqrt {(x - {{r}_{a}})(x - {{r}_{b}})(x - {{r}_{c}})(x - {{r}_{d}})} }}} \\ = \frac{g}{{(p - {{r}_{a}})(p - {{r}_{b}})}}\left[ {({{r}_{a}} - {{r}_{b}})\Pi \left( {{{\alpha }^{2}}\frac{{p - {{r}_{b}}}}{{p - {{r}_{a}}}},\nu ,k} \right)} \right. \\ \left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}} + \,(p - {{r}_{a}})F(\nu ,k)} \right]. \\ \end{gathered} $$

This integral is calculated on page 130 in [22]:

$$\begin{gathered} \int\limits_{{{r}_{a}}}^r {\frac{{{{x}^{2}}dx}}{{\sqrt {(x - {{r}_{a}})(x - {{r}_{b}})(x - {{r}_{c}})(x - {{r}_{d}})} }}} \\ = gr_{a}^{2}\int\limits_0^{{{u}_{1}}} {\frac{{{{{(1 - \alpha _{1}^{2}{\text{s}}{{{\text{n}}}^{2}}u)}}^{2}}}}{{{{{(1 - {{\alpha }^{2}}{\text{s}}{{{\text{n}}}^{2}}u)}}^{2}}}}du} , \\ \end{gathered} $$

where

$${\text{am}}{{u}_{1}} = \arcsin \sqrt {\frac{{({{r}_{b}} - {{r}_{d}})(r - {{r}_{a}})}}{{({{r}_{a}} - {{r}_{d}})(r - {{r}_{b}})}}} ,\quad \alpha _{1}^{2} = \frac{{{{r}_{a}} - {{r}_{d}}}}{{{{r}_{b}} - {{r}_{d}}}}\frac{{{{r}_{b}}}}{{{{r}_{a}}}}$$

and am denotes the amplitude of u₁.

This integral is calculated on page 205 in [22]:

$$\begin{gathered} \int {\frac{{{{{(1 - \alpha _{1}^{2}{\text{s}}{{{\text{n}}}^{2}}u)}}^{2}}}}{{{{{(1 - {{\alpha }^{2}}{\text{s}}{{{\text{n}}}^{2}}u)}}^{2}}}}} du \\ = \frac{{\alpha _{1}^{4}}}{{{{\alpha }^{4}}}}\left[ {{{V}_{0}} + 2\frac{{{{\alpha }^{2}} - \alpha _{1}^{2}}}{{\alpha _{1}^{2}}}{{V}_{1}} + \frac{{{{{({{\alpha }^{2}} - \alpha _{1}^{2})}}^{2}}}}{{\alpha _{1}^{4}}}{{V}_{2}}} \right], \\ \end{gathered} $$

where (see page 201 [22])

$${{V}_{0}} = \int {du = F(\nu ,k),} $$

$${{V}_{1}} = \int {\frac{{du}}{{1 - {{\alpha }^{2}}{\text{s}}{{{\text{n}}}^{2}}u}}} = \Pi ({{\alpha }^{2}},\nu ,k),$$

$$\begin{gathered} {{V}_{2}} = \int {\frac{{du}}{{{{{(1 - {{\alpha }^{2}}{\text{s}}{{{\text{n}}}^{2}}u)}}^{2}}}}} \\ = \frac{1}{{2({{\alpha }^{2}} - 1)(k - {{\alpha }^{2}})}}\left[ {{{\alpha }^{2}}E(u) + (k - {{\alpha }^{2}}){{u}_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}} \right. \\ + (2{{\alpha }^{2}}k + 2{{\alpha }^{2}} - {{\alpha }^{4}} - 3k)\Pi ({{\alpha }^{2}},\nu ,k) \\ \left. { - \frac{{{{\alpha }^{4}}{\text{sn}}u{\text{cn}}u{\text{dn}}u}}{{1 - {{\alpha }^{2}}{\text{s}}{{{\text{n}}}^{2}}u}}} \right], \\ \end{gathered} $$

snu, cnu, and dnu are the Jacobi elliptic integrals.

It is easy to represent the integral I_ϕ in the form tabulated integrals. For this purpose, let us expand I_ϕ as [23]

$$\begin{gathered} {{I}_{\phi }} = \int\limits_{{{r}_{a}}}^r {\frac{{a(2rM - {{Q}^{2}} - a\lambda )}}{{\Delta \sqrt R }}dr} \\ = a\sum\limits_{i = 1}^{i = 2} {\int\limits_{{{r}_{a}}}^r {\frac{{{{K}_{i}}dr}}{{(r - {{r}_{i}})\sqrt R }}} } \\ = ga\sum\limits_{i = 1}^2 {{{\Gamma }_{i}}[(1 - \beta _{i}^{2})\Pi (\nu ,\gamma _{i}^{2},k) + \beta _{i}^{2}F(\nu ,k)],} \\ \end{gathered} $$

where

$$\gamma _{i}^{2} = \frac{{{{r}_{a}} - {{r}_{d}}}}{{{{r}_{b}} - {{r}_{d}}}}\frac{{{{r}_{i}} - {{r}_{b}}}}{{{{r}_{i}} - {{r}_{a}}}},\quad \beta _{i}^{2} = \frac{{{{r}_{i}} - {{r}_{a}}}}{{{{r}_{i}} - {{r}_{b}}}},$$

$${{r}_{i}} = M + {{( - 1)}^{{i + 1}}}\sqrt {{{M}^{2}} - {{a}^{2}} - {{Q}^{2}}} ,\quad {{\Gamma }_{i}} = \frac{{{{K}_{i}}}}{{{{r}_{a}} - {{r}_{i}}}},$$

$${{K}_{i}} = M + {{( - 1)}^{{i + 1}}}\frac{{{{M}^{2}} - {{Q}^{2}}{\text{/}}2 - a\lambda {\text{/}}2}}{{\sqrt {{{M}^{2}} - {{a}^{2}} - {{Q}^{2}}} }},$$

Similarly, let us represent the integral I_t in the form of tabulated integrals as

$${{I}_{t}} = \int\limits_{{{r}_{a}}}^r {\left[ {\frac{{{{A}_{1}}}}{{r - {{r}_{1}}}} - \frac{{{{A}_{2}}}}{{r - {{r}_{2}}}} + 4{{M}^{2}} - {{Q}^{2}} + 2Mr + {{r}^{2}}} \right]\frac{{dr}}{{_{{ \pm r}}\sqrt R }}} ,$$

where

$${{A}_{1}}({{r}_{1}} - {{r}_{2}}) = (2M{{r}_{1}} - {{Q}^{2}})(r_{1}^{2} + {{a}^{2}} - a\lambda ),$$

$${{A}_{2}}({{r}_{1}} - {{r}_{2}}) = - (2M{{r}_{2}} - {{Q}^{2}})(r_{2}^{2} + {{a}^{2}} - a\lambda ).$$

APPENDIX D

To calculate the visibility function, we used the following tabulated integrals. To calculate the integral (10), we used the tabulated integral

$$\int\limits_0^{2\pi } {{{e}^{{ - 2\pi iu\rho \cos ({{\phi }_{\rho }} - {{\phi }_{u}})}}}d{{\phi }_{\rho }}} = 2\pi {{J}_{0}}(2\pi u\rho ).$$

To calculate the integrals (11)–(13), we used the tabulated integrals (page 484 in [24])

$$\int\limits_0^z {t{{J}_{0}}(ut)dt = \frac{z}{u}{{J}_{1}}(uz),} $$

$$\begin{gathered} \int\limits_0^z {{{t}^{2}}{{J}_{0}}(ut)dt} \\ = \frac{z}{{2{{u}^{2}}}}[2uz{{J}_{1}}(uz) - \pi {{J}_{1}}(uz){{H}_{0}}(uz) + \pi {{J}_{0}}(uz){{H}_{1}}(uz)], \\ \end{gathered} $$

$$\begin{gathered} \int\limits_0^z {{{t}^{3}}{{J}_{0}}(ut)dt = \frac{{{{z}^{2}}}}{{{{u}^{2}}}}(2{{J}_{2}}(uz) - uz{{J}_{3}}(uz))} \\ = \frac{{{{z}^{2}}}}{{{{u}^{2}}}}[uz{{J}_{1}}(uz) - 2{{J}_{2}}(uz)] \\ = \frac{{{{z}^{2}}}}{{{{u}^{2}}}}\left[ {uz{{J}_{1}}(uz) - \frac{4}{{uz}}{{J}_{1}}(uz) + 2{{J}_{0}}(uz)} \right], \\ \end{gathered} $$

where H is the Struve function. To calculate the integral (14), we used the tabulated integral

$$\frac{{2u}}{\pi }\int\limits_0^x {t{{J}_{1}}(ut)dt = x[{{J}_{1}}(ux){{H}_{0}}(ux) - {{J}_{0}}(ux){{H}_{1}}(ux)].} $$