Summary

DescriptionKerr.black.hole.shadow.and.horizons.gif	English: Shadow (black) & horizons and ergospheres (white) of a rotating black hole. The dimensionless spin parameter a/M=Jc/G/M² is running from 0 to 1. Units: G=M=c=1 (lengths are in GM/c²). The observer is assumed to be far away from the black hole and stationary with respect to the fixed stars.[1][2][3][4][5][6]
Date	2 November 2017
Source	Own work, Code
Author	Yukterez (Simon Tyran, Vienna)
Other versions

Equations

Natural dimensionless units:

${\displaystyle {\rm {G=M=c=1}}}$

Inner (-) and outer (+) ergospheres:

${\displaystyle {\rm {r_{E}^{\pm }={\sqrt {1-a^{2}\cos ^{2}\theta }}\pm 1,\ \{x,z\}=\{{\sqrt {a^{2}+{r_{E}^{\pm }}^{2}}}\sin \theta ,\ r_{E}^{\pm }\cos \theta \}}}}$

Inner (-) and outer (+) horizons:

${\displaystyle {\rm {r_{H}^{\pm }={\sqrt {1-a^{2}}}\pm 1,\ \{x,z\}=\{{\sqrt {a^{2}+{r_{H}^{\pm }}^{2}}}\sin \theta ,\ r_{H}^{\pm }\cos \theta \}}}}$

Shadow contours:

${\displaystyle {\rm {0=((-x)^{2}+z^{2}-x\ A)^{2}-B^{2}\ ((-x)^{2}+z^{2})}}}$

Limaςon parameter series expansion:

${\displaystyle {\rm {\alpha =}}}$${\displaystyle {\rm {-8892.68a^{10}+30413.2a^{9}-46107.4a^{8}+}}}$

${\displaystyle {\rm {\ \ \ \ \ \ +37064.7a^{7}-18685.4a^{6}+4666.5a^{5}-3894.54a^{4}+}}}$

${\displaystyle {\rm {\ \ \ \ \ \ \ +49.5645a^{3}-9672.25a^{2}+2.27392a+9669.01a\ \tan(a)}}}$

${\displaystyle {\rm {\beta }}}$${\displaystyle {\rm {=5.19058-0.343743a\ \tan(a)+0.0284803a-0.0470795a^{27.5224}\tan(a)}}}$

Fourier transformation for the observer's inclination angle θ:

${\displaystyle {\rm {A=\alpha \sin \theta +a\sin ^{3}\theta \cos ^{2}\theta /5}}}$

${\displaystyle {\rm {B=\beta +0.23\cos ^{4}\theta \ (1-{\sqrt {1-a^{4}}})}}}$

Code: Source (Mathematica Syntax)

References

1. ↑ Andreas de Vries: Shadows of rotating black holes
2. ↑ Hung-Yi Pu, Kiyun Yun, Ziri Younsi, Suk Jin Yoon: Odyssey
3. ↑ Max Planck Institute: Scientists to image event horizon of black hole
4. ↑ Claudio Paganini, Blazej Ruba, Marius Oancea: Null Geodesics on Kerr Spacetimes
5. ↑ Naoki Tsukamoto: Kerr-Newman and rotating regular black hole shadows in flat spacetime
6. ↑ Grenzebach, Perlick & Lämmerzahl: Photon Regions and Shadows of Kerr–Newman–NUT Black Holes

File Usage on Wikipedia

• de.wikipedia.org/wiki/Kerr-Metrik
• en.wikipedia.org/wiki/Kerr_metric

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