Summary

DescriptionOrbit around a rotating Kerr black hole.gif	English: Retrograde orbit around a black hole spinning with spin parameter a=Jc/G/M²=0.95. Initial conditions: local velocity v0=0.5, cartesian position x0=6.56906, y0=z0=0, orbital inclination angle i0=π/2-11/50=77.3949°, vertical launch angle=0 (which gives the local three-velocity-components vφ0=-sin(11/50)/2, vθ0=-cos(11/50)/2, vr0=0) - that corresponds to the conserved quantities total energy E=0.956545, axial angular momentum Lz=-0.830327, Carter constant Q=13.4126 and the Boyer-Lindquist-coordinates r0=6.5, θ0=π/2=90°, φ0=0. The observed (shapiro-delayed and frame-dragged) velocity at t0 is u0=0.409215. The bright blue pumpkin-shaped region outside of the BH is the outer ergosphere, the cyan region the BH's outer horizon, the oblated spheroid pink region the inner horizon and the innermost magenta region the inner ergosphere. The ring singularity is located on the equatorial kink of the inner ergosurface at R=a. Units: G=M=c=1. Deutsch: Retrograder Orbit um ein mit einem Spinparameter von a=Jc/G/M²=0.95 rotierendes schwarzes Loch. Startbedingungen: lokale Geschwindigkeit v0=0.5c, kartesische Position x0=6.56906, y0=z0=0, orbitaler Inklinationswinkel i0=π/2-11/50=77.3949°, vertikaler Abschusswinkel=0 (das ergibt die lokalen Dreiergeschwindigkeitskomponenten vφ0=-0.109115, vθ0=-0.487949, vr0=0) - was den Erhaltungsgrößen Gesamtenergie E=0.956545, axialer Drehimpuls Lz=-0.830327, Carter Konstante Q=13.4126 und den Boyer-Lindquist-Koordinaten r0=6.5, θ0=π/2=90°, φ0=0 entspricht. Die beobachtete (shapiroverzögerte und geframedragte) Geschwindigkeit bei t0 ist u0=0.409215. Die hellblaue äußere Zone ist die Ergosphäre, die cyane Region der äußere Ereignishorizont, das abgeflachte hellrote Spheroid der innere EH und die innerste violette Region die innere Ergosphäre. Die Ringsingularität befindet sich an deren Ausbuchtung in der Äquatorebene auf einem kartesischen Radius von R=a. Einheiten: G=M=c=1
Date	17 June 2017
Source	Own work
Author	Yukterez (Simon Tyran, Vienna)
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en

01) Coordinate time              08) Axial radius of gyration     15) Axial angular momentum       22) Framedragging delayed angular velocity
02) Proper time                  09) Poloidial radius of gyration 16) Polar angular momentum       23) Framedragging local velocity
03) Total time dilation          10) Radial coefficient           17) Radial momentum              24) Framedragging observed velocity
04) Gravitational time dilation  11) E kinetic                    18) Cartesian radius             25) Observed particle velocity
05) Boyer Lindquist radius       12) Potential energy component   19) Cartesian X-axis             26) Local escape velocity
06) BL Longitude in radians      13) Total particle energy        20) Cartesian Y-axis             27) Delayed particle velocity
07) BL Latitude in radians       14) Carter Constant              21) Cartesian Z-axis             28) Local particle velocity

de

01) Koordinatenzeit              08) Axialer Gyrationsradius      15) Axialer Drehimpuls           22) Framedrag verzögerte Winkelgeschwindigkeit
02) Eigenzeit des Testpartikels  09) Poloidialer Gyrationsradius  16) Polarer Drehimpuls           23) Framedrag lokale Transversalgeschwindigkeit
03) Insgesamte Zeitdilatation    10) Radialer Vorfaktor           17) Radialer Impuls              24) Framedrag beobachtete Transversalgeschwindigkeit
04) Gravitative  Zeitdilatation  11) E kinetisch                  18) Kartesischer Radius          25) Beobachtete Totalgeschwindigkeit
05) Boyer Lindquist Radius       12) Potentielle Energie          19) Kartesische X-Achse          26) Lokale Fluchtgeschwindigkeit
06) BL Längengrad in Radianten   13) Totale Energie               20) Kartesische Y-Achse          27) Verzögerte Geschwindigkeit
07) BL Breitengrad in Radianten  14) Carter Konstante             21) Kartesische Z-Achse          28) Lokale Geschwindigkeit relativ zum ZAMO

Equations of motion

en

All formulas come in natural units:

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

Shorthand Terms:

${\displaystyle {\rm {\Sigma =a^{2}\cos ^{2}\theta +r^{2}\ ,\ \ \Delta =a^{2}+r^{2}-2r\ ,\ \ \chi =\left(a^{2}+r^{2}\right)^{2}-a^{2}\ \sin ^{2}\theta \ \Delta }}}$

Metric components, covariant form:

${\displaystyle {g_{\rm {tt}}={\rm {\frac {\Delta -a^{2}\ \sin ^{2}\theta }{\Sigma }}}\ ,\ \ g_{\rm {t\phi }}={\rm {\frac {2\ a\ r\ \sin ^{2}\theta }{\Sigma }}}\ ,\ \ g_{\rm {\phi \phi }}={\rm {-{\frac {\chi \ \sin ^{2}\theta }{\Sigma }}}}\ ,\ \ g_{\rm {rr}}={\rm {-{\frac {\Sigma }{\Delta }}}}\ ,\ \ g_{\rm {\theta \theta }}=-\Sigma }}$

Contravariant components:

${\displaystyle {g^{\rm {tt}}={\rm {\frac {\chi }{\Delta \ \Sigma }}}\ ,\ \ g^{\rm {t\phi }}={\rm {\frac {2\ a\ r}{\Delta \ \Sigma }}}\ ,\ \ g^{\rm {\phi \phi }}={\rm {-{\frac {\Delta -a^{2}\ \sin ^{2}\theta }{\Delta \ \Sigma \ \sin ^{2}\theta }}}}\ ,\ \ g^{\rm {rr}}={\rm {-{\frac {\Delta }{\Sigma }}}}\ ,\ \ g^{\rm {\theta \theta }}=-{\frac {1}{\Sigma }}}}$

Coordinate time by proper time (dt/dτ):

${\displaystyle {\rm {{\dot {t}}={\frac {2\ E\ r\ \left(a^{2}+r^{2}\right)-2\ a\ L_{z}\ r}{\Delta \ \Sigma }}+E={\frac {\varsigma }{\sqrt {1-v^{2}}}}}}}$

Radial coordinate time derivative (dr/dτ):

${\displaystyle {\rm {{\dot {r}}={\frac {\Delta \ p_{r}}{\Sigma }}}}}$

Time derivative of the covariant momentum's r-component (pr/dτ):

${\displaystyle {\rm {{\dot {p}}_{r}={\frac {(r-1)\left(\mu \ \left(a^{2}+r^{2}\right)-k\right)+2\ E^{2}\ r\left(a^{2}+r^{2}\right)-2\ a\ E\ L_{z}+\Delta \ \mu \ r}{\Delta \ \Sigma }}-{\frac {2\ p_{r}^{2}\ (r-1)}{\Sigma }}}}}$

Relation to the local velocity:

${\displaystyle {\rm {p_{r}={\frac {v_{r}}{\sqrt {1-\mu ^{2}v^{2}}}}{\sqrt {\frac {\Sigma }{\Delta }}}}}}$

Latitudinal time derivative (dθ/dτ):

${\displaystyle {\rm {{\dot {\theta }}={\frac {p_{\theta }}{\Sigma }}}}}$

Time derivative of the covariant momentum's θ-component (pθ/dτ):

${\displaystyle {\rm {{\dot {p}}_{\theta }={\frac {\sin \theta \ \cos \theta \left({\frac {L_{z}^{2}}{\sin ^{4}\theta }}-a^{2}\left(E^{2}-\mu ^{2}\right)\right)}{\Sigma }}}}}$

Relation to the local velocity:

${\displaystyle {\rm {p_{\theta }={\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1-\mu ^{2}v^{2}}}}}}}$

Longitudinal time derivative (dФ/dτ):

${\displaystyle {\rm {{\dot {\phi }}={\frac {2\ a\ E\ r+L_{z}\ \csc ^{2}\theta \ (\Sigma -2r)}{\Delta \ \Sigma }}}}}$

Time derivative of the covariant momentum's Ф-component (pФ/dτ):

${\displaystyle {\rm {{\dot {p}}_{\phi }=0}}}$

Carter-constant:

${\displaystyle {\rm {Q=p_{\theta }^{2}+\cos ^{2}\theta \left(a^{2}(\mu ^{2}-E^{2})+{\frac {L_{z}^{2}}{\sin ^{2}\theta }}\right)=a^{2}\ (\mu ^{2}-E^{2})\ \sin ^{2}I+L_{z}^{2}\ \tan ^{2}I}}}$

Carter k:

${\displaystyle {\rm {k=a^{2}\left(E^{2}-\mu ^{2}\right)+L_{z}^{2}+Q}}}$

Total energy:

${\displaystyle {{\rm {E}}=g_{\rm {tt}}\ {\dot {\rm {t}}}+g_{\rm {t\phi }}\ {\dot {\phi }}={\rm {{\sqrt {{\frac {(\Sigma -2\ r)\left({\dot {\theta }}^{2}\ \Delta \ \Sigma +{\dot {r}}^{2}\ \Sigma +\Delta \ \mu ^{2}\right)}{\Delta \ \Sigma }}+{\dot {\phi }}^{2}\ \Delta \ \sin ^{2}\theta }}={\sqrt {\frac {\Delta \ \Sigma }{(1-\mu ^{2}v^{2})\ \chi }}}+\Omega \ L_{z}}}}}$

Angular momentum on the Ф-axis:

${\displaystyle {\rm {L_{z}}}=-g_{\phi \phi }\ {\dot {\phi }}-g_{\rm {t\phi }}\ {\dot {\rm {t}}}={\rm {{\frac {\sin ^{2}\theta \ ({\dot {\phi }}\ \Delta \ \Sigma -2\ a\ E\ r)}{\Sigma -2\ r}}={\frac {v_{\phi }\ {\bar {R}}}{\sqrt {1-\mu ^{2}v^{2}}}}}}}$

with the radius of gyration

${\displaystyle {\rm {{\bar {R}}_{\phi }}}={\sqrt {-g_{\phi \phi }}}={\sqrt {\frac {\chi }{\Sigma }}}\ \sin \theta }$

Frame Dragging angular velocity (dФ/dt):

${\displaystyle \omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {\rm {2\ a\ r}}{\chi }}}$

Gravitational time dilation (dt/dτ):

${\displaystyle \varsigma ={\sqrt {g^{\rm {tt}}}}={\sqrt {\frac {\chi }{\Delta \ \Sigma }}}}$

Local velocity on the r-axis:

${\displaystyle {\rm {{\frac {v_{r}}{\sqrt {1-\mu ^{2}v^{2}}}}={\dot {r}}\ {\sqrt {\frac {\Sigma }{\Delta }}}}}}$

Local velocity on the θ-axis:

${\displaystyle {\rm {{\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1-\mu ^{2}v^{2}}}}={\dot {\theta }}\ \Sigma }}}$

Local velocity on the Ф-axis:

${\displaystyle {\frac {\rm {v_{\phi }}}{\sqrt {1-\mu ^{2}{\rm {v^{2}}}}}}={\frac {\rm {L_{z}}}{\rm {{\bar {R}}_{\phi }}}}}$

with the cartesian coordinates:

${\displaystyle {\rm {x={\sqrt {r^{2}+a^{2}}}\sin \theta \ \cos \phi \ ,\ y={\sqrt {r^{2}+a^{2}}}\sin \theta \ \sin \phi \ ,\ z=r\cos \theta \quad }}}$

The observed velocity β is given by:

${\displaystyle {\rm {\beta ={\sqrt {(dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2}}}}}}$

The local escape velocity is given by the relation:

${\displaystyle {\rm {\varsigma =1/{\sqrt {1-v_{\rm {esc}}^{2}}}\ \to \ v_{\rm {esc}}={\sqrt {\varsigma ^{2}-1}}/\varsigma }}}$

Sources:^{[1][2][3][4][5][6]}

References

1. ↑ Pu, Yun, Younsi & Yoon: General-relativistic radiative transfer in Kerr spacetime, p. 2+
2. ↑ Janna Levin & Gabe Perez-Giz: A Periodic Table for Black Hole Orbits, p. 30+
3. ↑ Scott A. Hughes: Nearly horizon skimming orbits of Kerr black holes, p. 5+
4. ↑ Janna Levin & Gabe Perez-Giz: The Phase Space Portrait, p. 2+
5. ↑ Misner, Thorne & Wheeler (MTW): The Bible archive copy at the Wayback Machine, p. 897+
6. ↑ Simon Tyran: Kerr Orbits / Gravitationslinsen

de

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