DescriptionKerr photon orbits with orbital inclination.gif	English: All possible photon-orbits around a black hole rotating with the spin-parameter a=Jc/G/M²=1. The position of photon and ZAMO is shown for t=150GM/c³ coordinate time. Initial position: θ0=π/2, φ0=0. Deutsch: Alle möglichen Photonenorbits um ein mit dem Spinparameter a=Jc/G/M²=1 rotierendes schwarzes Loch. Gezeigt wird die Position eines Photons und eines ZAMO nach einer Koordinatenzeit von t=150GM/c³. Die Startposition ist auf θ0=π/2, φ0=0.
Date	23 July 2017
Source	Own work
Author	Yukterez (Simon Tyran, Vienna)
Other versions

01) a Spin parameter            08) δ local equatorial          15) L Axial angular momentum    22) ω Frame dragging delayed angular velocity
    of  the central mass            inclination angle               conserved quantity              observed at infinty
02) r Boyer-Lindquist radius    09) δ observed equatorial       16) L Poloidial component       23) v Frame dragging local velocity
    constant for photon orbits      inclination angle               of the angular momentum         equals 1 at the outer ergosurface
03) φ Longitude                 10) δ frame drag angle          17) p Radial component          24) Ω Frame dragging observed velocity
    measured from infinity          difference local-observed       of the momentum                 in cartesian coordinates
04) θ Latitude                  11) E kinetic energy            18) R Radius                    25) v Observed particle velocity
    0=northpole, π=southpole        local energy of the photon      cartesian coordinate            in the bookeepers frame of reference
05) ς Grav. time dilation       12) E potential energy          19) x X-axis                    26) v Local escape velocity
    depending on r and θ            total-kinetic                   cartesian coordinate            equals 1 at the outer horizon
06) t Coordinate time           13) E total energy              20) y Y-axis                    27) v Delayed particle velocity
    of the distant bookeeper        conserved quantity              cartesian coordinate            differential velocity vs a local ZAMO
07) λ Affine parameter          14) Q Carter constant           21) z Z-axis                    28) v Local particle velocity
    takes the place of τ if μ=0     conserved quantity              cartesian coordinate            relative velocity vs a local ZAMO

For a given a and r and starting from θ0=π/2 the required initial orbital inclination angle δ0 for a photon's circular orbit can be found^[1] by setting

${\displaystyle {\rm {{\frac {\zeta _{1}+\zeta _{2}-\zeta _{3}}{\zeta _{4}}}=0}}}$

and solving for δ0. The real solutions of the polynomial give one possible orbit in the positive poloidial direction, and one other in the opposite z-direction (since the metric is axially symmetric the sign of the coaxial angular momentum can be both). The shorthand terms are:

${\displaystyle {\rm {\zeta _{1}=a^{4}\ (r-1)+a^{2}\ r\ (r\ (3\ r-5)+6)+2\ (r-3)\ r^{4}}}}$

${\displaystyle {\rm {\zeta _{2}=4\ a\ {\sqrt {a^{2}+(r-2)\ r}}\ \left(a^{2}+3\ r^{2}\right)\ \cos(\delta )}}}$

${\displaystyle {\rm {\zeta _{3}=a^{2}\ (r+3)\ \left(a^{2}+(r-2)\ r\right)\ \cos(2\ \delta )}}}$

${\displaystyle {\rm {\zeta _{4}=2\ r\ \left(a^{2}+(r-2)\ r\right)\ \left(a^{2}\ (r+2)+r^{3}\right)}}}$

All photon-orbits have a constant Boyer-Lindquist-radius.^{[2] [3]}

All formulas come in natural units:

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

Coordinate time t by proper time τ (dt/dτ), where τ becomes the affine parameter λ for massless particles:

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

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:

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

Latitudinal time derivative (dθ/dτ):

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

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

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

Relation to the local velocity:

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

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:

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

Total energy:

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

Angular momentum on the Ф-axis:

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

with the radius of gyration

${\displaystyle {\rm {{\bar {R}}={\sqrt {\frac {\chi }{\Sigma }}}\ \sin \theta }}}$

Frame Dragging angular velocity (dФ/dt):

${\displaystyle {\rm {\omega ={\frac {2\ a\ r}{\chi }}}}}$

Gravitational time dilation (dt/dτ):

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

Local velocity on the r-axis:

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

Local velocity on the θ-axis:

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

Local velocity on the Ф-axis:

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

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:

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

Shorthand Terms:

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

Sources:^{[4][5][6][7][8][9]}

Für eine deutschsprachige Version der Bewegungsgleichungen geht es hier entlang

1. ↑ Simon Tyran: Kreisbahnen in der Kerr-Raumzeit
2. ↑ Stein Leo: Kerr Spherical Photon Orbits
3. ↑ Ed Teo: Spherical Photon Orbits around a Kerr Black Hole doi:10.1023/A:1026286607562
4. ↑ Pu, Yun, Younsi & Yoon: General-relativistic radiative transfer in Kerr spacetime, p. 2+
5. ↑ Janna Levin & Gabe Perez-Giz: A Periodic Table for Black Hole Orbits, p. 30+
6. ↑ Scott A. Hughes: Nearly horizon skimming orbits of Kerr black holes, p. 5+
7. ↑ Janna Levin & Gabe Perez-Giz: The Phase Space Portrait, p. 2+
8. ↑ Misner, Thorne & Wheeler (MTW): The Bible archive copy at the Wayback Machine, p. 897+
9. ↑ Simon Tyran: Kerr Orbits / Gravitationslinsen

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