File:Kerr photon orbits with orbital inclination.gif

Summary[edit]

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

Display[edit]

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

Inclination angle by radius[edit]

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]}

Equations of motion[edit]

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]}

de[edit]

Für eine deutschsprachige Version der Bewegungsgleichungen geht es hier entlang

References[edit]

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

Licensing[edit]

I, the copyright holder of this work, hereby publish it under the following license:

This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.

You are free:

• to share – to copy, distribute and transmit the work
• to remix – to adapt the work

Under the following conditions:

• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

https://creativecommons.org/licenses/by-sa/4.0CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 truetrue