Summary

DescriptionHorizon skimming Kerr orbit.gif	English: Photon orbit below the ergosphere and above the horizon of an extremal Kerr black hole. Spin parameter: a=1, direction of motion: prograde. Initial conditions: radius r=1.448GM/c², polar angle θ=90°, local equatorial inclination angle i=66.1391°, constants of motion: energy E=0.475744hf, axial angular momentum Lz=0.856004GMhf/c³, Carter constant Q=1.75361GMHf/c³. Left: x,z-projection, right: x,y-projection. For a close particle orbit see here.
Date	23 June 2017
Source	Own work
Author	Yukterez (Simon Tyran, Vienna)
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01) Coordinate time (GM/c^3)         11) BL r coordinate (GM/c^2)         21) Radius of gyration (GM/c^2)      31) Observed framedragging rate (c^3/G/M)
02) Affine parameter (GM/c^3)        12) BL φ coordinate (radians)        22) Cartesian radius (GM/c^2)        32) Local framedragging velocity (c)
03) 1st derivative (dt/dτ)           13) BL θ coordinate (radians)        23) BH Irreducible mass (M)          33) Cartesian framedragging velocity (c)
04) Grav. time dilation (dt/dτ)      14) dr/dτ (c)                        24) Kinetic energy (hf)              34) Proper velocity (c, dl/dτ)
05) Local energy (dt/dτ, mc^2)       15) dφ/dτ (c^3/G/M)                  25) Potential energy (hf)            35) Observed velocity (c, d{x,y,z}/dt)
06) Cartesian radius (GM/c^2)        16) dθ/dτ (c^3/G/M)                  26) Total energy (hf)                36) Escape velocity (c)
07) x Axis (GM/c^2)                  17) d^2r/dτ^2 (c^6/G/M)              27) Carter constant (GMhf/c^3)       37) Local r velocity (c)
08) y Axis (GM/c^2)                  18) d^2φ/dτ^2 (c^6/G^2/M^2)          28) φ angular momentum (GMhf/c^3)    38) Local θ velocity (c)
09) z Axis (GM/c^2)                  19) d^2θ/dτ^2 (c^6/G^2/M^2)          29) θ angular momentum (GMhf/c^3)    39) Local φ velocity (c)
10) travelled distance (GM/c^2)      20) Spin parameter (GM^2/c)          30) Radial momentum (hf/c)           40) Total local velocity (c)

Equations of motion

All formulas come in natural units:

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

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:

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

Latitudinal time derivative (dθ/dτ):

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

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

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

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 (I is the orbital inclination angel):

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

Carter k (constant):

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

Total energy (constant):

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

Angular momentum on the Ф-axis (constant):

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

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: ^{[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

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