Sammanfattning
Display
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:
G
=
M
=
c
=
1
{\displaystyle {\rm {G=M=c=1}}}
Coordinate time by proper time (dt/dτ):
t
˙
=
2
E
r
(
a
2
+
r
2
)
−
2
a
L
z
r
Δ
Σ
+
E
=
ς
1
−
v
2
{\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):
r
˙
=
Δ
p
r
Σ
{\displaystyle {\rm {{\dot {r}}={\frac {\Delta \ p_{r}}{\Sigma }}}}}
Time derivative of the covariant momentum's r-component (pr/dτ):
p
˙
r
=
(
r
−
1
)
(
μ
(
a
2
+
r
2
)
−
k
)
+
2
E
2
r
(
a
2
+
r
2
)
−
2
a
E
L
z
+
Δ
μ
r
Δ
Σ
−
2
p
r
2
(
r
−
1
)
Σ
{\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:
p
r
=
v
r
1
+
μ
v
2
Σ
Δ
{\displaystyle {\rm {p_{r}={\frac {v_{r}}{\sqrt {1+\mu \ v^{2}}}}{\sqrt {\frac {\Sigma }{\Delta }}}}}}
Latitudinal time derivative (dθ/dτ):
θ
˙
=
p
θ
Σ
{\displaystyle {\rm {{\dot {\theta }}={\frac {p_{\theta }}{\Sigma }}}}}
Time derivative of the covariant momentum's θ-component (pθ/dτ):
p
˙
θ
=
sin
θ
cos
θ
(
L
z
2
/
sin
4
θ
−
a
2
(
E
2
+
μ
)
)
Σ
{\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:
p
θ
=
v
θ
Σ
1
+
μ
v
2
{\displaystyle {\rm {p_{\theta }={\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1+\mu \ v^{2}}}}}}}
Longitudinal time derivative (dФ/dτ):
ϕ
˙
=
2
a
E
r
+
L
z
csc
2
θ
(
Σ
−
2
r
)
Δ
Σ
{\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):
p
˙
ϕ
=
0
{\displaystyle {\rm {{\dot {p}}_{\phi }=0}}}
Carter-constant (I is the orbital inclination angel):
Q
=
p
θ
2
+
cos
2
θ
(
a
2
(
μ
2
−
E
2
)
+
L
z
2
sin
2
θ
)
=
a
2
(
μ
2
−
E
2
)
sin
2
I
+
L
z
2
tan
2
I
{\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):
k
=
a
2
(
E
2
+
μ
)
+
L
z
2
+
Q
{\displaystyle {\rm {k=a^{2}\left(E^{2}+\mu \right)+L_{z}^{2}+Q}}}
Total energy (constant):
E
=
(
Σ
−
2
r
)
(
θ
˙
2
Δ
Σ
+
r
˙
2
Σ
−
Δ
μ
)
Δ
Σ
+
ϕ
˙
2
Δ
sin
2
θ
=
Δ
Σ
(
1
+
μ
v
2
)
χ
+
Ω
L
z
{\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):
L
z
=
sin
2
θ
(
ϕ
˙
Δ
Σ
−
2
a
E
r
)
Σ
−
2
r
=
v
ϕ
R
¯
1
+
μ
v
2
{\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
R
¯
=
χ
Σ
sin
θ
{\displaystyle {\rm {{\bar {R}}={\sqrt {\frac {\chi }{\Sigma }}}\ \sin \theta }}}
Frame Dragging angular velocity (dФ/dt):
ω
=
2
a
r
χ
{\displaystyle {\rm {\omega ={\frac {2\ a\ r}{\chi }}}}}
Gravitational time dilation (dt/dτ):
ς
=
χ
Δ
Σ
{\displaystyle {\rm {\varsigma ={\sqrt {\frac {\chi }{\Delta \ \Sigma }}}}}}
Local velocity on the r-axis:
v
r
1
+
μ
v
2
=
r
˙
Σ
Δ
{\displaystyle {\rm {{\frac {v_{r}}{\sqrt {1+\mu \ v^{2}}}}={\dot {r}}\ {\sqrt {\frac {\Sigma }{\Delta }}}}}}
Local velocity on the θ-axis:
v
θ
Σ
1
+
μ
v
2
=
θ
˙
Σ
{\displaystyle {\rm {{\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1+\mu \ v^{2}}}}={\dot {\theta }}\ \Sigma }}}
Local velocity on the Ф-axis:
v
ϕ
1
+
μ
v
2
=
L
z
R
¯
ϕ
{\displaystyle {\frac {\rm {v_{\phi }}}{\sqrt {1+\mu \ {\rm {v^{2}}}}}}={\frac {\rm {L_{z}}}{\rm {{\bar {R}}_{\phi }}}}}
with the cartesian coordinates:
x
=
r
2
+
a
2
sin
θ
cos
ϕ
,
y
=
r
2
+
a
2
sin
θ
sin
ϕ
,
z
=
r
cos
θ
{\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:
β
=
(
d
x
/
d
t
)
2
+
(
d
y
/
d
t
)
2
+
(
d
z
/
d
t
)
2
{\displaystyle {\rm {\beta ={\sqrt {(dx/dt)^{2}+(dy/dt)^{2}+(dz/dt)^{2}}}}}}
The local escape velocity is given by the relation:
ς
=
1
/
1
−
v
e
s
c
2
→
v
e
s
c
=
ς
2
−
1
/
ς
{\displaystyle {\rm {\varsigma =1/{\sqrt {1-v_{\rm {esc}}^{2}}}\ \to \ v_{\rm {esc}}={\sqrt {\varsigma ^{2}-1}}/\varsigma }}}
Shorthand Terms:
Σ
=
a
2
cos
2
θ
+
r
2
,
Δ
=
a
2
+
r
2
−
2
r
,
χ
=
(
a
2
+
r
2
)
2
−
a
2
sin
2
θ
Δ
{\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
↑ Pu, Yun, Younsi & Yoon: General-relativistic radiative transfer in Kerr spacetime , p. 2+
↑ Janna Levin & Gabe Perez-Giz: A Periodic Table for Black Hole Orbits , p. 30+
↑ Scott A. Hughes: Nearly horizon skimming orbits of Kerr black holes , p. 5+
↑ Janna Levin & Gabe Perez-Giz: The Phase Space Portrait , p. 2+
↑ Misner, Thorne & Wheeler (MTW): The Bible archive copy at the Wayback Machine , p. 897+
↑ Simon Tyran: Kerr Orbits / Gravitationslinsen
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inner ergosphere and ring singularity
svenska Ingen bildtext har definierats
engelska Photon orbit around an extremal Kerr black hole
tyska Photonenorbit um ein maximal rotierendes schwarzes Loch