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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:
G
=
M
=
c
=
1
{\displaystyle {\rm {G=M=c=1}}}
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 }}}
Metric components, covariant form:
g
t
t
=
Δ
−
a
2
sin
2
θ
Σ
,
g
t
ϕ
=
2
a
r
sin
2
θ
Σ
,
g
ϕ
ϕ
=
−
χ
sin
2
θ
Σ
,
g
r
r
=
−
Σ
Δ
,
g
θ
θ
=
−
Σ
{\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:
g
t
t
=
χ
Δ
Σ
,
g
t
ϕ
=
2
a
r
Δ
Σ
,
g
ϕ
ϕ
=
−
Δ
−
a
2
sin
2
θ
Δ
Σ
sin
2
θ
,
g
r
r
=
−
Δ
Σ
,
g
θ
θ
=
−
1
Σ
{\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):
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
2
v
2
Σ
Δ
{\displaystyle {\rm {p_{r}={\frac {v_{r}}{\sqrt {1-\mu ^{2}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
2
)
)
Σ
{\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:
p
θ
=
v
θ
Σ
1
−
μ
2
v
2
{\displaystyle {\rm {p_{\theta }={\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1-\mu ^{2}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:
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:
k
=
a
2
(
E
2
−
μ
2
)
+
L
z
2
+
Q
{\displaystyle {\rm {k=a^{2}\left(E^{2}-\mu ^{2}\right)+L_{z}^{2}+Q}}}
Total energy:
E
=
g
t
t
t
˙
+
g
t
ϕ
ϕ
˙
=
(
Σ
−
2
r
)
(
θ
˙
2
Δ
Σ
+
r
˙
2
Σ
+
Δ
μ
2
)
Δ
Σ
+
ϕ
˙
2
Δ
sin
2
θ
=
Δ
Σ
(
1
−
μ
2
v
2
)
χ
+
Ω
L
z
{\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:
L
z
=
−
g
ϕ
ϕ
ϕ
˙
−
g
t
ϕ
t
˙
=
sin
2
θ
(
ϕ
˙
Δ
Σ
−
2
a
E
r
)
Σ
−
2
r
=
v
ϕ
R
¯
1
−
μ
2
v
2
{\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
R
¯
ϕ
=
−
g
ϕ
ϕ
=
χ
Σ
sin
θ
{\displaystyle {\rm {{\bar {R}}_{\phi }}}={\sqrt {-g_{\phi \phi }}}={\sqrt {\frac {\chi }{\Sigma }}}\ \sin \theta }
Frame Dragging angular velocity (dФ/dt):
ω
=
−
g
t
ϕ
g
ϕ
ϕ
=
2
a
r
χ
{\displaystyle \omega =-{\frac {g_{t\phi }}{g_{\phi \phi }}}={\frac {\rm {2\ a\ r}}{\chi }}}
Gravitational time dilation (dt/dτ):
ς
=
g
t
t
=
χ
Δ
Σ
{\displaystyle \varsigma ={\sqrt {g^{\rm {tt}}}}={\sqrt {\frac {\chi }{\Delta \ \Sigma }}}}
Local velocity on the r-axis:
v
r
1
−
μ
2
v
2
=
r
˙
Σ
Δ
{\displaystyle {\rm {{\frac {v_{r}}{\sqrt {1-\mu ^{2}v^{2}}}}={\dot {r}}\ {\sqrt {\frac {\Sigma }{\Delta }}}}}}
Local velocity on the θ-axis:
v
θ
Σ
1
−
μ
2
v
2
=
θ
˙
Σ
{\displaystyle {\rm {{\frac {v_{\theta }\ {\sqrt {\Sigma }}}{\sqrt {1-\mu ^{2}v^{2}}}}={\dot {\theta }}\ \Sigma }}}
Local velocity on the Ф-axis:
v
ϕ
1
−
μ
2
v
2
=
L
z
R
¯
ϕ
{\displaystyle {\frac {\rm {v_{\phi }}}{\sqrt {1-\mu ^{2}{\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 }}}
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
de
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inner ergosphere and ring singularity
ייִדיש נא להוסיף משפט שמסביר מה הקובץ מייצג
ענגליש Retrograde obit around a spinning black hole
דײַטש Retrograder Orbit um ein rotierendes schwarzes Loch