Popis
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
Licence
Já, držitel autorských práv k tomuto dílu, ho tímto zveřejňuji za podmínek následující licence:
Dílo smíte:
šířit – kopírovat, distribuovat a sdělovat veřejnosti
upravovat – pozměňovat, doplňovat, využívat celé nebo částečně v jiných dílech
Za těchto podmínek:
uveďte autora – Máte povinnost uvést autorství, poskytnout odkaz na licenci a uvést, pokud jste provedli změny. Toho můžete docílit jakýmkoli rozumným způsobem, avšak ne způsobem naznačujícím, že by poskytovatel licence schvaloval nebo podporoval vás nebo vaše užití díla.
zachovejte licenci – Pokud tento materiál jakkoliv upravíte, přepracujete nebo použijete ve svém díle, musíte své příspěvky šířit pod stejnou nebo slučitelnou licencí jako originál. https://creativecommons.org/licenses/by-sa/4.0 CC BY-SA 4.0 Creative Commons Attribution-Share Alike 4.0 true true
File usage
187
189
8
8
758
500
inner ergosphere and ring singularity
čeština Přidejte jednořádkové vysvětlení, co tento soubor představuje
angličtina Photon orbit around an extremal Kerr black hole
němčina Photonenorbit um ein maximal rotierendes schwarzes Loch