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Equations The frame-dragging angular velocity is
ω
=
d
ϕ
d
t
=
−
g
t
ϕ
g
ϕ
ϕ
=
a
[
a
2
Λ
(
a
2
+
r
2
)
cos
2
θ
+
a
2
Λ
r
2
+
6
r
+
Λ
r
4
−
3
℧
2
]
a
2
sin
2
θ
[
a
2
(
Λ
r
2
−
3
)
+
6
r
+
Λ
r
4
−
3
r
2
−
3
℧
2
]
+
a
2
Λ
(
a
2
+
r
2
)
2
cos
2
θ
+
3
(
a
2
+
r
2
)
2
{\displaystyle \omega ={\frac {\rm {d\phi }}{\rm {dt}}}=-{\frac {g_{\rm {t\phi }}}{g_{\rm {\phi \phi }}}}={\rm {\frac {a\ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]}{a^{2}\ \sin ^{2}\theta \ [a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]+a^{2}\ \Lambda \ \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\ \left(a^{2}+r^{2}\right)^{2}}}}}
{
r
,
θ
,
ϕ
}
{\displaystyle {\rm {\{r,\theta ,\phi \}}}}
ergospheres )
ν
=
g
t
ϕ
g
t
ϕ
=
−
a
2
sin
2
θ
[
a
2
Λ
(
a
2
+
r
2
)
cos
2
θ
+
a
2
Λ
r
2
+
6
r
+
Λ
r
4
−
3
℧
2
]
2
(
a
2
Λ
cos
2
θ
+
3
)
(
a
2
+
r
2
−
a
2
sin
2
θ
)
2
[
a
2
(
Λ
r
2
−
3
)
+
6
r
+
Λ
r
4
−
3
r
2
−
3
℧
2
]
{\displaystyle \nu ={\sqrt {g_{\rm {t\phi }}\ g^{\rm {t\phi }}}}={\rm {\sqrt {-{\frac {a^{2}\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]^{2}}{\left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}[a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]}}}}}}
v
=
1
−
1
/
g
t
t
=
3
(
a
2
Λ
cos
2
θ
+
3
)
(
a
2
+
r
2
−
a
2
sin
2
θ
)
2
[
a
2
(
Λ
r
2
−
3
)
+
Λ
r
4
−
3
r
2
+
6
r
−
3
℧
2
]
(
a
2
Λ
+
3
)
2
(
a
2
cos
2
θ
+
r
2
)
{
a
2
Λ
(
a
2
+
r
2
)
2
cos
2
θ
+
3
(
a
2
+
r
2
)
2
+
a
2
sin
2
θ
[
a
2
(
Λ
r
2
−
3
)
+
Λ
r
4
−
3
r
2
+
6
r
−
3
℧
2
]
}
+
1
{\displaystyle {\rm {v}}={\sqrt {1-1/g^{\rm {tt}}}}={\rm {\sqrt {{\frac {3\left(a^{2}\Lambda \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}\left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]}{\left(a^{2}\Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)\{a^{2}\Lambda \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\left(a^{2}+r^{2}\right)^{2}+a^{2}\sin ^{2}\theta \left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]\}}}+1}}}}
ν
=
1
{\displaystyle \nu =1}
g
t
t
=
0
{\displaystyle g_{\rm {tt}}=0}
r
{\displaystyle {\rm {r}}}
Boyer Lindquist radii of the ergospheres and setting
g
r
r
=
0
{\displaystyle g^{\rm {rr}}=0}
here and here
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Date/Time Thumbnail Dimensions User Comment current 17:26, 11 September 2023 800 × 400 (42 KB) imagescommonswiki>Yukterez increase contrast to make the bright parts more visible on uncalibrated monitors
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_Horizons_%26_Ergospheres.gif)
![{\displaystyle \omega ={\frac {\rm {d\phi }}{\rm {dt}}}=-{\frac {g_{\rm {t\phi }}}{g_{\rm {\phi \phi }}}}={\rm {\frac {a\ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]}{a^{2}\ \sin ^{2}\theta \ [a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]+a^{2}\ \Lambda \ \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\ \left(a^{2}+r^{2}\right)^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dda8e28849c12cda4142724ba3596931fa5020ad)
![{\displaystyle \nu ={\sqrt {g_{\rm {t\phi }}\ g^{\rm {t\phi }}}}={\rm {\sqrt {-{\frac {a^{2}\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\Lambda \ r^{2}+6\ r+\Lambda \ r^{4}-3\ \mho ^{2}]^{2}}{\left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}[a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\ r^{2}-3\ \mho ^{2}]}}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4081e3f031366394e1b09362e3ee4f8c725527b8)
![{\displaystyle {\rm {v}}={\sqrt {1-1/g^{\rm {tt}}}}={\rm {\sqrt {{\frac {3\left(a^{2}\Lambda \cos ^{2}\theta +3\right)\left(a^{2}+r^{2}-a^{2}\sin ^{2}\theta \right)^{2}\left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]}{\left(a^{2}\Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)\{a^{2}\Lambda \left(a^{2}+r^{2}\right)^{2}\cos ^{2}\theta +3\left(a^{2}+r^{2}\right)^{2}+a^{2}\sin ^{2}\theta \left[a^{2}\left(\Lambda r^{2}-3\right)+\Lambda r^{4}-3r^{2}+6r-3\mho ^{2}\right]\}}}+1}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9147e151af87916136f383850ec6704aead3b6a9)
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