Solution of Einstein field equations
The Kerr–Newman–de–Sitter metric (KNdS)[ 1] [ 2] stationary solutions of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding an electrically charged, rotating mass embedded in an expanding universe. It generalizes the Kerr–Newman metric by taking into account the cosmological constant
Λ
{\displaystyle \Lambda }
In (+, −, −, −) signature and in natural units of
G
=
M
=
c
=
k
e
=
1
{\displaystyle {\rm {G=M=c=k_{e}=1}}}
[ 3] [ 4] [ 5] [ 6]
g
t
t
=
−
3
[
a
2
sin
2
θ
(
a
2
Λ
cos
2
θ
+
3
)
+
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
)
{\displaystyle g_{\rm {tt}}={\rm {-{\frac {3\ [a^{2}\ \sin ^{2}\theta \left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)+a^{2}\left(\Lambda \ r^{2}-3\right)+\Lambda \ r^{4}-3\ r^{2}+6\ r-3\mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}}
g
r
r
=
−
a
2
cos
2
θ
+
r
2
(
a
2
+
r
2
)
(
1
−
Λ
r
2
3
)
−
2
r
+
℧
2
{\displaystyle g_{\rm {rr}}={\rm {-{\frac {a^{2}\ \cos ^{2}\theta +r^{2}}{\left(a^{2}+r^{2}\right)\left(1-{\frac {\Lambda \ r^{2}}{3}}\right)-2\ r+\mho ^{2}}}}}}
g
θ
θ
=
−
3
(
a
2
cos
2
θ
+
r
2
)
a
2
Λ
cos
2
θ
+
3
{\displaystyle g_{\rm {\theta \theta }}={\rm {-{\frac {3\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}{a^{2}\ \Lambda \ \cos ^{2}\theta +3}}}}}
g
ϕ
ϕ
=
9
{
1
3
(
a
2
+
r
2
)
2
sin
2
θ
(
a
2
Λ
cos
2
θ
+
3
)
−
a
2
sin
4
θ
[
(
a
2
+
r
2
)
(
1
−
Λ
r
2
/
3
)
−
2
r
+
℧
2
]
}
−
(
a
2
Λ
+
3
)
2
(
a
2
cos
2
θ
+
r
2
)
{\displaystyle g_{\rm {\phi \phi }}={\rm {\frac {9\ \{{\frac {1}{3}}\left(a^{2}+r^{2}\right)^{2}\sin ^{2}\theta \left(a^{2}\ \Lambda \cos ^{2}\theta +3\right)-a^{2}\sin ^{4}\theta \ [\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}]\}}{-\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}
g
t
ϕ
=
3
a
sin
2
θ
[
a
2
Λ
(
a
2
+
r
2
)
cos
2
θ
+
a
2
Λ
r
2
+
Λ
r
4
+
6
r
−
3
℧
2
]
(
a
2
Λ
+
3
)
2
(
a
2
cos
2
θ
+
r
2
)
{\displaystyle g_{\rm {t\phi }}={\rm {\frac {3\ a\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+\Lambda \ r^{4}+6\ r-3\ \mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}}}}
with all the other metric tensor components
g
μ
ν
=
0
{\displaystyle g_{\mu \nu }=0}
a
{\displaystyle {\rm {a}}}
℧
{\displaystyle {\rm {\mho }}}
Λ
=
3
H
2
{\displaystyle {\rm {\Lambda =3H^{2}}}}
[ 7]
H
{\displaystyle {\rm {H}}}
Hubble parameter . The electromagnetic 4-potential is
A
μ
=
{
3
r
℧
(
a
2
Λ
+
3
)
(
a
2
cos
2
θ
+
r
2
)
,
0
,
0
,
−
3
a
r
℧
sin
2
θ
(
a
2
Λ
+
3
)
(
a
2
cos
2
θ
+
r
2
)
}
{\displaystyle {\rm {A_{\mu }=\left\{{\frac {3\ r\ \mho }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}},\ 0,\ 0,\ -{\frac {3\ a\ r\ \mho \ \sin ^{2}\theta }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}}\right\}}}}
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}}}}}
and the local frame-dragging velocity relative to constant
{
r
,
θ
,
ϕ
}
{\displaystyle {\rm {\{r,\theta ,\phi \}}}}
ergosphere )
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}]}}}}}}
The escape velocity (the speed of light at the horizons) relative to the local corotating zero-angular momentum observer is
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}}}}
The conserved quantities in the equations of motion
x
¨
μ
=
−
∑
α
,
β
(
Γ
α
β
μ
x
˙
α
x
˙
β
+
q
F
μ
β
x
˙
α
g
α
β
)
{\displaystyle {\rm {{\ddot {x}}^{\mu }=-\sum _{\alpha ,\beta }\ (\Gamma _{\alpha \beta }^{\mu }\ {\dot {x}}^{\alpha }\ {\dot {x}}^{\beta }+q\ {\rm {F}}^{\mu \beta }\ {\rm {\dot {x}}}^{\alpha }}}\ g_{\alpha \beta })}
where
x
˙
{\displaystyle {\rm {\dot {x}}}}
four velocity ,
q
{\displaystyle {\rm {q}}}
specific charge and
F
{\displaystyle {\rm {F}}}
Maxwell–Faraday tensor
F
μ
ν
=
∂
A
μ
∂
x
ν
−
∂
A
ν
∂
x
μ
{\displaystyle {\rm {{\ F}_{\mu \nu }={\frac {\partial A_{\mu }}{\partial x^{\nu }}}-{\frac {\partial A_{\nu }}{\partial x^{\mu }}}}}}
are the total energy
E
=
−
p
t
=
g
t
t
t
˙
+
g
t
ϕ
ϕ
˙
+
q
A
t
{\displaystyle {\rm {E=-p_{t}}}=g_{\rm {tt}}{\rm {\dot {t}}}+g_{\rm {t\phi }}{\rm {\dot {\phi }}}+{\rm {q\ A_{t}}}}
and the covariant axial angular momentum
L
z
=
p
ϕ
=
−
g
ϕ
ϕ
ϕ
˙
−
g
t
ϕ
t
˙
−
q
A
ϕ
{\displaystyle {\rm {L_{z}=p_{\phi }}}=-g_{\rm {\phi \phi }}{\rm {\dot {\phi }}}-g_{\rm {t\phi }}{\rm {\dot {t}}}-{\rm {q\ A_{\phi }}}}
The overdot stands for differentiation by the testparticle's proper time
τ
{\displaystyle \tau }
affine parameter , so
x
˙
=
d
x
/
d
τ
,
x
¨
=
d
2
x
/
d
τ
2
{\displaystyle {\rm {{\dot {x}}=dx/d\tau ,\ {\ddot {x}}=d^{2}x/d\tau ^{2}}}}
To get
g
r
r
=
0
{\displaystyle g_{\rm {rr}}=0}
d
t
=
d
u
−
d
r
(
a
2
Λ
/
3
+
1
)
(
a
2
+
r
2
)
(
a
2
+
r
2
)
(
1
−
Λ
r
2
/
3
)
−
2
r
+
℧
2
{\displaystyle {\rm {dt=du-{\frac {dr\left(a^{2}\ \Lambda /3+1\right)\left(a^{2}+r^{2}\right)}{\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}}}}}}
d
ϕ
=
d
φ
−
a
d
r
(
a
2
Λ
/
3
+
1
)
(
a
2
+
r
2
)
(
1
−
Λ
r
2
/
3
)
−
2
r
+
℧
2
{\displaystyle {\rm {d\phi =d\varphi -{\frac {a\ dr\left(a^{2}\ \Lambda /3+1\right)}{\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}}}}}}
and get the metric coefficients
g
u
r
=
−
3
a
2
Λ
+
3
{\displaystyle g_{\rm {ur}}={\rm {-{\frac {3}{a^{2}\ \Lambda +3}}}}}
g
r
φ
=
3
a
sin
2
θ
a
2
Λ
+
3
{\displaystyle g_{\rm {r\varphi }}={\rm {\frac {3\ a\sin ^{2}\theta }{a^{2}\ \Lambda +3}}}}
g
u
u
=
g
t
t
,
g
θ
θ
=
g
θ
θ
,
g
φ
φ
=
g
ϕ
ϕ
,
g
u
φ
=
g
t
ϕ
{\displaystyle g_{\rm {uu}}=g_{\rm {tt}}\ ,\ \ g_{\theta \theta }=g_{\theta \theta }\ ,\ \ g_{\rm {\varphi \varphi }}=g_{\rm {\phi \phi }}\ ,\ \ g_{\rm {u\varphi }}=g_{\rm {t\phi }}}
and all the other
g
μ
ν
=
0
{\displaystyle g_{\mu \nu }=0}
vector potential
A
μ
=
{
3
r
℧
(
a
2
Λ
+
3
)
(
a
2
cos
2
θ
+
r
2
)
,
3
r
℧
a
2
(
Λ
r
2
−
3
)
+
6
r
+
Λ
r
4
−
3
(
r
2
+
℧
2
)
,
0
,
−
3
a
r
℧
sin
2
θ
(
a
2
Λ
+
3
)
(
a
2
cos
2
θ
+
r
2
)
}
{\displaystyle {\rm {A_{\mu }=\left\{{\frac {3\ r\ \mho }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\cos ^{2}\theta +r^{2}\right)}},{\frac {3\ r\ \mho }{a^{2}\left(\Lambda \ r^{2}-3\right)+6\ r+\Lambda \ r^{4}-3\left(r^{2}+\mho ^{2}\right)}},\ 0,\ -{\frac {3\ a\ r\ \mho \sin ^{2}\theta }{\left(a^{2}\ \Lambda +3\right)\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}\right\}}}}
Defining
t
¯
=
u
−
r
{\displaystyle {\rm {{\bar {t}}=u-r}}}
45
∘
{\displaystyle 45^{\circ }}
{
t
¯
,
r
}
{\displaystyle \{{\rm {{\bar {t}},\ r\}}}}
spacetime diagram .
Horizons and ergosheres in the KNdS metric for different M:Λ ratios. The black hole related surfaces are color coded as in here . Left: horizons, right: ergosheres for M=1, a=9/10, ℧=2/5, Λ=1/9. At this point the black hole's outer ergosphere has joined the cosmic one to form two domes around the black hole. Unstable orbit at r=2 with the black hole and cosmic parameters as in the image above. The horizons are at
g
r
r
=
0
{\displaystyle g^{\rm {rr}}=0}
g
t
t
|
|
g
u
u
=
0
{\displaystyle g_{\rm {tt}}||g_{\rm {uu}}=0}
Kerr and Kerr–Newman metrics, the horizons have constant Boyer-Lindquist
r
{\displaystyle {\rm {r}}}
θ
{\displaystyle \theta }
This gives 3 positive solutions each (including the black hole's inner and outer horizons and ergospheres as well as the cosmic ones) and a negative solution for the space at
r
<
0
{\displaystyle {\rm {r<0}}}
antiverse [ 8] [ 9] ring singularity , which is part of the probably unphysical extended solution of the metric.
With a negative
Λ
{\displaystyle \Lambda }
Anti–de–Sitter variant with an attractive cosmological constant), there are no cosmic horizon and ergosphere, only the black hole-related ones.
In the Nariai limit[ 10] Schwarzschild–de–Sitter metric to which the KNdS reduces with
a
=
℧
=
0
{\displaystyle {\rm {a=\mho =0}}}
Λ
=
1
/
9
{\displaystyle \Lambda =1/9}
The Ricci scalar for the KNdS metric is
R
=
−
4
Λ
{\displaystyle {\rm {R=-4\Lambda }}}
Kretschmann scalar is
K
=
{
220
a
12
Λ
2
cos
(
6
θ
)
+
66
a
12
Λ
2
cos
(
8
θ
)
+
12
a
12
Λ
2
cos
(
10
θ
)
+
a
12
Λ
2
cos
(
12
θ
)
+
{\displaystyle {\rm {K=\{220a^{12}\Lambda ^{2}\cos(6\theta )+66a^{12}\Lambda ^{2}\cos(8\theta )+12a^{12}\Lambda ^{2}\cos(10\theta )+a^{12}\Lambda ^{2}\cos(12\theta )+}}}
462
a
12
Λ
2
+
1080
a
10
Λ
2
r
2
cos
(
6
θ
)
+
240
a
10
Λ
2
r
2
cos
(
8
θ
)
+
24
a
10
Λ
2
r
2
cos
(
10
θ
)
+
{\displaystyle {\rm {462a^{12}\Lambda ^{2}+1080a^{10}\Lambda ^{2}r^{2}\cos(6\theta )+240a^{10}\Lambda ^{2}r^{2}\cos(8\theta )+24a^{10}\Lambda ^{2}r^{2}\cos(10\theta )+}}}
3024
a
10
Λ
2
r
2
+
1920
a
8
Λ
2
r
4
cos
(
6
θ
)
+
240
a
8
Λ
2
r
4
cos
(
8
θ
)
+
8400
a
8
Λ
2
r
4
−
{\displaystyle {\rm {3024a^{10}\Lambda ^{2}r^{2}+1920a^{8}\Lambda ^{2}r^{4}\cos(6\theta )+240a^{8}\Lambda ^{2}r^{4}\cos(8\theta )+8400a^{8}\Lambda ^{2}r^{4}-}}}
1152
a
6
cos
(
6
θ
)
−
11520
a
6
+
1280
a
6
Λ
2
r
6
cos
(
6
θ
)
+
12800
a
6
Λ
2
r
6
+
207360
a
4
r
2
−
{\displaystyle {\rm {1152a^{6}\cos(6\theta )-11520a^{6}+1280a^{6}\Lambda ^{2}r^{6}\cos(6\theta )+12800a^{6}\Lambda ^{2}r^{6}+207360a^{4}r^{2}-}}}
138240
a
4
r
℧
2
+
11520
a
4
Λ
2
r
8
+
16128
a
4
℧
4
−
276480
a
2
r
4
+
368640
a
2
r
3
℧
2
+
{\displaystyle {\rm {138240a^{4}r\mho ^{2}+11520a^{4}\Lambda ^{2}r^{8}+16128a^{4}\mho ^{4}-276480a^{2}r^{4}+368640a^{2}r^{3}\mho ^{2}+}}}
6144
a
2
Λ
2
r
10
−
104448
a
2
r
2
℧
4
+
3
a
4
cos
(
4
θ
)
[
165
a
8
Λ
2
+
960
a
6
Λ
2
r
2
+
2240
a
4
Λ
2
r
4
−
{\displaystyle {\rm {6144a^{2}\Lambda ^{2}r^{10}-104448a^{2}r^{2}\mho ^{4}+3a^{4}\cos(4\theta )[165a^{8}\Lambda ^{2}+960a^{6}\Lambda ^{2}r^{2}+2240a^{4}\Lambda ^{2}r^{4}-}}}
256
a
2
(
9
−
10
Λ
2
r
6
)
+
256
(
90
r
2
−
60
r
℧
2
+
5
Λ
2
r
8
+
7
℧
4
)
]
+
24
a
2
cos
(
2
θ
)
[
33
a
10
Λ
2
+
{\displaystyle {\rm {256a^{2}(9-10\Lambda ^{2}r^{6})+256(90r^{2}-60r\mho ^{2}+5\Lambda ^{2}r^{8}+7\mho ^{4})]+24a^{2}\cos(2\theta )[33a^{10}\Lambda ^{2}+}}}
210
a
8
Λ
2
r
2
+
560
a
6
Λ
2
r
4
−
80
a
4
(
9
−
10
Λ
2
r
6
)
+
128
a
2
(
90
r
2
−
60
r
℧
2
+
5
Λ
2
r
8
+
{\displaystyle {\rm {210a^{8}\Lambda ^{2}r^{2}+560a^{6}\Lambda ^{2}r^{4}-80a^{4}(9-10\Lambda ^{2}r^{6})+128a^{2}(90r^{2}-60r\mho ^{2}+5\Lambda ^{2}r^{8}+}}}
7
℧
4
)
+
256
r
2
(
−
45
r
2
+
60
r
℧
2
+
Λ
2
r
8
−
17
℧
4
)
]
+
36864
r
6
−
73728
r
5
℧
2
+
{\displaystyle {\rm {7\mho ^{4})+256r^{2}(-45r^{2}+60r\mho ^{2}+\Lambda ^{2}r^{8}-17\mho ^{4})]+36864r^{6}-73728r^{5}\mho ^{2}+}}}
2048
Λ
2
r
12
+
43008
r
4
℧
4
}
÷
{
12
[
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{\displaystyle {\rm {2048\Lambda ^{2}r^{12}+43008r^{4}\mho ^{4}\}\div \{12[a^{2}\cos(2\theta )+a^{2}+2r^{2}]^{6}\}{\text{.}}}}}
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![{\displaystyle g_{\rm {tt}}={\rm {-{\frac {3\ [a^{2}\ \sin ^{2}\theta \left(a^{2}\ \Lambda \ \cos ^{2}\theta +3\right)+a^{2}\left(\Lambda \ r^{2}-3\right)+\Lambda \ r^{4}-3\ r^{2}+6\ r-3\mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f2b5b59138ef22eecb2c6c151fd5c5d8ebbd28de)
![{\displaystyle g_{\rm {\phi \phi }}={\rm {\frac {9\ \{{\frac {1}{3}}\left(a^{2}+r^{2}\right)^{2}\sin ^{2}\theta \left(a^{2}\ \Lambda \cos ^{2}\theta +3\right)-a^{2}\sin ^{4}\theta \ [\left(a^{2}+r^{2}\right)\left(1-\Lambda \ r^{2}/3\right)-2\ r+\mho ^{2}]\}}{-\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\cos ^{2}\theta +r^{2}\right)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07ebc2bba7c71aa235ff94d29110470feb661727)
![{\displaystyle g_{\rm {t\phi }}={\rm {\frac {3\ a\ \sin ^{2}\theta \ [a^{2}\ \Lambda \left(a^{2}+r^{2}\right)\cos ^{2}\theta +a^{2}\ \Lambda \ r^{2}+\Lambda \ r^{4}+6\ r-3\ \mho ^{2}]}{\left(a^{2}\ \Lambda +3\right)^{2}\left(a^{2}\ \cos ^{2}\theta +r^{2}\right)}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92a804c925442fcf46553ca63a60b8b0a57f75d0)
![{\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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![{\displaystyle {\rm {256a^{2}(9-10\Lambda ^{2}r^{6})+256(90r^{2}-60r\mho ^{2}+5\Lambda ^{2}r^{8}+7\mho ^{4})]+24a^{2}\cos(2\theta )[33a^{10}\Lambda ^{2}+}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18d96d215a7193f2d8796b38933f32365b103d18)
![{\displaystyle {\rm {7\mho ^{4})+256r^{2}(-45r^{2}+60r\mho ^{2}+\Lambda ^{2}r^{8}-17\mho ^{4})]+36864r^{6}-73728r^{5}\mho ^{2}+}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bf4ad3936d29159700186fadd9df4023f331b3ff)
![{\displaystyle {\rm {2048\Lambda ^{2}r^{12}+43008r^{4}\mho ^{4}\}\div \{12[a^{2}\cos(2\theta )+a^{2}+2r^{2}]^{6}\}{\text{.}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3fae35385d21f47c2076cdbd0066e8ba4c28c98f)
