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 .
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
[
a
2
cos
(
2
θ
)
+
a
2
+
2
r
2
]
6
}
.
{\displaystyle {\rm {2048\Lambda ^{2}r^{12}+43008r^{4}\mho ^{4}\}\div \{12[a^{2}\cos(2\theta )+a^{2}+2r^{2}]^{6}\}{\text{.}}}}}
![{\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)
