Solution of Einstein field equations
The Kerr–Newman–de–Sitter metric (KNdS)
[1]
[2] is the one of the most general
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}}}
the KNdS metric is
[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}
, where
a
{\displaystyle {\rm {a}}}
is the black hole's spin parameter,
℧
{\displaystyle {\rm {\mho }}}
its electric charge, and
Λ
=
3
H
2
{\displaystyle {\rm {\Lambda =3H^{2}}}}
[7] the cosmological constant with
H
{\displaystyle {\rm {H}}}
as the time-independent
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 \}}}}
positions (the speed of light at the
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}}}}
is the
four velocity ,
q
{\displaystyle {\rm {q}}}
is the test particle's
specific charge and
F
{\displaystyle {\rm {F}}}
the
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 }
or the photon's
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}
coordinates we apply the transformation
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}
, with the electromagnetic
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}}}
ingoing lightlike worldlines give a
45
∘
{\displaystyle 45^{\circ }}
light cone on a
{
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}
and the ergospheres at
g
t
t
|
|
g
u
u
=
0
{\displaystyle g_{\rm {tt}}||g_{\rm {uu}}=0}
.
This can be solved numerically or analytically. Like in the
Kerr and
Kerr–Newman metrics, the horizons have constant Boyer-Lindquist
r
{\displaystyle {\rm {r}}}
, while the ergospheres' radii also depend on the polar angle
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}}}
in the
antiverse
[8]
[9] behind the
ring singularity , which is part of the probably unphysical extended solution of the metric.
With a negative
Λ
{\displaystyle \Lambda }
(the
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] the black hole's outer horizon and ergosphere coincide with the cosmic ones (in the
Schwarzschild–de–Sitter metric to which the KNdS reduces with
a
=
℧
=
0
{\displaystyle {\rm {a=\mho =0}}}
that would be the case when
Λ
=
1
/
9
{\displaystyle \Lambda =1/9}
).
The
Ricci scalar for the KNdS metric is
R
=
−
4
Λ
{\displaystyle {\rm {R=-4\Lambda }}}
, and the
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{.}}}}}
See also
References
Types Size Formation Properties Issues Metrics Alternatives Analogs Lists Related Notable