Properties of Kerr Spacetime
Properties of Kerr Spacetime
Kerr spacetime describes a rotating black hole. The line element in Boyer–Lindquist coordinates is given by
(t,r,θ,ϕ)
d=1ddtdϕ+d+Σd+++θd
2
s
2Mr
Σ
2
t
4Marθ
2
sin
Σ
Σ
Δ
2
r
2
θ
2
r
2
a
2Mrθ
2
a
2
sin
Σ
2
sin
2
ϕ
with and . is the mass of the black hole and is the angular momentum.
Δ=2Mr+
2
r
2
a
Σ=+θ
2
r
2
a
2
cos
M
a
The roots of the function define the event horizon and the inner horizon , where =M±. The region between the event horizon and the static limit =M+θ is called the ergosphere.
Δ
r
+
r

r
±
2
M
2
a
r
0
2
M
2
a
2
cos
The two marginally stable timelike circular geodesics are defined by the radii =M3+∓, where =1++ and =+. An object on the smaller radius rotates with the Kerr black hole, whereas an object on the larger radius rotates in the opposite direction.
r
ms
Z
2
(3)(3++2)
Z
1
Z
1
Z
2
Z
1
1/3
1
2
a
2
M
1/3
1+
a
M
1/3
1
a
M
Z
2
3
2
a
2
M
2
Z
1
The direct () and retrograde (+) photon orbits are defined via =2M1+cosarccos.
r
po±
2
3
∓a
M
Here, the mass of the black hole is taken as .
M=1