Properties of Kerr Spacetime

a

0.5

specific radii
static limit r 0 θ = π 2	=	2.
event horizon r +	=	1.86603
inner horizon r -	=	0.133975

marginally stable orbits
rotation	=	4.233
counter-rotation	=	7.55458

photon orbits
direct	=	2.3473
retrograde	=	3.53209

Kerr spacetime describes a rotating black hole. The line element in Boyer–Lindquist coordinates

(t,r,θ,ϕ)

is given by

d

2

s

=-1-

2Mr

Σ

d

2

t

-

4Mar

2

sin

θ

Σ

dtdϕ+

Σ

Δ

d

2

r

+Σd

2

θ

+

2

r

+

2

a

+

2M

2

a

r

2

sin

θ

Σ

2

sin

θd

2

ϕ

with

Δ=

2

r

-2Mr+

2

a

and

Σ=

2

r

+

2

a

2

cos

θ

.

M

is the mass of the black hole and

a

is the angular momentum.

The roots of the function

Δ

define the event horizon

r

+

and the inner horizon

r

-

, where

r

±

=M±

2

M

-

2

a

. The region between the event horizon and the static limit

r

0

=M+

2

M

-

2

a

2

cos

θ

is called the ergosphere.

The two marginally stable timelike circular geodesics are defined by the radii

r

ms

=M3+

Z

2

∓

(3-

Z

1

)(3+

Z

1

+2

Z

2

)



, where

Z

1

=1+

1/3

1-

2

a

2

M

1/3

1+

a

M

+

1/3

1-

a

M

and

Z

2

=

3

2

a

2

M

+

2

Z

1

. An object on the smaller radius rotates with the Kerr black hole, whereas an object on the larger radius rotates in the opposite direction.

The direct (-) and retrograde (+) photon orbits are defined via

r

po±

=2M1+cos

2

3

arccos

∓a

M

.

Here, the mass of the black hole is taken as

M=1

.