The Schwarzschild reference frame is static outside the Black Hole, so even though space is curved, and time is slowed down close to the Black Hole, it is much like the absolute space of Newton. But we will need a generalized reference frame in the case of rotating Black Holes. Roy Kerr generalized the Schwarzschild geometry to include rotating stars, and especially rotating Black Holes. Most stars are rotating, so it is natural to expect newly formed Black Holes to process significant rotation too.
In Boyer-Lindquist coordinates the Kerr metric
is written:
where the coordinate functions are given (with G=c=1):
the specific angular momentum is:
The physical value of J is for a star like the sun:
corresponding to a=0.185 M.
If a=0 we have the Schwarzschild case for a nonrotating Black Hole (or star).
We define FIDucial Observers (FIDOs) as little (experimental) physicists locatedat each point in spacetime measuring all possible physical quantities in their local proper units. They'll get a hard job in the Kerr geometry. To keep their job, they have to follow the geometry which actually rotates with increasing speed towards the center. How can this be? All physical objects are dragged into circular motion by the Black Hole's rotation. Our FIDOs (which are supposed to be at rest) will follow the (absolute) space around the rotating hole. The Boyer-Lindquist coordinates naturally includes this rotating coordinate system, so in the Kerr reference frame, the geometry actually swirls like the air in a tornado. The angular velocity of a FIDO as viewed from infinity is:
Figure 8: The coordinate system rotates with the hole (because of 
).
One straight radius is deformed into a spiral after some time. From left to right:
a=0,0; a=0,5 and a=1,0.
The redshift factor 
is given by the coordinate functions:
The metric has some general properties which makes it different from the Schwarzschild
metric. It has two horizons (where 
and 
becomes singular), and
a static limit inside which nothing can remain at rest:
and the static limit is called the
ergosphere.
The Kerr horizon has a different value than the Schwarzschild horizon.
As the rotation increases, the inner horizon increases (from 0 to 1M) and the outer
horizon decreases (from 2M to 1M). They are coinciding when a=1.
The static limit depends on 
and defines a surface which is attached to
the horizon at the poles. In the equator plane ( 
) 
.
Figure 9: The curvature in Kerr geometry depends on a. The horizon
decreases from 2M to 1M.
The transition from one 3-D space section to another is carried out in the Schwarzschild reference frame by shifting along the time lines orthogonal to the spatial sections. In the Kerr reference frame it is more complicated, and handled by the Killing vector.
When looking at just one slice of space, we are again interested in the curvature close to
the Black Hole. I will not try to calculate the curvature tensor of the Kerr metric. I will just assume, that
the shift in gravitational radius: 
holds in the equator plane.
If we make this assumption, the curvature is:
The equations of motion of particles that move along geodesics are much more complicated in the Kerr reference frame because the lack of symmetry. With the computer program used earlier I will limit the investigation to numerical calculations. If we perform the same experiment as in the Schwarzschild case - sending lightrays towards the Black Hole with different impact parameters - we observe the symmetry is broken: Figure 3.3 shows 6 values of a:
Figure 10: Different impact parameters. The rotation parameter a goes from 0 to 1
in steps of 
. In the first plot (a=0), the geodesics are symmetric from left
to right, and stops at the horizon 
. By increasing a, the symmetry breaks, and the
rotation of the black hole (rotating clockwise) "drags" the light in the same direction as
the spacetime right outside the hole's horizon. The horizon is 
for a=1.