1. The Rotating Black Hole
Syed Ali Raza
General Relativity Project

2. Overview Kerr Metric for rotating Black holes
Kerr Metric for extreme angular momentum
Static Limit and the ergosphere
Radial and Tangential motion of light
Comparison of non-spinning and extreme-spin black holes
Plunging
Negative Energy
Penrose process

3. Kerr Metric in Boyer Lindquist coordinates
Angular Momentum Parameter
Boyer Lindquist Coordinates

4. Metric only holds in the equatorial plane.
Exact analytic solution, describes the rotating black hole completely
Kerr metric in the limit of zero angular momentum reduces to the schwarzschild metric
Schwarzschild Metric for stationary Black Hole
Kerr Metric for rotating Black Hole

5. First New Feature
Calculate horizon radius , point of no return
When the coefficient of dr^2 blows up
Coefficient of dr^2 is dependent on angular momentum
Non rotating black hole r_H = 2M
Rotating black hole has two singularities

6. The Kerr Metric for extreme angular momentum
For a real value of r_H, the maximum value can be a=M
Maximum angular momentum J=M^2
Only one value of r_H, inner and outer horizons have merged
The Kerr metric for a = M

7. Second New Feature
presence of the product
This term is responsible for frame dragging
the spacetime is swept around by the rotating black hole

8. Third New Feature (Static Limit)
Coefficient of dt^2 goes to zero
The static limit is r_S = 2M
it is independent of the angular momentum parameter a
The static limit gets its name from the prediction that for radii smaller than r_S,
but greater than that of the horizon r_H, the observer cannot remain at rest.
- Ergosphere

9. We will look at the behavior of light in ergosphere:
Flash light in the tangential direction (dr = 0)
For light, proper time on adjacent events of the path is zero
Divide metric by dt^2

10. - At the static limit, we have two solutions
Light flashed in the direction of rotation will move at the speed of 2nd solution
Light flashed in the other direction remains stationary
The dragging of the hole has become so strong that even light can not move
in the opposite direction
The dragging of the hole has become so strong that even light can not move
Inside the ergosphere light in either of the tangential direction
would be dragged along the direction of the black hole

11. Fourth New Feature
Extract energy from spinning black hole
Rotating black hole loses mass and angular momentum to become a
stationary black hole.
Can not extract energy from non rotating black hole
Discussed the Penrose process later

12. Radial and Tangential motion in light
Radial motion of light
At static limit
Zero at the static limit
Imaginary at radii less than the static limit
Which means no real radial motion is possible inside the ergosphere
Tangential velocity of light

15. Plunging
- Near the non rotating black hole the simplest motion was a radial plunge, but
what about in the spinning black hole's case
By setting the angular momentum term in the table equal to zero
- This shows that even for a particle with zero angular momentum, when it
comes near a spinning black hole, it circulates around it
- By setting E/m= 1 for a stone with zero angular momentum we get the following

16. We can integrate and plot it
At r = M, horizon, dr = 0 and so fixed radial motion

17. Negative Energy: The Penrose Process
a lump of matter enters into the ergosphere of the black hole, and once it enters
the ergosphere, it is split into two
The momentum of the two pieces of matter can be arranged so that one piece escapes
to infinity, whilst the other falls past the outer event horizon into the hole
The escaping piece of matter can possibly have greater mass-energy than the original
infalling piece of matter, whereas the infalling piece has negative mass-energy
- The process results in a decrease in the angular momentum of the black hole, and
that reduction corresponds to a transference of energy whereby the momentum
lost is converted to energy extracted
if the process is performed repeatedly, the black hole can eventually lose all of its
angular momentum, becoming non-rotating, i.e. a Schwarzschild black hole

19. We saw that the energy can be negative
Now we want to find conditions at which the energy is negative
We first do it with E/m = 0
For r > 2M, angular velocity is negative
For r< 2M, angular velocity is positive
Corresponding tangential velocity
By plugging back and comparing, the condition for negative energies is:

21. Questions !