The Rotating Black Hole

Syed Ali RazaGeneral Relativity Project

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

Kerr Metric in Boyer Lindquist coordinates

Angular Momentum Parameter

Boyer Lindquist Coordinates

- 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

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

- Coefficient of dr^2 is dependent on angular momentum

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

Second New Feature

- presence of the product

- This term is responsible for frame dragging

- the spacetime is swept around by the rotating black hole

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

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

- 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 in the opposite direction

- Inside the ergosphere light in either of the tangential direction would be dragged along the direction of the black hole

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

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

Plunging

- Near the non rotating black hole the simplest motion was a radial plunge, butwhat 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

- We can integrate and plot it

- At r = M, horizon, dr = 0 and so fixed radial motion

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

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:

Questions !