Syed Ali Raza
General Relativity Project
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Kerr Metric for rotating Black holes
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Kerr Metric for extreme angular momentum
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Static Limit and the ergosphere
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Radial and Tangential motion of light
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Comparison of non-spinning and extreme-spin black holes
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Plunging
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Negative Energy
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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, 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
- 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 !