The Rotating Black Hole

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

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 • Coefficient of dr^2 is dependent on angular momentum

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 • in the opposite direction • Inside the ergospherelight 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 • infallingpiece 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 !