How to "overspin" a rotating black hole? If the value of the parameter
a = J / M
in the Kerr metric exceeds a certain threshold, then the event horizons disappear. In the formula, M is the ADM mass of the black hole and J is its angular momentum.
Various authors have tried to prove that you cannot over-spin a black hole by throwing light or other material into it.
Let us have a black hole for which the quantity J / M is at the threshold value. Such a black hole is called extremal.
In the link we have a paper, probably written by Edwin F. Taylor, where Figure 2 seems to indicate that the tangential speed of light to the spinning direction of an extremal black hole is less than the speed of light far away.
Pulling a Schwarzschild black hole with a rope
___
/ \ __________ rope
\___/ -->
black hole F
Let us lower a strong rope very close to the horizon of a Schwarzschild black hole.
The radial metric is stretched, but that does not affect our reasoning. The speed of light is very slow at the left end of the rope, as measured by a faraway observer. The end of the rope is "stuck in the syrup" of the gravity field of the black hole.
Let us pull the rope with a large force F for a time t. We give to the system the black hole & the rope a momentum
p = F t.
We give it the energy
E = F v t,
where v is the average speed of the right end of the rope. How is the momentum distributed between the black hole and the rope?
Let the mass of the rope be m. Since the black hole weighs a lot, the rope receives almost all of the energy E.
Let v' be the final velocity of the rope when it is completely far away from the black hole. We have
E = F v t > 1/2 m v'²,
because the rope had some negative potential energy.
p' = m v' < sqrt( 2 F v t m ).
If v is "small", then the black hole receives almost all of the momentum p.
We get another estimate from the fact that v' < c. The final momentum of the rope is at most
p' < m c + F t v / c.
If m is very small, then p' / p < v / c. If such a lightweight, very strong rope exists, we can use it as a tether to pull a black hole around.
Let us analyze what is the fundamental mechanism of pulling the black hole. The end of the rope acquires a lot of inertia from the black hole. When we pull the rope, we pull that inertia, too. When the rope moves away, the momentum absorbed by that inertia remains with the black hole.
Accelerating the spin by pulling on ropes whose end is close to the black hole
Let us put ropes around the black hole and pull them very fast, at almost the speed of light (the light speed far away).
F
---> pull
----------------------- rope
r
● rotating extremal black hole
<-- direction of rotation
rope ------------------------
pull <---
F
Does the field of the black hole resist the pull? We are not sure. The radial metric is stretched by the black hole, and clocks tick slower. Does this contribute some resisting force for the pulling?
Let us assume that the black hole resists the pull of each rope with a force F.
Let us calculate the mass-energy and the angular momentum which we input to the system black hole & its gravity field.
The mass-energy m which we input is approximately
m = 2 F c t / c² = 2 F t / c,
where c is the speed of light and t is the time which we pull.
The angular momentum j which we input is approximately
j = 2 F t r,
where r is the distance of rope ends from the black hole. The ratio
j / m = c r.
By choosing r large we can make the ratio as large as we like.
Does this guarantee that we can push the ratio J / M of the whole system over the threshold?
Not necessarily. As we speed up the system, it may lose some angular momentum in gravitational waves. We need to analyze this in more detail.
Lowest orbits around an extremal spinning black hole touch the horizon: can we disassemble the black hole?
According to Wikipedia, the innermost stable circular orbit (ISCO) touches the horizon of an extremal black hole.
Maybe adding still more spin, making the system superextremal, then causes particles to fly away from the forming horizon?
We have been claiming that the event horizon of a Schwarzschild black hole is never "completed" because clock slow down so much when close to the forming horizon.
In principle, it should be possible to reverse the process and make the falling matter to fly away. Over-spinning an extremal black hole might do the trick.
Once the topmost matter flies away, lower layers of the forming black hole are exposed. If we continue over-spinning the system, we may be able to disassemble the forming black hole entirely.
Conclusions
We need to study overspinning in detail. Is the excess angular momentum carried away in gravitational waves?
If over-spinning really is a mechanism to disassemble a black hole, that refutes black hole thermodynamics which was developed by Jacob Bekenstein and others in the early 1970s. The entropy of a black hole can be quite low if it is built from large chunks of matter.
No comments:
Post a Comment