Black Holes: Accretion Disks /Increase/ Angular Momentum?

[MattRob]

I've been reading Kip Thorne's "Black Holes and Time Warps," and it mentioned something rather counter-intuitive; apparently, when material forms an accretion disk and falls into a spinning black hole, it increases the angular momentum of it.

Now, let's take a gas cloud, and put a spinning black hole in it.

The gas cloud has zero angular momentum as a whole ([itex]L_{cloud} = 0 [/itex]), and the black hole itself has [itex]L_{hole}[/itex] angular momentum.

From what I know of classical mechanics, the momentum of the system shouldn't change; once the black hole accretes all the gas of the cloud, it will, according to classical mechanics, spin slower, and the total momentum of the system will stay the same.

[itex]∑L_{final} = L_{cloud} + L_{hole} = L_{hole} = ∑L_{initial} [/itex]
by classical mechanics.

So, am I understanding this correctly, that that classical mechanics' description of the system doesn't agree with General Relativity, and that a full GR description of the same spinning black hole and gas cloud will end with the entire system having more angular momentum?

[itex]∑L_{final} > L_{hole initial} + L_{cloud initial} = L_{hole initial} = ∑L_{initial}[/itex]
by General Relativity.

If that's so, then how? The book didn't go into detail much on how that's the case, but I find it really odd/interesting and would like some insight on how it happens.

 

[pervect]

So, am I understanding this correctly, that that classical mechanics' description of the system doesn't agree with General Relativity, and that a full GR description of the same spinning black hole and gas cloud will end with the entire system having more angular momentum?

No. The full GR description will make calculating the angular momentum much more problematical, though it can be defined in asymptotically flat space-times. But once you get over the hurdle of actually defining the angular momentum, it should be conserved. In some situations, such as inpiraling binaries, gravitational waves (GW) can carry away angular momentum, so you might have to add the GW angular momentum into the total to get a conserved number.

 

[MattRob]

No. The full GR description will make calculating the angular momentum much more problematical, though it can be defined in asymptotically flat space-times. But once you get over the hurdle of actually defining the angular momentum, it should be conserved. In some situations, such as inpiraling binaries, gravitational waves (GW) can carry away angular momentum, so you might have to add the GW angular momentum into the total to get a conserved number.

Okay, I'm re-reading this bit again, I think my original bafflement confused me from understanding it properly...

For context, this portion of the book is on quasars and their discovery.
To quote;

How strong will the swirl of space be near the gigantic [black] hole? In other words, how fast will gigantic holes spin? James Bardeen deduced the answer: He showed mathematically that gas accreting into the hole from its disk should gradually make the hole spin faster and faster. By the time the hole has swallowed enough inspiraling gas to double its mass, the hole should be spinning at nearly its maximum possible rate - the rate beyond which centrifugal forces prevent any further speedup (Chapter 7). Thus, gigantic holes should typically have near-maximal spins.

-Black Holes and Time Warps by Kip Thorne

I guess my question then is simply how? There must be a lot of interesting interactions going on in the accretion disk in terms of GWs, then?

 

[pervect]

I haven't seen any detailed calculations, but my understanding is that there is some residual angular momentum in the interstellar media, it's not zero due to tiny imbalances. This is similar to the solar system, the sun and the planets all have nonzero spins, though the condensed gravitationally from the interstellar media. This process was rather similar in its major outline to the black hole case, t's just in the black hole case, the collapse radius is smaller, so the spin is faster.

The paper by Bardeen might have more info. I'm afraid I don't know much more possibly someone else can help.

 

[sardar]

I can confirm that your understanding is correct. Classical mechanics does not fully describe the behavior of black holes, and General Relativity provides a more accurate explanation of the phenomenon you have described.

In classical mechanics, the total angular momentum of a system is conserved, meaning it cannot change. However, in General Relativity, the concept of angular momentum is more complex and is not necessarily conserved. When matter falls into a spinning black hole, it can increase the overall angular momentum of the system, as you have correctly stated.

This is due to the effects of spacetime curvature near the black hole. As matter falls towards the black hole, it is accelerated and gains energy. This energy can be converted into angular momentum, resulting in an increase in the overall angular momentum of the system.

Additionally, the accretion disk itself can also contribute to the increase in angular momentum. As matter orbits closer and closer to the black hole, it experiences a phenomenon known as frame dragging. This causes the matter to rotate at the same rate as the black hole, resulting in an increase in angular momentum for the entire system.

Overall, the behavior of black holes and their accretion disks is complex and cannot be fully explained by classical mechanics. General Relativity provides a more accurate and comprehensive understanding of these phenomena. I hope this helps to provide some insight into the counter-intuitive behavior you have encountered.