Unravelling the Mysteries of the Kerr Black Hole Ergosphere

[Marilyn67]

TL;DR Summary

The entrainment of frames of reference in the ergosphere of a Kerr black hole (Lense-Thirring effect) and the speed of light ?

I have a problem understanding what is going on in the region called the ergosphere of a "fast" Kerr black hole.

- Relativity teaches us that no frame of reference can have relative displacements greater than the speed of light, ok.
- The ergosphere of a fast Kerr black hole can contain light rays and even relativistic particles.

From the perspective of a distant observer, isn't there a problem with relative displacement greater than the speed of light ?
(here the speed composition does not apply, right ?)

Can we talk about time loops ?
I've never heard of it in this case, but why ?

What difference is there with the famous cylinder of Tipler (hypothetical, of infinite length, ok..) which authorizes the trip on the past ?

Thank you in advance for your answers !

Cordially,
Marilyn

 

[Dale]

Relativity teaches us that no frame of reference can have relative displacements greater than the speed of light, ok.

That is true globally in special relativity or only locally in general relativity.

From the perspective of a distant observer, isn't there a problem with relative displacement greater than the speed of light ?

No. That is non-local in GR so the limitation does not apply.

 

[Marilyn67]

Thanks @Dale,

I understood your answer.
I know that general relativity is very complex and very counterintuitive.

My second question concerns the hypothetical Tipler cylinder.
It seems to me that Tipler originally proposed a cylinder of finite length. (?)

Apart from length, what is the difference between the space-time of a Tipler cylinder (allowing CTCs) and the space-time of an extreme Kerr black hole?
What are the theoretical elements that should be added to an extreme Kerr black hole to obtain a tilting of the light cones allowing CTCs?

Physicists often speak of large masses and high rotational speeds when discussing the issue of CTCs.

I don't understand where the limit of the possible is (in theory of course).

 

[Dale]

According to a theorem by Hawking any closed timelike curve “time machine” must be either infinite or involve negative energy density

 

[Marilyn67]

Ok @Dale,

It's not won !

 

[PeterDonis]

Relativity teaches us that no frame of reference can have relative displacements greater than the speed of light

What do you mean by this?

The ergosphere of a fast Kerr black hole can contain light rays and even relativistic particles.

Yes. So can any region of spacetime.

From the perspective of a distant observer, isn't there a problem with relative displacement greater than the speed of light ?

Meaning, for particles inside the ergosphere? Not at all. The unusual thing about the ergosphere is that there are no static objects, i.e, no objects that are not rotating around the black hole. Any object inside the ergosphere must be rotating about the hole, in the same sense that the hole itself is rotating. As the object gets closer and closer to the hole's horizon, the range of allowed angular velocities around the hole gets narrower and narrower, centered on the angular velocity of the horizon itself.

 

[Marilyn67]

Thank you @PeterDonis, for your argued answer as usual

Cordially,
Marilyn

 

[mitchell porter]

This honors thesis talks about CTCs for both Kerr black hole and Tipler cylinder. You can find more about the Kerr case if you google 'Kerr metric CTC'.

 

[Marilyn67]

Thanks @mitchell porter,

Oddly, we are not talking about negative energy or infinite lengths in this document.

I read it carefully and it seems that the notion of CTC is approached only under the horizon.
I will reread this document in much more depth.

Thank you for this document !

Marylin

 

[Dale]

Oddly, we are not talking about negative energy or infinite lengths in this document.

Yes, I thought that was interesting too. I wonder how that fits in with Hawking’s theorem.

 

[Marilyn67]

Hello @Dale,

This study is completely beyond me.

We first speak of co-rotation and then of counter-rotation.
(it seems to me that a counter-rotation is impossible, even for light, right ?).

We are also talking about an effect that would be rather secondary.

The conclusion leaves the reader in doubt.
However, the study seems serious and very pointed.

I don't know what you understood.
I don't understand the conclusion (Chapter 7, p 47)

 

[PeterDonis]

I wonder how that fits in with Hawking’s theorem.

Which theorem of Hawking's do you mean?

The presence of CTCs in the extreme interior of the maximally extended Kerr solution and in the infinite cylinder van Stockum solution has been well known for decades and doesn't contradict any of the classical GR theorems. IIRC Hawking & Ellis discusses the Kerr case.

 

[PeterDonis]

This study is completely beyond me.

The thesis is certainly talking about highly counterintuitive spacetime geometries. However, there is no doubt about the presence of CTCs in the two spacetime geometries considered.

For the Kerr case, bear in mind that the CTCs are only present in the extreme interior (well inside the inner horizon), and do not allow you to go inside the hole, "time travel" to some time in the far past, and then come back out again into the external universe long before you went in.

For the van Stockum cylinder, AFAIK CTCs are only known to exist if the cylinder is infinite. The thesis mentions that Tipler conjectured that a finite cylinder with sufficient rotation would also cause CTCs in the exterior vacuum around it, but AFAIK that conjecture has never been proved. (The analogy mentioned with Kerr spacetime in the thesis seems very weak to me since Kerr spacetime is a vacuum solution, whereas the presence of the CTCs in the van Stockum solution depends on the presence of stress-energy in the rotating cylinder.)

 

[Marilyn67]

Hello @PeterDonis

For the Kerr case, bear in mind that the CTCs are only present in the extreme interior (well inside the inner horizon)

That's what seemed to me on first reading, but the conclusion is more evasive, and I don't understand what side effect the conclusion is talking about.

I found a link (sorry, it's in French), and I don't know if it's consistent (page 2) :

http://nrumiano.free.fr/Fetoiles/int_noir2.html

 

[PeterDonis]

the conclusion is more evasive

What "conclusion" are you talking about? Can you give a specific reference (page number/quote)?

 

[Marilyn67]

Yes, chapter 7, page 47, of the thesis

 

[PeterDonis]

chapter 7, page 47, of the thesis

What conclusion stated there do you think is "more evasive"?

Note, also, that this is just a thesis, not a textbook or peer-reviewed paper. Theses are reviewed, but not to the same standards that textbooks or peer-reviewed papers are. There is no dispute whatever about the fact that CTCs exist in the extreme interior of the Kerr spacetime geometry or in the infinite van Stockum cylinder geometry.

 

[Dale]

Which theorem of Hawking's do you mean?

The presence of CTCs in the extreme interior of the maximally extended Kerr solution and in the infinite cylinder van Stockum solution has been well known for decades and doesn't contradict any of the classical GR theorems. IIRC Hawking & Ellis discusses the Kerr case.

I don’t have details of the theorem. Apparently it was in Hawking’s famous chronology protection conjecture paper, although it is not actually part of the conjecture itself.

I don’t have the paper, so my knowledge of this is entirely second-hand through Wikipedia. My understanding is that the proof is that any finite time machine must violate the weak energy condition.

 

[PeterDonis]

any finite time machine must violate the weak energy condition

Ah, that theorem. I believe it appears in Hawking's "chronology protection conjecture" paper, which unfortunately I have not been able to find online.

The van Stockum infinite cylinder obviously doesn't violate the theorem because it's infinite. I believe the reason Kerr doesn't violate the theorem is that it also counts as "infinite" as that term is defined in the premises of the theorem (basically, IIRC, because to get to the deep interior of Kerr you have to be working with the maximal analytic extension, and that extension doesn't have just one deep interior region but an infinite number of them), but I would need to take some time to refresh my memory about the details.

 

[Marilyn67]

Thank you all for your very precise answers !

I will reread this thesis with a clear head.

Cordially,
Marilyn