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The region beyond the ring singularity in the maximal Kerr spacetime is described as having closed timeline curves. Why and/or how is the question.enter image description here

Now if you look a Kruskal-Szkeres Diagram (or a Penrose Diagram as above) you can see that the Kerr singularity (right) is timelike but the Schwarzschild singularity is spacelike.

Inside the Schwarzschild event horizon curves with constant longitude, latitude, and areal radius are actually spacelike so areal radius is actually a time direction. So you could claim that since r is the timelike direction there are curves that start and stop at the same t (since t is a spacelike direction) but I've never seen anyone claim there are CTC inside the event horizon of a Schwarzschild solution.

And even if we interpreted it like that, the region in Kerr where r (not areal r in Kerr, but the usual r for Kerr) is timelike is the region between the two horizons. And the lines making an X to the right of our universe are the outer horizon (see Penrose diagram) whereas the lines making an X to the left of the right most singularity is the inner horizon so the region between where r is timelike isn't connected to the singularity except in its infinite past (where we won't go).

So over where the singularity is, the singularity is a vertical line and is r=0 so r looks pretty spacelike there. We can avoid the singularity since that vertical line is r=0 which includes the whole disk that has the ring as its edge.

So we can get to the region the diagram labels as the weird space. And people usually just cite Hawking and Ellis for the existance of closed timelike curves instead of working it out but on page 164 the existence of closed timelike curves is asserted, but then it seems like it is just a discussion of the ergoregion and the two horizons, but I don't see any more mention of closed timelike curves until the section on Gödel's solution which is a different solution, not the Kerr solution.

So I'd like to know why and/or how there are closed timelike curves in the negative r region of the Kerr solution. And if someone knows why people cite Hawking and Ellis for that fact that would be interesting too.

Timaeus
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  • I am sure you are aware that none of the objects you are categorizing here is actually physically realized? They are all just different border cases of how general relativity breaks and we don't have the slightest idea of how to repair it correctly. Just in case you weren't aware, I think you may want to think about that a little before you go off to create a new theory that mathematically models the breakdown of another... rather than the actual behavior of nature. – CuriousOne Jun 13 '15 at 05:40
  • @CuriousOne What really kept me thinking about this was how many sources cited Hawking and Ellis but I couldn't find it there. This happens to me frequently and if I can get enough data to find out if it is my fault somehow I'd want that data. And I see situations where CTCs can develop but don't have to, cases where they can be forced to occur if you restrict to analytic solutions but can be avoided if you use non analytic solutions and part of me even wondered about terminology if by timelike they meant t direction or if they meant tangents having the sign that is the minority in the metric. – Timaeus Jun 13 '15 at 05:49
  • @CuriousOne Learning to read the literature, compare and contrast and communicate with standard terminology are all important things in science inasmuch as science is a group endeavour. Sure, understanding the universe is awesome too, but I'd want to have the skills to communicate my ideas to others whether it is about this or any other topic and to understand what people intended to communicate. – Timaeus Jun 13 '15 at 05:52
  • Solutions to equations are solutions to equations. They are not automatically physics. – CuriousOne Jun 13 '15 at 05:52
  • @CuriousOne If I thought what goes inside horizons was useful I'd have probably already figured this out. But I'm more fascinated by why people repeatedly cite something that I can't find in the citation and I think that I can learn something from people that have already studied it. And after I learn it I might then know enough to learn why people cited it the way they did, thus learning what I wanted to know, which is about methods of citation. The specific reference tag didn't seem quite right since I want to know why they cite not what they cite but I do want to know why they cite that way – Timaeus Jun 13 '15 at 05:59
  • I am mostly fascinated by people who talk about physics rather than about themselves, but tastes differ. – CuriousOne Jun 13 '15 at 06:04
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    I don't see anything wrong with this question. The Kerr metric is a solution to Einstein's equations and it seems to me perfectly proper to ask about the properties of the solution. To downvote or VTC this question on the grounds that the Kerr metric doesn't reflect reality is making the (rather arrogant) assumption that you know what reality is. – John Rennie Jun 13 '15 at 06:19
  • @JohnRennie The objection could be that I asked why people say it has CTCs rather than directly asking about the Physics. And while the Kerr solution is a legitimate topic of study going beyond an event horizon is less physical and beyond both horizons means you are in the causal future of a singularity so even less physical and through the ring is in a region of CTCs so even less physical for even more reasons. And I could have owned that up front, so maybe it is my fault. I didn't mean to mislead anyone, I just haven't figured out how to say all that without distracting from the question. – Timaeus Jun 13 '15 at 06:27
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    @Timaeus: the Penrose diagrams show only the $u$ and $v$ coordinates and ignore the angular coordinates. The CTCs (one class of them at least) require moving in a ring in the equatorial plane, i.e. constant $u$ and $v$ at $\theta = \pi/s$, so you can't plot them on the diagram you show. On your diagram they would be represented by a single point. – John Rennie Jun 13 '15 at 06:53
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    @Timaeus: It is at the bottom of page 162. – MBN Jun 13 '15 at 07:31
  • Timaeus: "[...] a Penrose Diagram as above [...] but the Schwarzschild singularity is [...]" -- I note that the diagram you included contains a mis-spelling of the surname of Karl Schwarzschild. Please consider including the diagram in editable form, e.g. using the appropriate MathJax commands, so it may be edited accordingly. (Also, this might help in distinctly denoting certain vertices in the diagram, for further reference.) – user12262 Jun 13 '15 at 07:34
  • @JohnRennie: I know what physicists have measured and what they have not. Nothing in this diagram has ever been measured. It's pure fiction as far as physics is concerned, moreover, it's non-testable fiction, which makes it not even false, but it makes it not even science. I would say the same thing to Hawking or Penrose if they would have posted this here. Fair? – CuriousOne Jun 13 '15 at 08:01
  • @CuriousOne: Are you saying that the question is not on topic for this site? – MBN Jun 13 '15 at 08:29
  • @MBN: I am saying that it is necessary to point out that it's not even false. Unfortunately, that doesn't make it true. While there is nothing wrong with elaborating on the non-physical solutions of Einstein's equations, doing so doesn't make them physical. If somebody asks for the meaning of the infinite self-energy of a classical point charge, what's the correct answer? It's that the theory breaks down. If somebody asks for the meaning of the singularity in black holes, what's the correct answer? It's that the theory breaks down. Sue me if I am wrong about that. – CuriousOne Jun 13 '15 at 08:35
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    @CuriousOne: The question is very clear, where are the CTC in the Kerr solution and why people cite Hawking and Ellis. The theory doesn't break down, it gives a logically consistent answer. I don't see what you rant is all about! – MBN Jun 13 '15 at 08:53
  • @MBN: I am looking forward to your citation of the experimental confirmation of the Kerr solution. – CuriousOne Jun 13 '15 at 09:32
  • @CuriousOne: Let me ask you again. Do you think this question is not suitable for this site? Do you post similar comments on all questions about black holes? What about topics as string theory? – MBN Jun 13 '15 at 12:17
  • Let us continue this discussion in chat. – MBN Jun 13 '15 at 13:33
  • @MBN: Let me ask you, again, for citations of experimental or observational black hole papers that contain data with which we can decide whether this is just intellectual nonsense, or not. – CuriousOne Jun 13 '15 at 18:53

3 Answers3

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I"m pretty sure that this discussion does appear in Hawking and Ellis, though I admit that it's been a while since I looked. It's not done through a Penrose diagram, though.

The argument really comes down to the fact that for sufficiently small $r$, $d\phi$ is timelike. But, by construction, the orbits of $\phi$ are closed curves. When $d\phi$ is spacelike outside the horizon, this just generates the axisymmetry of the Kerr solution. But, for these small values of $r$, it becomes timelike, and therefore, these orbits represent closed timelike curves. You avoid intersection with the horizon so long as your value of $\theta$ doesn't put you on the same plane as the ring singularity, so these curves are not incomplete geodesics or anything like that.

You can't see this on any of those penrose diagrams, because they all suppress $\theta$ and $\phi$

Zo the Relativist
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  • Oh yes, the $\phi$ orbit is really the most elegant way to do this. It should be just noted that $d \phi$ can be timelike only for $r<0$ and some $\theta$ reasonably around $\pi/2$. – Void Jun 13 '15 at 17:24
  • It is at the bottom of page 162. – MBN Jun 13 '15 at 19:32
  • The claim that it is only in the case of negative r that dϕ can be timelike and CTCs are present is true only for the simple Kerr spacetime. In the Kerr-Newman spacetime, they are already realized in certain positive r ranges. @Void – Attila Janos Kovacs Nov 18 '22 at 23:57
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This is really one google search away, see e.g. page 26 (marked 64) here.

As already noted by John Rennie, Penrose diagrams are not suited for the analysis of Kerr CTCs because they show a $\phi = const., \theta=\pi/2$ slice of the global structure. The $r<0$ region is however accessible only through $\theta \neq \pi/2$. The Boyer-Lindquist coordinates actually misrepresent the central singularity but you can see the singularity "unwrapped" locally by understanding Boyer-Lindquist as oblate spheroidal coordinates.

The $r<0$ region can be essentially covered by the $r>0$ Kerr metric with $M \to -M$. Here you find cases where the $g_{tt}>0$ and you can thus choose a time-like four-velocity to point in the negative time direction with respect to $r \to \infty$. It is obvious that curves that spend some time in this region and then go "outside" towards $r \to \infty$ can be CTCs.

The Gödel solution is so often cited in this context because it is the historically first solution in which this rather unsatisfactory possibility of relativity was shown and discussed.

Void
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  • I think your answer's the best and upvoted it, but, re your 3rd paragraph, wouldn't it be more correct to say that CTCs, being timelike, occupy some space in the Kerr BH (where they're "closed") and extend somewhere else as well? – Edouard Oct 09 '18 at 16:48
  • @Edouard CTCs means "closed time-like curves". In the usual sense of the word, you cannot close a curve and then extend it somewhere else. – Void Oct 10 '18 at 07:39
  • I'd meant "closed" as an adjective, not verbally. I'm picturing the universe as past- and future-eternal, with that balancing (required, for past eternality, by the BGV Theorem) between expansion and contraction being maintained by changes in the scale of space and time, and resulting in an impossibility of distinguishing between them from within any of its iterations, that would be temporal but might be represented as concentric within each other. Like Kerr, I'm considering each rotating BH's inner event horizon to be a singularity. – Edouard Oct 12 '18 at 16:32
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Timaeus : I can't answer your question, but would like to comment. However what I want to say is to big for a comment, so I'm using the answer facility. Apologies in advance, feel free to downvote.

The region beyond the ring singularity in the maximal Kerr spacetime is described as having closed timelike curves.

Let's imagine we're out in space, at a safe distance from a non-spinning black hole. What's the speed of light at the event horizon? Zero. From where we're standing the the "coordinate" speed of light is zero. (See John Rennie saying that here). Now let's say our black hole spins at half the speed of light. But half of zero is zero. So it isn't spinning. Or it's spinning infinitely faster than light. Something's not right. Something's not right with the closed timelike curve too. See this page of Palle Yourgrau's book A World Without Time: The Forgotten Legacy of Gödel and Einstein:

"Wheeler, unfortunately has conflated a temporal circle with a cycle, precisely missing the force of Gödel's conclusion that the possibility of closed, future-directed, timelike curves, ie time travel, proves that space-time is a space, not a time in the intuitive sense. Whereas a circle is a figure in space, a cycle is a journey undertaken along a circular path, one that can repeated, in Wheeler's words, "over and over again". Exactly how many times, one wants to ask Wheeler, is the journey supposed to be repeated? The question clearly cannot be answered, since the time traveller's journey is not over time, along the closed timelike curve: it is the curve itself".

The moot point is that you don't travel along a worldline. You travel through space over time. The worldline is a static representation of this. So a CTC doesn't represent time "travel". So what does it represent? Groundhog Day? No. If your CTC worldline is 24 hours long it's more like Mayfly Day. Your life is 24 hours long and it's causeless, you hatch from your own egg or something. Much as I love all those science fiction movies, I'm afraid time travel is a fantasy.

Let's move on to the Penrose diagrams, wherein "space is uni-directional within the horizon, just as time is uni-directional outside the horizon)". Whoa, hang on a minute. Einstein said a gravitational field is a place where "the speed of light is spatially variable". And if the speed of light is zero at the event horizon, how can it go slower than that? And since nothing can go faster than light, how can an object ever pass the horizon? And how can we say it will inevitably hit the singularity if this only occurs at future infinity? And where has this unidirectional space come from? A gravitational field is not a place where space is moving inwards, we do not live in some Chicken-Little world. A gravitational field is a place where space is "neither homogeneous nor isotropic". So what's all this about connecting two separate universes? Where does all this Schwarzschild wormhole stuff come from, wherein "particles from the interior white hole region can escape into either universe"? From Einstein's 1935 paper with Rosen? No way. Einstein refers to a singularity at r=2M, at the event horizon. That's where the gravitational field ends, because light can't go slower than stopped. That's the boundary of the space. And this paper is The Particle Problem in General Relativity. It's about particles like the electron, and how they can't be point-particle singularites. It isn't about wormholes to another universe. How did we get to an anti-gravity universe, and to three other universes? Methinks what we have here contradicts Einstein whilst appealing to his authority, and that it isn't just time travel that's a fantasy.

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