Closed timelike curves in the region beyond the ring singularity in the maximal Kerr spacetime

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Solution 1

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.

Solution 2

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 $\phi$

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Updated on June 14, 2020

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  • Timaeus
    Timaeus over 3 years

    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.

    • CuriousOne
      CuriousOne over 8 years
      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.
    • Timaeus
      Timaeus over 8 years
      @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
      Timaeus over 8 years
      @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.
    • CuriousOne
      CuriousOne over 8 years
      Solutions to equations are solutions to equations. They are not automatically physics.
    • Timaeus
      Timaeus over 8 years
      @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
    • CuriousOne
      CuriousOne over 8 years
      I am mostly fascinated by people who talk about physics rather than about themselves, but tastes differ.
    • John Rennie
      John Rennie over 8 years
      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.
    • Timaeus
      Timaeus over 8 years
      @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.
    • John Rennie
      John Rennie over 8 years
      @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.
    • MBN
      MBN over 8 years
      @Timaeus: It is at the bottom of page 162.
    • user12262
      user12262 over 8 years
      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.)
    • CuriousOne
      CuriousOne over 8 years
      @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?
    • MBN
      MBN over 8 years
      @CuriousOne: Are you saying that the question is not on topic for this site?
    • CuriousOne
      CuriousOne over 8 years
      @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.
    • MBN
      MBN over 8 years
      @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!
    • CuriousOne
      CuriousOne over 8 years
      @MBN: I am looking forward to your citation of the experimental confirmation of the Kerr solution.
    • MBN
      MBN over 8 years
      @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
      MBN over 8 years
    • CuriousOne
      CuriousOne over 8 years
      @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.
  • Void
    Void over 8 years
    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 $\pi/2$.
  • MBN
    MBN over 8 years
    It is at the bottom of page 162.
  • Edouard
    Edouard about 5 years
    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?
  • Void
    Void about 5 years
    @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.
  • Edouard
    Edouard about 5 years
    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.