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Study materials about Closed timelike curves (CTCs)

  1. Jun 1, 2012 #1
    I'm looking for some study materials regarding Closed time-like curves (CTCs). Be it a book, paper or anything other. It is highly accepted, but I'm particularly looking for a book that includes it and similar topics.
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  3. Jun 1, 2012 #2

    George Jones

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    At the popular level, you might be interested in an excellent, non-technical reference on time travel, the second edition (make sure that it's the second edition) of Time Machines: Time Travel in Physics, Metaphysics, and Science Fiction by Paul Nahin. This is a wonderful book that is written for the educated layperson.


    Physicist (and relativist) Kip Thorne wrote a foreword for the second edition of this book, and here's a quote from this foreword: "It now is not only the most complete documentation of time travel in science fiction; it is also the most thorough review of serious scientific literature on the subject - a review that, remarkably, is scientifically accurate and at the same time largely accessible to a broad audience of nonspecialists."

    More technical is

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  4. Jun 1, 2012 #3
    Thank you very much George Jones! :)
  5. Jun 1, 2012 #4


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    Chronology condition: There are no closed timelike curves.

    Causality condition: Closed null curves can exist even when the chronology condition is satisfied, this motivates the definition of the causality condition. The causality condition is satisfied when there are no closed causal curves.

    Strong causality: the spacetime is not on the verge of violating the causality condition. This is of particular importance for the singularity theorems.

    Future (past) distinguishing: [itex](M, g)[/itex] is future distinguishing at [itex]p \in M[/itex] if [itex]I+(p) \not= I+(q)[/itex] for [itex]q \not= p[/itex] and [itex]q \in M[/itex].

    There are other ways for a spacetime to have causal problems, in fact there is a whole hierarchy of causality conditions.

    Stable causality is the mother of all causality conditions as it prohibits all other causality conditions (due to Hawking). A spacetime is stably causal if an arbitrary small perturbation of the metric doesn't result in a violation of the chronology condition.

    Books.."spacetime and Singularities An Introduction" and "The Large Scale Structure of spacetime".
    Last edited: Jun 1, 2012
  6. Jun 1, 2012 #5


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    There is one higher restriction on causality than stable causality. Global hyperbolicity (basically that there exist Cauchy surfaces which foliate the spacetime) ensures stable causality as well as a well-defined initial value formulation in GR.
  7. Jun 1, 2012 #6


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    Yep your right, it's just in "spacetime and Singularities An Introduction" on page 110 they imply stable causality is the one that excludes all others "Many such possibilities exist and we would like to impose some condition on our spacetimes which will prohibit all of them. Fortunately, the insight of Stephen Hawking has provided us with such a condition..."

    But you are right and also from another source

    "Global hyperbolicity is the most important condition on Causality, which lies at the top of the so-called causal hierarchy of spacetimes and is involved in problems as Cosmic Censorship, predictability etc."

    What they say in the book is misleading.
    Last edited: Jun 1, 2012
  8. Jun 1, 2012 #7


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    However, in 'Spacetime and Singularities An Introduction' they go on to say "The assumptions we have made of our spacetimes so far (time-orientability and stable causailty) have, from a physical point-of-view been rather easy to live with. The next condition [Global hyperbolicity] we propose is quite a bit stronger and not so easy to justify physically."
    Last edited: Jun 1, 2012
  9. Jun 2, 2012 #8
    Thanks to all of you!
  10. Jun 2, 2012 #9
    Something to add, Carroll Guth and Fahri published this paper in 1992 showing that a CTC in 2+1 spacetimes cannot exist.
  11. Jun 3, 2012 #10
    That's a good one Mark, thanks!
  12. Jun 3, 2012 #11


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    Actually, looking back on something I wrote about the singularity theorems (ages ago) I wrote

    Globally hyperbolic [itex]\rightarrow[/itex] causally simple [itex]\rightarrow[/itex] stably causality [itex]\rightarrow[/itex] strong causality [itex]\rightarrow[/itex] ....

    Here is an interesting paper: http://www.mat.univie.ac.at/~esiprpr/esi1876.pdf. They claim to make global hyperbolicity more mathematically simpler and physically clearer. They also discuss, and gives references about, the causal ladder.

    p.s. another book is Wald's chapter 8.
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