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How do I know that a conformal transformation exist? + Global properties of spacetime

  1. Oct 15, 2012 #1
    I have two 2-dimensional space-times. One of them is flat the other one has not-vanishing curvature (Riemann tensor). But they seem to have a similar global and causal structure.
    Of course, because of the 2-dimensional case they are local conformally flat.

    I am looking for a relation between them that could explain the similar causal structure and I think that a conformal transformation would be nice.

    1) How do I know (prove) whether there exists a (global) conformal transformation between them?
    Is there a way to prove that there exists one (I do not need the transformation mapping itself, only the proof of existence)?


    2) Are there other global properties of space-times that are worth discussing? What are the
    typical global properties (my guess: Horizons, causal light cone structure...what else)?
     
  2. jcsd
  3. Oct 15, 2012 #2

    bcrowell

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    Re: How do I know that a conformal transformation exist? + Global properties of space

    By 2-dimensional, you mean 1+1?

    I guess conical singularities are not possible in 1+1 dimensions, so there's probably no way to get any singularities of any kind.

    In n+1 dimensions with n>1, Penrose diagrams only work if there's symmetry. In 1+1 dimensions, I guess you should be able to represent the entire conformal structure using a Penrose diagram, regardless of symmetry. Have you tried drawing Penrose diagrams?

    Are they asymptotically flat?
     
  4. Oct 15, 2012 #3
    Re: How do I know that a conformal transformation exist? + Global properties of space

    By 2-dimensions I mean a 2-dim manifold with Lorentz metric (-,+).

    As a mathematician, I am not familiar with the Penrose diagrams, so I didn't draw them.
    I thought there might be a necessary requirement/ premise or assumption so that we can conclude a "conformal relation".

    But, what kind of information would a Penrose diagram reveal?
    -> Provided the Penrose diagrams are identical, would that be a "proof" for a conformal relation (or causal and global similarity) between those two spacetimes?

    If so, it would probably make sense to learn about the Penrose diagrams.


    Here are the two spacetimes/ metrics that seem to have the same global and causal properties:
    http://tbf.me/a/OikJJ
     
  5. Oct 15, 2012 #4
    Re: How do I know that a conformal transformation exist? + Global properties of space

    They are not asymptotically flat because of the identification of one coordinate; I call it asymptotically flat under identification. But without the identification they would be both asymptotically flat.


     
  6. Oct 16, 2012 #5
    Re: How do I know that a conformal transformation exist? + Global properties of space

    i) Sorry, I posted the wrong metrics. Here are the correct two metrics:
    http://tbf.me/a/Bu4jVZ


    ii) Here are the two penrose diagrams (I believe they are correct):
    http://tbf.me/a/BntYLL

    In both diagrams the red line is a horizon (chronology/ cauchy).
    Could someone help me read/ compare the diagrams?
    Do the diagrams show that both spacetimes have the same causal and global structure (which should be the case)? Why?

    THANK YOU!
     
    Last edited: Oct 16, 2012
  7. Oct 16, 2012 #6

    zonde

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    Re: How do I know that a conformal transformation exist? + Global properties of space

    Isn't curvature of 2-dimensional manifold a Gaussian curvature?
    And if so then I believe you can't have global transformation between flat and curved space that preserves distances and angles. Or maybe conformal transformation requires only preservation of angles but not distances?
     
  8. Oct 16, 2012 #7

    bcrowell

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    Re: How do I know that a conformal transformation exist? + Global properties of space

    But isn't that only in a Riemannian space?
     
  9. Oct 17, 2012 #8
    Re: How do I know that a conformal transformation exist? + Global properties of space

    Yes, in 2 dimensions the Gaussian curvature (K) can be directly derived from the Riemann curvature (R) by K=-R.
    Thus, there is no isometry between those two spaces. But conformal transformations are only angle preserving and do not preserve distances. So I believe that there must be a conformal transformation.
    (?)
     
  10. Oct 17, 2012 #9

    zonde

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    Re: How do I know that a conformal transformation exist? + Global properties of space

    Hmm, obviously you can't have conformal transformation between surface of sphere and flat plane so I believe there should be additional conditions to allow for conformal transformation.

    Another thought. Before you look for a proof that (under certain conditions) there is conformal transformation between the two it would be reasonable to look if it works for some simple case, right?
     
  11. Oct 17, 2012 #10

    zonde

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    Re: How do I know that a conformal transformation exist? + Global properties of space

    I am not sure what is Riemannian space.
    I asked about this because I believe I more or less understand what is Gaussian curvature so I tried to tie the question with the thing that I know.
     
  12. Oct 18, 2012 #11
    This is false. The surface of a sphere is conformally flat, so a conformal transform. can turn it to a flat plane.
     
  13. Oct 18, 2012 #12

    zonde

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    Re: How do I know that a conformal transformation exist? + Global properties of space

    Indeed. Turns out this can be done using stereographic projection.
     
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