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Curved space-time and dimensions

  1. Jun 23, 2008 #1
    I don't know anything about GTR, nor do I know anything about differential geometry. But I have one maybe stupid question:
    As far as I know space-time in general relativity is represented by a pseudo-riemannian manifold. And according to Whitney's (or Nash's? - don't know who is in charge here) embedding theorem every (Riemannian) manifold can be embedded (isometrically) into an euclidic R^n.
    So, if we now have some curved space-time manifold and we find the smallest possible embedding R^n. Could the extra dimensions in the embedding space have any physical meaning? Or are they even known to?
    In other words: Would it perhaps make any sense to suppose some "virtual" processes to happen within that embedding space, I mean, just like calculating with complex numbers but only taking real results "for real", that kind of thing?
    I'm just a curious pseudo-mathematician trying to understand some physics and asking questions that come to my mind. Sorry if it's stupid.
    Sincerely,
    Unkraut
     
  2. jcsd
  3. Jun 23, 2008 #2

    Fredrik

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    They definitely don't have any meaning in general relativity. Einstein's equation defines a relationship between the metric and the stress-energy tensor (i.e. matter), but doesn't relate any of that to anything outside of space-time.

    I suppose that what you have in mind is some kind of duality between mathematical objects that "live" in the manifold and mathematical objects that live in an embedding space. I have never heard of anything like that, but I don't see a way to rule out that something like that could be useful as a mathematical tool.
     
  4. Jun 23, 2008 #3
    Okay, thanks. That's what I expected.
     
  5. Jun 23, 2008 #4

    Fredrik

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    Someone who knows this stuff better than I do suggested that I should inform you that embedding theorems for Lorentzian manifolds are quite different from embedding theorems for Riemannian manifolds.

    This is how he explained it to me:

    Whitney embedding theorem: any smooth m-dimensional RIEMANNIAN manifold you are likely to meet can be embedded in 2m-dimensional euclidean space.

    Wikipedia is actually pretty good for RIEMMANNIAN embeddings, but has nothing on LORENTZIAN embeddings, where the known bounds are MUCH worse. You can Google for an old sci.physics.research post which gives a citation for Lorentzian embeddings.

    One other thing which might be worth mentioning is that one can sometimes "represent" higher dimensional manifolds with special properties (e.g. special kinds of solutions to the EFE) as lower dimensional manifolds with abstract coordinates, and satisfying a simpler criterion. For example you can look for a paper by H. J. Schmidt, formerly editor of Gen. Rel. Grav.
     
  6. Jun 24, 2008 #5

    George Jones

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