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Locally flat - what does it mean?

  1. Feb 1, 2010 #1
    What does it mean in general relativity that space is locally flat. It means that in neighbourhood of each point we can chose such coordinates that the metric is flat or we can do it only in a point (not in neighbourhood - open set).
     
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  3. Feb 1, 2010 #2

    Ben Niehoff

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    It means that given any point, we may choose a set of coordinates in which the metric is flat up to first order (i.e., the metric is flat at the point, and its first derivatives all vanish). The second derivatives cannot be made to vanish in general (since the second derivatives are basically the same thing as curvature).

    Stated another way, it means that given any positive epsilon, we may find a neighborhood around the point where the deviation from flatness is smaller than epsilon.
     
  4. Feb 1, 2010 #3

    George Jones

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    Unfortunately, the term "locally flat" is not used consistently, and you have identified the two main uses of the term. The first definition is used by mathematically careful references, but the second definition is used by many physicists.

    An example ot the former, the book Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry by Peter Szekeres, writes:

    "A manifold M with affine connection is said to be locally flat if for every point p in M there is a chart (U; x^i) with such that all the components of the connection vanish throughout U. This implies of course both torsion tensor and curvature tensor vanish throughout U, ..."
     
    Last edited: Feb 1, 2010
  5. Feb 1, 2010 #4

    George Jones

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    Other mathematically careful reference that have similar definition for local flatness include: Analysis, Manifolds and Physics; Tensor Analysis on Manifolds.
     
  6. Feb 1, 2010 #5

    Ben Niehoff

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    How in the world does this even make sense? Wouldn't such a manifold automatically be globally flat (though with possibly non-trivial topology)?
     
  7. Feb 1, 2010 #6

    George Jones

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    And if the manifold has non-trivial topology, then there isn't necessarily a global coordinate system (in which the components of the connection are zero), hence the concept is formulated in terms of coordinate neighbourhoods in manner in which "local" is used for other mathematical concepts, for example, "locally compact".

    Now, let me ask you a question. What restriction does your definition place on (semi)Riemannian manifolds? None; every (semi)Riemannian manifold satisfies your definition. Each definition is, in some sense, trivial. :smile:
     
  8. Feb 2, 2010 #7
    A quote from me explains what LF means in general:

     
    Last edited: Feb 3, 2010
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