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4 dimensional spacetime manifold question

  1. Mar 22, 2006 #1
    I'm having trouble understanding exactly what this manifold is. Let me draw an analogy: Say I have a flat map of the world. The map is a two-dimensional surface with a coordinate chart on it. However, its embedded in a higher three-dimensional space.

    So by analogy, is the four dimensional spacetime manifold of Einstein equivalent to our three spatial dimensions ("the map") that is embedded in a higher fourth time dimension, or are all four dimensions and related coordinates (x,y,z,t) part of the surface of the "map"?

    If that were the case, and the "map" is curved, then does that mean there must be at least five dimensions?
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  3. Mar 23, 2006 #2


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  4. Mar 23, 2006 #3
  5. Mar 24, 2006 #4


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  6. Mar 29, 2006 #5
    Embedding Problem

    It can be useful in practice to embed manifolds into (pseudo)Euclidean spaces of higher dimension since the tangent spaces at all points in our manifold will be (pseudo)Euclidean hyperplanes.

    Curvature is intrinsic in the sense that these tangent hyperplanes are different at different points of our manifold regardless of how we do the embedding. Flat manifolds have the property that the tangent hyperplane is the same at all points on the manifold.
  7. Apr 2, 2006 #6
    Different in the sense that they each have a different slope at each point of tangency?
  8. Apr 2, 2006 #7


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    It seems to me that one would do this for purposes of visualization only (i.e. to draw the tangent planes of various sample points)..as long as one knows how to read the visualization. Embedding isn't necessary... especially since one is interested in intrinsic quantities.

    Comparison of vectors in different tangent planes of a manifold requires a connection on the manifold. Embedding isn't necessary... and it may even be distracting or misleading.
  9. Apr 2, 2006 #8


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    Embedding is a crutch that you have to learn to get rid off. It doesn't generalize very well, particularly in the cases where we have manifolds that lack much symmetry. Very often we are forced to multiply by two the amount of dimensions (so the amount of degrees of freedom starts growing drastically).

    Moreover, I have no good visualization of what those embedded spaces should look like either, so its not clear what it buys you.
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