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Interpretation of Flamm Paraboloid

  1. Jul 26, 2009 #1
    In the Schwarzschild Metric, the curvature of space around the gravitating mass can be described by the Flamm Paraboloid:
    [tex]w(r) = 2 \sqrt{r_{s} (r - r_{s})}[/tex]​
    Unlike the Newtonian depiction of Gravitational Potential Wells (U = - G M / r) which decrease inwards, the Flamm Paraboloid increases outwards.

    QUESTION: Does this mean, that rather than mass "bending the fabric of Spacetime 'downwards'" -- a la the "Rubber Sheet" analogy -- that mass actually bends the fabric of Spacetime about it "upwards" ?

    ANALOGY: Take a long blade of grass. It's straight representing a flat 1D "Lineland" space. Now, bend the blade of grass at some spot in the middle. That represents the curving of space caused by a massive body, "at the point of bend". But, the result is not so much that the "point of bend" bends downwards, but that both tips of the blade of grass bend upwards.

    Can this be construed as an accurate interpretation of the Flamm Paraboloid ? Perhaps, if you "embed" a roughly Schwarzschild-esque solution, for a star (say), in a larger Cosmological fabric of Spacetime, then those "tips of the blade of grass" are "anchored" into that larger fabric, so that when the star tries to bend those tips upward, it actually "pushes itself downwards" ??

    I have tried to illustrate my questions w/ the attached figure below:

    Attached Files:

  2. jcsd
  3. Jul 27, 2009 #2


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    Flamm's paraboloid only represents the spatial curvature not the space-time curvature.

    It doesn't matter if you visualize it downwards or upwards. What matters are the distances within the 2D paraboloid surface, which are greater near the mass than further away. The 3rd dimension of the pictures is irrelevant for someone living within the 2D paraboloid surface which represents our 3D space.
    Last edited: Jul 27, 2009
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