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The Fourth Dimension

  1. Apr 21, 2012 #1
    To start, lets say I took a bowling pin and passed it through a two-dimensional universe. An observer in the two-dimensional universe would see a two dimensional slice of the bowling pin expanding and contracting as the bowling pin passed through their universe. Similarly, a 4-dimensional object passing through a 3-dimensional universe would appear to be an 3 dimensional object changing size and shape.

    When a balloon is inflated, it changes size. Could the inflated balloon and the deflated balloon be different cross-sections of the same four dimensional object passing through out three-dimensional universe?

    Could all motion simply be changing cross sections of four dimensional objects passing through our universe. Think of the two-dimensional universe again. If you took a 'V' and passed it through the two-dimensional universe, the observer would see a single object appear and then two objects break apart and move away from each other.
     
  2. jcsd
  3. Apr 21, 2012 #2
    The history of the positions of a point throughout all moments in time forms a one-dimensional object in spacetime - a World Line. Likewise, the world lines emerging from all the points of a one-dimensional space-like curve delimit a two-dimensional "World sheet". If the space-like curve was a loop, the World sheet would be the envelope of a cylinder. Analogously, the world lines emerging from a space-like surface element would fill in the interior of this cylinder.

    Now, we come to a point that I can no longer envisage intuitively. If you consider a closed space-like surface (a ball, for example), it cannot be encircled by a closed line uniquely (you may draw infinitely many circles on the surface of a ball), and I don't know how to envisage the 4-dimensional hypersurface formed by the world lines emerging from these points.

    In any case, this hypersurface envelopes a region in 4-dimensional spacetime filled by the world lines emerging from the points interior to the closed surface (points inside the ball in our example).
     
    Last edited: Apr 21, 2012
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