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Closed 1D world
Some point-shaped being lives on a circle. His limited 1D vision makes
the circle appear to him as a straight line. To actually see the extrinsic curvature he would need to
have 2D vision. He is looking at a point in the distance in his circular
world. Now he turns around and looks down his circle in the opposite direction. He sees
"another" point but this is actually the same point again, looked at
from the "backside".
Closed 2D world
You can extend this to 2D worlds, 3D worlds and
so on. The basic principle remains the same. In any n-dimensional closed world
or manifold (i.e., closed in yet another higher dimension),
every point always shows two projections to our "eyes", provided our
spatial vision is limited to those same n dimensions. The
projections follow geodesics, 1D trajectories like the circular world
of the point creature. Once our spatial vision is extended with one more
dimension, it becomes clear immediately that we are looking at a single point
only.
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World perception of a flatlander |
Closed 3D world
Also for our own 3D world, assumed closed in the 4th dimension, this means that
there should be "linked" points in pairs. The obvious candidate is
electrical charge which always comes in pairs and is conserved. It does not mean
that the link is necessarily between positive and negative charges in the same
atom. On the contrary, the linked charges are very likely to be separated by
large distances in 3D space.
Why then the asymmetry in mass between protons
and electrons? Protons are compound hadrons while electrons are
point-like leptons. Although a bit hard to visualize in 4D, a 3D analogue may
suffice to get the idea. It's kind of looking at the top of a tetrahedron from
above (electron) versus looking at the base of it from below (proton). The corners then
more or less represent its 4 dimensions.
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4D geometry of positive and
negative charges |
This has some interesting implications. The "spontaneous" emission
of a photon by a charge has its origin in the simultaneous absorption
of a photon by its linked companion charge. Since the distance between the two
linked charges is arbitrary (albeit depending on the geodesic connection between
them), this means an instantaneous transformation of information in
3D. That brings the EPR experiment to mind where something similar happens with
two "linked" photons. The seemingly instantaneous information transfer in this experiment
indicates that, again, a higher dimension must be involved in the link.
In the next two sections, a similar absorption process as found with charges and
photons will be worked out, but
now the 5th dimension plays the role as linking and closing dimension, like the 4th
did with electrical charges.
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