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| In this page, I attempt to understand the main principle
of General Relativity (curved spacetime) by examining a simple 2-dimensional
universe, having one length dimension and one time dimension. |
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We start by showing a snapshot of the universe.
The red line in the figure represents a planet in our universe. Notice
that the line of space is bent by the presence of this planet. The small
blue dot represents a small object of negligible mass compared to the planet.
Our goal is to determine what the shape of spacetime in this universe must
be. |
| The first natural shape to speculate is obtained
simply by extending the given line shape in the time dimension, producing
the graph as shown on the right. However, one immediately notices a problem
with this universe. Because the surface does not curve with respect to
time, the geodesic (straightest path on a curved surface) of the blue object through spacetime
is a straight line parallel to the planet's worldline. We would like the
geodesic to represent the object's actual path as it accelerates toward
the planet. |
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One way to do this is to make spacetime curve
around a single point in spacetime. This may look similar to diagrams you
have seen in books about gravity. The difference is, the books generally
portray only space curvature. We are trying to represent spacetime curvature.
It seems natural that the shape of space should not be changing with respect
to time, since the planet, which is the only significant determiner of
the shape of spacetime, is stationary from our reference frame. All snapshots
in time should look similar. Here, we have spacetime favoring a particular
event, and flattening out before and after that event. If the planet is
still there, however, we should not have spacetime flattening out. |
| I propose the following shape of spacetime. Notice
how the blue curve is now directed toward the planet, but we have also
preserved the constant shape of space at all moments in time. There is
one potential problem with this model: The shape of spacetime ought to
be flat at far distances from the planet. Here, spacetime is rolled up
into a cylinder like shape, which is hardly the same as flat spacetime.
Or is it? If you examine the topologies of a flat surface as opposed to
a "rolled up" one, you will find that to observers within the surface
there is no difference between the two. To the inhabitants of this universe,
spacetime is, for all practical purposes, relatively flat far away from the
planet. Now, you may be worried about the "bump" in spacetime caused by
the planet causing the universe to intersect itself. I remind you that
our decision to embed the surface in 3-dimensional space was arbitrary.
We could have just as well embedded it in 4-space. This could easily remove
any possibility of self-intersections. And really, we only embed the surface
in a Euclidean space to better visualize what is happening with the curvature
of the universe. Once we know how the universe is curved, it really isn't
necessary to think of it as being "in" any space. |
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| Although the surface I propose seems to make sense intuitively,
I have no idea if it agrees with a physicist's model of curved spacetime.
If you or someone you know can shed more light on this subject, let me
know. My email is berglund@math.gatech.edu.
Thanks! |