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Unit 4

Topology's Twists and Turns

4.9 The Shape of Space

LIFE IN A MANIFOLD

Let us return, finally, to our initial question: does the universe go on forever? To the thinker in ancient times, the size of the Earth must have been unfathomable. Determining the shape of the Earth by a brute-force exploration approach would have been a distinct impossibility. Like any manifold, the surface of the Earth appears to be flat everywhere to those of us on its surface. Still, early philosophers and thinkers were able to gauge, more-or-less correctly, the size and shape of the Earth. The most famous of these efforts was made by the Greek philosopher and mathematician, Eratosthenes, who very cleverly calculated the circumference of the Earth by comparing shadows at different latitudes. This ingenious exercise established facts that were empirically verified many centuries later by the first round-the-world explorers.

In trying to comprehend the size and shape of our universe, we are faced with a similar dilemma. As far as we can tell with local measurements, space appears to be the three-dimensional analog of an infinite flat plane. To verify this empirically, we would have to set out in a theoretically–impossible-to–build, faster–than-light spaceship to explore the furthest reaches of the visible universe. This is even less of an option to us than sailing around the world would have been to the Greeks. We can search for other evidence and an alternative verification method, however, just as Eratosthenes studied shadows rather than attempting to sail around the world.

When we inquire into the shape of space, we are seeking to know whether the universe has some sort of interesting topological structure. That is, if we were somehow able to explore the furthest reaches, as the Flatlander did in her world, would we find that certain routes lead us back to where we started? At the heart of this question is the idea of connectivity. If the universe is simply connected, then it would be analogous to the surface of a sphere in three dimensions. This would mean that any loop of string in space could be reeled in to a point with no problems (at least no theoretical problems). This is the way that Riemann and Einstein imagined space to be.

Another possibility, however, is that the universe is multi-connected. The simplest shape for a multi-connected universe would be a 3-torus.

SpaceWhat would it be like to exist in this kind of universe? Well, if you traveled forward far enough, you would end up where you started. Actually, the same thing would happen no matter which direction you traveled. What if this space were so large that it was impossible to travel far enough to return to your starting point? If you would simply look around, you would find some clues as to the nature of your space. If you looked forward, you would see your back; if you looked to the left, you would see your right side; and if you looked up, you would see the soles of your feet.

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HEAVENLY CLUES

By making visual observations, such as those just noted, and analyzing how the copies of yourself that you see are arranged, you could begin to deduce the large-scale topological structure of your universe. So, in terms of galaxies, we could scan the night sky, looking for copies of our own galaxy. A problem arises, however: we don't know what our galaxy looks like from the outside, so how would we know if we were looking at some distant image of it? Another problem is that space is so large that light takes a very long time to reach us from other galaxies. Consequently, the images that we see are not of the galaxies as they are now, but rather as they were when the light that reaches us left them hundreds, thousands, or even millions of years ago. Relating this to our earlier 3-torus example, this situation is equivalent to seeing your back as it appeared when you were ten years younger, or perhaps your side when you were five years younger, or, possibly, the soles of you feet as they looked twenty years ago. It would not be evident that you are even seeing an image of yourself.

The MICROWAVE SKY WMAP Item 1776 / NASA/WMAP Science Team, The MICROWAVE SKY WMAP (2006). Courtesy of NASA/WMAP Science Team.


In order to hypothesize about the shape of space, astronomers have to study something more basic than images of galaxies. So, they study things such as the average distance between galaxies. These average distances can then be collated into a distribution, and that distribution can be matched up against theoretical ones corresponding to different shapes of the universe. Space scientists also study the cosmic background radiation, which is a form of radiation that is left over from the Big Bang. As it turns out, this type of radiation is not uniformly distributed in space, so it provides a reference base for exploring the night sky. Astronomers can also search the night sky for spots that have the same temperature features and, possibly, other similarities. The discovery of such regions that share characteristics would be a significant step toward reaching a better understanding of the shape of our universe.

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