Hubble discovered that all galaxies were fl ying away from us, isotropically in all directions. Th at seems to suggest that we are at the center of the universe. Are we?
A long time ago we also thought that we were at the center of the universe because the sun, the moon, and the stars all evolved around us. Galileo proclaimed otherwise, and got himself into trouble and excommunicated, but he was right. Although the moon does evolve around the earth, everything else seems to do so only because the earth rotates about its own axis.
Unless we are very egoistic, there is no reason to resurrect the notion of our central place in the universe at this late date. If not, then how can we explain the fact that all the galaxies are fl ying away from us, isotropically in all directions?
It turns out that this could be understood if we assume the universe to be homogeneous, and the galaxies to be uniformly distributed. Th is assumption, sometimes known as the cosmological
principle, is consistent with our knowledge of the distribution of
galaxies and the cosmic microwave background (see Chap. 14). In that case, every point in the universe is equivalent, and every one of them may be considered to be the center of the expansion.
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To see how that works, imagine our universe to be a very big loaf of raisin bread, with every raisin representing a galaxy. The rise of the bread in an oven is like the expansion of the universe. No matter which galaxy (raisin) you imagine yourself to be living on, when the bread rises every other raisin is moving away from you, isotropically in all directions; so each can be thought to be the center of expansion.
Figure 19: A small loaf of raisin bread.
Now that we know the universe to be homogeneous, let us ask how large it is, whether it is flat or curved, and what possible shapes it could have.
It is hard to know how large our whole universe is because we cannot see beyond the horizon. The farthest galaxy visible to us is the one whose light emitted at the beginning of the universe 13.7 billion years ago has just reached us. If the universe were static, that galaxy is 13.7 billion light years away and that is the size of the observable universe, or our visible horizon. Since the universe is expanding, that galaxy would have moved away after light emission, so by now it has to be more than 13.7 billion light years from us. Calculation shows that it is some 78 billion light years away, which is then the actual size of the observable universe at
the present. This size grows in the future and shrinks in the past. Whatever the number is, the point is that at any given time we can see only so far away, so it is hard to know how large the entire universe is.
However, with the homogeneity of the universe, there is potentially a way to find out its size provided it is finite in extent.
Before the age of airplanes and steam ships, the earth also appeared to be immeasurably vast, yet almost two thousand years ago the Greek mathematician Eratosthenes (276 to 194 BCE) found a way to determine its size by measuring the curvature of earth’s surface.[1] If we can measure the curvature of the universe, then
we might be able to determine its size in a similar manner. There is, however, a significant difference between the two: the surface of the earth is two-dimensional but the universe is three- dimensional. We are familiar with two-dimensional curvatures by looking down from the third dimension, but we cannot go into a fourth-dimensional space to look down at our three dimensions. Moreover, the immediate space around us is flat, so it is hard to understand what is meant by a curved three dimension, and why the universe would bother to be curved at all.
In order to build up a feeling for the real universe, let us pretend it to be two-dimensional and without a boundary. If it is flat, then it is like an infinitely large flat rubber sheet, being stretched in all directions at all times. If it has a positive curvature, then it is finite like the surface of the balloon shown in Fig. 20. The surface gets flatter as the balloon gets larger. If it has a negative curvature, then it is harder to picture, but everywhere it looks a bit like the center of a saddle, curving up one way in one direction and the opposite way in the orthogonal direction.
An animal in this universe lives on the balloon and is unaware of the presence of a third dimension. Nevertheless, assuming it to be intelligent, it can still figure out its universe to have a positive
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curvature by using geometry. According to Euclidean geometry, the kind that we learn in school, the sum of the three angles of a triangle is 180 degrees. As the following example shows, this will no longer be the case for a triangle on a curved surface. If the curvature is uniformly positive, like a sphere, then the sum is more than 180 degrees. If it is uniformly negative, then the sum is less than 180 degrees.
To see why the sum of the three angles of a triangle is more than 180 degrees on a positively curved surface like a balloon, paint on it the longitude and latitude lines to make it look like the globe in Fig. 21. Consider the triangle with one apex at the north pole, and two others on the equator. The three sides of the triangle are taken to be the two longitude lines from the north pole to the two places on the equator, and a section of the equator joining the two. In this triangle, each of the two angles at the equator is 90 degrees, making their sum of the three angles 180 degrees plus the angle at the north pole.
Note that there are no straight lines on a balloon or any curved surface, so the best we can do for the three sides of a triangle is to take the shortest lines between the apexes. These are the proper generalization of straight lines on a curved surface because straight lines are the shortest lines between two points on a flat surface. The shortest line between two points in any dimension is called a geodesic line. Straight lines are geodesics on a flat surface, the longitude lines and the equator are geodesics on a globe.
Now we come to the real universe in three dimensions. The notion that such a universe may not be “flat” came from Einstein, who showed us that gravity can curl up space and space-time.[2]
Curvature in a three-dimensional space can again be determined from geometry in a somewhat similar way.
Using the cosmic microwave radiation, one finds to within about 2% of possible errors that our three-dimensional universe is approximately flat. Since the allowed error is still fairly large, the universe is still allowed to have either a small positive or a small negative curvature.
If the universe is exactly flat, then it looks like an infinitely large loaf of raisin bread which continues to rise all the time. If
Figure 21: The triangle on a globe as described in the text, with the three apexes circled in red.
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it has a negative curvature, then it resembles a three-dimensional generalization of the middle of a saddle. It is not so easy to picture what that is so we will not attempt to do so here. If it has a positive curvature, then it looks like a three-dimensional generalization of the balloon: a 3-sphere.
Mathematically, the balloon is called a 2-sphere. Every point on its surface is equidistant from a point in the third dimension — the center of the balloon. The three-dimensional analog is a 3-sphere: every point on its surface is equidistant from a center
point in the fourth dimension. The problem is, we do not live in four spatial dimensions, so it is hard to picture what a 3-sphere really looks like.
Other than a sphere, one might ask what other shapes the universe could look like, assuming that it has a single connected piece, has no boundary, and is finite in extent.
In the case of a two-dimensional universe, the answer is completely known. It could be a distorted sphere, obtained from a 2-sphere by pushing it in at some points and pulling it out at some other points, carefully not tearing it in the process. Mathematically, such an object is known as a topological 2-sphere.
It could also be shaped like a donut or a distorted donut, or several of them stuck together.
The genus of an object is the number of independent closed curves that can be drawn on it. Two closed curves are not considered independent if one can be continuously deformed to become another. Neither is a curve considered independent if it can be continuously deformed to a point. A 2-sphere has genus 0, a donut has genus 2, and n donuts stuck together has genus 2n.
For the real universe in three dimensions, the problem is much harder. A famous conjecture by the well-known mathematician Henri Poincaré, first put forward in 1900, subsequently modified in 1904, postulated that the only three-dimensional object with
genus 0 is the topological 3-sphere. Many high-powered mathe- maticians have worked on this famous conjecture for the past one hundred years, but its proof did not come until very recently. For his achievement in proving the conjecture, the Russian mathematician Grigori Perelman was offered a 2006 Fields Medal. But, he declined to accept it! The Fields Medal in mathematics is like the Nobel Prize in physics, awarded to those with extraordinary achievements, but unlike the Nobel Prize, it is offered only once every four years and the recipient has to be under forty years of age. This is the first time in history that anybody has ever turned down the award.
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