Geometry of the Universe (Review - Taken from Class Readings)

Today, many physicists believe that we are the carp swimming in our tiny pond, blissfully unaware of invisible, unseen universes hovering just above us in hyperspace. We spend our life in three spatial dimensions, confident that what we can see with our telescopes is all there is, ignorant of the possibility of 10 dimensional hyperspace. Although these higher dimensions are invisible, their "ripples" can clearly be seen and felt. We call these ripples gravity and light.

Relativity physics began by considering four-dimensional collections, with three dimensions for space and one for time. Recently, in the attempt to unify Einstein's theory of relativity and the theory of subatomic particles, modern physics has become much more complicated. Some current models keep track of seven dimensions that act like space and four that act like time, to give an 11-dimensional configuration space. Another important model uses a configuration space with 26 dimensions.

Looking up at the sky on a clear night, we feel we can see forever. There seems to be no end to the stars and galaxies. Maybe not. Like a hall of mirrors, the apparently endless universe might be deluding us. The cosmos could, in fact, be finite. The illusion of infinity would come about as light wrapped all the way around space, perhaps more than once--creating multiple images of each galaxy. (People are looking for these repeated patterns in the sky).

Many cosmologists expect the universe to be finite. Historically, a finite universe ran into its own obstacle: the apparent need for an edge. But every edge has another side, so we would need to know what was on the other side. So why not redefine the "universe" to include that other side? German mathematician Georg F. B. Riemann solved the riddle in the mid-19th century. As a model for the cosmos, he proposed the hypersphere--the three-dimensional surface of a four-dimensional ball, just as an ordinary sphere is the two-dimensional surface of a three-dimensional ball. It was the first example of a space that is finite yet has no problematic boundary.

One might still ask what is outside this finite universe. But this question supposes that the ultimate physical reality must be a Euclidean space of some dimension. That is, it presumes that if space is a hypersphere, then that hypersphere must sit in a four-dimensional Euclidean space, allowing us to view it from the outside. Nature, however, need not cling to this notion. It would be perfectly acceptable for the universe to be a hypersphere and not be embedded in any higher-dimensional space. Such an object may be difficult to visualize, because we are used to viewing shapes from the outside. But there need not be an "outside."

In lower dimensions, a finite Euclidean space must have the topology of either a 2-torus or a Klein bottle (while a Klein bottle is 2 dimensional, we have viewed it in 3 dimensions as a shape with intersections. Since it doesn't have intersections, we would need to go to 4 dimensions to view it properly.)

As a possible shape of the universe, there are only 10 Euclidean possibilities--namely, the 3-torus and nine simple variations on it, such as gluing together opposite faces with a quarter turn or with a reflection, instead of straight across. The number of spherical possibilities are infinite, but have been classified completely. There are infinitely many possible topologies for a finite hyperbolic three-dimensional universe. Their rich structure is still the subject of intense research and the classification is still an open problem today. One example, discovered by Jeff Weeks, may be constructed by identifying pairs of faces of an 18-sided polyhedron.

The universe is (presumably) "globally" is a geometric 3-manifold in the same sense in which we say that globally Earth is a sphere (and spherical geometry is the appropriate geometry for intercontinental airplane flights) even though it is clear almost anywhere on the earth that locally there are many hills and valleys that make Earth not locally isometric to a sphere. However, the highest point on earth (Mount Everest) is 8.85 km above sea level and the lowest point on the floor of the ocean (the Mariana Trench) is 10.99 km below sea level the difference is about 0.3% of the 6368 km radius of Earth (variations in the radius are of the same magnitude). It is known that locally our physical universe is definitely NOT a geometric 3-manifold, since local geometry is affected by mass. This has been confirmed in real-life experiments.

Some attempts at determining the geometry:

  • Measure the angles of a large triangle in our universe. (If the sum is less than 180 degrees then the universe is hyperbolic, greater than 180 degrees then spherical, and equal to 180 then Euclidean).
  • Assuming that stars are distributed uniformly in space (not clear if this is true) and then looking for patterns of stars at different distances to determine the geometry.
  • Assuming that certain types of stars or galaxies have a fixed amount of brightness, and that the brightness of a shining object in Euclidean space is inversely proportional to the square of the distance to the object, measuring several of these at various distances would determine the geometry.
  • Cosmic microwave background radiation (WMAP) and circles in the sky.