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Geometry of the Universe PDF Print E-mail
Thursday, 19 November 2009 11:10

  • general relativity allows for spacetime to be curved, thus the whole Universe may have a non-flat geometry
  • three possible shapes are allowed, flat, positive or negative curvature
 Can the Universe be finite in size? If so, what is ``outside'' the Universe? The answer to both these questions involves a discussion of the intrinsic geometry of the Universe.

There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a spherical Universe (positive curvature) or a hyperbolic Universe (negative curvature). Note that this curvature is similar to spacetime curvature due to stellar masses except that the entire mass of the Universe determines the curvature. So a high mass Universe can have positive curvature, a low mass Universe might have negative curvature.

  • different tests are avalable to determine the curvature of the Universe, such as measuring triangles or parallel lines
 All three geometries are classes of what is called Riemannian geometry, based on three possible states for parallel lines
  • never meeting (flat or Euclidean)
  • must cross (spherical)
  • always divergent (hyperbolic)

    or one can think of triangles where for a flat Universe the angles of a triangle sum to 180 degrees, in a closed Universe the sum must be greater than 180, in an open Universe the sum must be less than 180.

    • note that curvature or geometry of the Universe does not determine how it is connected, which is its topology
    		 Standard cosmological observations do not say anything about how those volumes fit together to give the universe its overall shape--its topology. The three plausible cosmic geometries are consistent with many different topologies. For example, relativity would describe both a torus (a doughnutlike shape) and a plane with the same equations, even though the torus is finite and the plane is infinite. Determining the topology requires some physical understanding beyond relativity.
    • a finite Universe, if wrapped, would appear infinite like a box of mirrors
    		 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. A mirror box evokes a finite cosmos that looks endless. The box contains only three balls, yet the mirrors that line its walls produce an infinite number of images. Of course, in the real universe there is no boundary from which light can reflect. Instead a multiplicity of images could arise as light rays wrap around the universe over and over again. From the pattern of repeated images, one could deduce the universe's true size and shape.
    • topologies need not be simple, for example a Moebius strip
    		 Topology shows that a flat piece of spacetime can be folded into a torus when the edges touch. In a similar manner, a flat strip of paper can be twisted to form a Moebius Strip.
    • or a Klien bottle
    		 The 3D version of a moebius strip is a Klein Bottle, where spacetime is distorted so there is no inside or outside, only one surface.
    • deep space observations indicate that the Universe is simply connected
    		 The usual assumption is that the universe is, like a plane, "simply connected," which means there is only one direct path for light to travel from a source to an observer. A simply connected Euclidean or hyperbolic universe would indeed be infinite. But the universe might instead be "multiply connected," like a torus, in which case there are many different such paths. An observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a huge crowd.
    • however, a large Universe may be connected in complex ways that are not visible to our limited observations
    		 One possible finite geometry is donutspace or more properly known as the Euclidean 2-torus, is a flat square whose opposite sides are connected. Anything crossing one edge reenters from the opposite edge (like a video game see 1 above). Although this surface cannot exist within our three-dimensional space, a distorted version can be built by taping together top and bottom (see 2 above) and scrunching the resulting cylinder into a ring (see 3 above). For observers in the pictured red galaxy, space seems infinite because their line of sight never ends (below). Light from the yellow galaxy can reach them along several different paths, so they see more than one image of it. A Euclidean 3-torus is built from a cube rather than a square.
    • even simple topologies lead to complex connections
    		 A finite hyperbolic space is formed by an octagon whose opposite sides are connected, so that anything crossing one edge reenters from the opposite edge (top left). Topologically, the octagonal space is equivalent to a two-holed pretzel (top right). Observers who lived on the surface would see an infinite octagonal grid of galaxies. Such a grid can be drawn only on a hyperbolic manifold--a strange floppy surface where every point has the geometry of a saddle (bottom).
    • and all this is connected in 4D spacetime, not simply in 3D space
    • the key to understand the shape of the Universe is its history and dynamics
    		 Its important to remember that the above images are 2D shadows of 4D space, it is impossible to draw the geometry of the Universe on a piece of paper, it can only be described by mathematics. All possible Universes are finite since there is only a finite age and, therefore, a limiting horizon. The geometry may be flat or open, and therefore infinite in possible size (it continues to grow forever), but the amount of mass and time in our Universe is finite.
     


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