Cosmology and Geometry (5) ...

The force of primary importance in cosmology is gravity, because it acts relentlessly over long distances. In his theory of general relativity, Einstein showed that mass caused space to curve and objects travelling in that curved space have their paths deflected, exactly as if a force had acted on them. Thus, in general relativity, which is the best theory of gravitation that we have, the effect of gravity is reduced to the effect of curved spacetime on the motion of particles. Let us recall briefly some basic ideas concerning curved spaces that we introduced in the module on general relativity in Chapter 4.
Curvature of Space in Two Dimensions
We may distinguish three qualitatively different classes of curvature, as illustrated in the adjacent figure for surfaces that have two dimensions. The flat surface is said to have zero curvature, the spherical surface is said to have positive curvature, and the saddle-shaped surface is said to have negative curvature.
Curvature of Four-Dimensional Spacetime
The spacetime of general relativity has three space-like dimensions and one time-like dimension, not just two space dimensions as in our example above. This is difficult to visualize but it can be described mathematically by the same methods that mathematicians use to describe the two-dimensional surfaces shown above. In this case, we can also speak of three classes of curvature that are the analogs of the classes just discussed for two-dimensional surfaces.

The Large-Scale Geometry of the Universe
Since space itself is curved, there are three general possibilities for the geometry of the Universe. Each of these possibilites is tied intimately to the amount of mass (and thus to the total strength of gravitation) in the Universe, and each implies a different past and future for the Universe:

  • If space has negative curvature, there is insufficient mass to cause the expansion of the Universe to stop. The Universe in that case has no bounds and will expand forever. This is termed an open universe.
  • If space has no curvature (it is flat), there is exactly enough mass to cause the expansion to stop, but only after an infinite amount of time. Thus, the Universe has no bounds in that case and will also expand forever, but with the rate of expansion gradually approaching zero after an infinite amount of time. This is termed a flat universe or a Euclidian universe (because the usual geometry of noncurved surfaces that we learn in high school is called Euclidian geometry).
  • If space has positive curvature, there is more than enough mass to stop the present expansion of the Universe. The Universe in this case is not infinite, but it has no end (just as the area on the surface of a sphere is not infinite but there is no point on the sphere that could be called the "end"). The expansion will eventually stop and turn into a contraction. Thus, at some point in the future the galaxies will stop receding from each other and begin approaching each other as the Universe collapses on itself. This is called a closed universe.

    • Which of these scenarios is correct is still unknown because we have been unable to determine exactly how much mass is in the Universe.

    Technically Speaking: How Can the Infinite Expand?

    From the preceding, it is possible that the Universe is infinite. One conceptual problem that this generates is that of how something that is already infinite gets "bigger". This is a difficult concept, because the concept of infinity itself is difficult to grasp. However, we may think of a mathematical analogy. Imagine the set of all positive integers, 1, 2, 3, ... This is an infinite set since given an integer we can always think of one that is larger, and the spacing between the integers is clearly 1. Now imagine multiplying all the integers in our infinite set by 2, giving 2, 4, 6, ... This is again an infinite set, but now the spacing between integers has been doubled to 2. This then is an example of something being infinite, and yet capable of "expansion".

    The Cosmic Scale Factor
    The "size" of the Universe is usually expressed in terms of a quantity R(t) that is a function of time t and is called the cosmic scale factor. It is a measure of how much the spacing between galaxies (like the dots on the balloon) increases with time as a result of expanding spacetime. We may think of it loosely like a "radius" for the Universe, but that really has meaning only if the geometry of the Universe is closed. The expansion of the Universe is particulary simple when expressed in comoving coordinates. In that case, only the scale factor changes with time.