The term shape of the Universe can most usefully refer either to the shape of a comoving spatial section of the Universe (a loose term for this is the shape of space) or more generally, to the shape of the whole of spacetime.

The shape of space (a comoving spatial section of the Universe)
To understand concepts of the shape of the universe, according to the standard big bang model, the reader should, ideally, first develop his/her intuition of manifolds, and more specifically, of Riemannian manifolds.
However, those definitions are somewhat abstract.
Here is an attempted shortcut to developing that intuition.
The reader's ordinary notions of space and time are likely to be wrong, they are psychological constructions developed from common sense and folk physics[?]. These notions are useful for ordinary living, as they closely approximate reality over distances and times that are at human scales, but this does not make them real.
An easy way to convince oneself that one's intuition is wrong in at least some ways is to think of a globe of the Earth with the South Pole at the "top". Does this seem "wrong"? It is clear that any point on the Earth is just as much the "top" as any other point (though it could be argued that the highest point somewhere near the Equator is the top, since this is furthest from the centre of the Earth), so clearly there is something wrong in one's spatial intuition if a globe with the South Pole at the "top" seems wrong.
One way of developing correct intuition is to ignore one's existing intuition and start from scratch, from very simple logic.
The reader should imagine starting off with a very abstract definition of a set, which is more or less just a collection of points, and then adding more and more definitions. These definitions include ways in which the points relate to each other, and eventually include some concepts so that this set has some properties which are like the common notions of a space.
It is then proposed that the reader accept the use of twodimensional spaces as analogies for real, threedimensional space, since this way the third dimension of his/her intuition can be used as a psychological tool for imagining different possibilities for twodimensional spaces. The reader should remember that the use of a dimension for intuitionbuilding does not imply that it has any physical meaning. It is merely one way, among many, of thinking about spaces of different curvature and topology.
Comoving coordinates are necessary for thinking about the shape of the Universe. In comoving coordinates, we can think of the Universe as static, despite the fact that in reality it is expanding. This is simply a useful way of separating geometry (shape) from dynamics (expansion).
In simple words, this is the question of whether or not Pythagoras' theorem, is correct, or equivalently, whether or not parallel lines remain equidistant from one another, in the space one is talking about.
If we put Pythagoras' theorem in the form
then:
The first and third of these are relatively easy to imagine with twodimensional analogies. The first is an infinite flat plane. The third is the surface of an ordinary sphere.
In simple words, this is the question which ignores Pythagoras' theorem.
Three different twodimensional spaces which are all flat spaces, in all of which Pythagoras' theorem is true, are
Each of these is globally very different.
The third is finite in 2volume, i.e. surface area, but has no edges and Pythagoras' theorem is true everywhere in it.
The Twin paradox leads to a new paradox in the context of the global shape of space. See the external references below for more on this.
We know neither the local nor the global shape of space. We do know that the local shape is approximately flat, just like the Earth is approximately flat. We do not yet know the topology of the universe, and maybe never will.
The shape of the whole of spacetime
remains to be written
Intuition building:
Texts:
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