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Shape of the universe

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 space-time.

The shape of space (a comoving spatial section of the Universe)

Language/Intuition Prerequisites

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 yourself that your intuition is wrong in at least
one way is to imagine a globe of the Earth with the South Pole at the "top".
Does this seem "wrong"? If the North Pole is at the "top", does this seem
"right"? How is it possible that somehow "North" is "right" and "South" is
"wrong"? It is clear that physically, there should be symmetry between North
and South, neither should be favoured. So clearly there is something wrong
in your spatial intuition if you have the feeling that a globe with the
South Pole at the "top" seems wrong. Try looking at a map of your
country/region with South at the top and the effect should be clear.

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 two-dimensional spaces
as analogies for real, three-dimensional space, since this way the third
dimension of his/her intuition can be used as a psychological tool for
imagining different possibilities for two-dimensional spaces. The reader
should remember that the use of a dimension for intuition-building 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 space

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).

Local geometry (curvature) versus global geometry (topology)

Local geometry (curvature)

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

[h = \sqrt{x^2 + y^2} ]

then:

* a flat space (zero curvature) is one for which this is true
* an hyperbolic space (negative curvature) is one for which
[h > \sqrt{x^2 + y^2} ]
* a spherical space (positive curvature) is one for which
[h < \sqrt{x^2 + y^2} ]

The first and third of these are relatively easy to imagine with
two-dimensional analogies. The first is an infinite flat plane. The third is
the surface of an ordinary sphere.

Global geometry (topology)

In simple words, this is the question which ignores Pythagoras' theorem.

Three different two-dimensional spaces which are all flat spaces, in all of
which Pythagoras' theorem is true, are

* an infinite flat plane
* an infinitely long cylinder
* a 2-torus, i.e. a cylinder with two ends which are defined to be stuck
to each other ("identified" with each other)

Each of these is globally very different.

The third is finite in 2-volume, i.e. surface area, but has no edges and
Pythagoras' theorem is true everywhere in it.