Shape of the universe

The term shape of the Universe can most usefully refer either to the geometry (curvature and topology) 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.

Table of contents

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

1.1 Language/Intuition Prerequisites
1.2 Comoving space
1.3 Local geometry (curvature) versus global geometry (topology)

1.3.1 Local geometry (curvature)
1.3.2 Global geometry (topology)

1.4 What is the shape of space of our Universe?

2 The shape of the whole of space-time
3 External references

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.

For example, common sense and everyday observation tells us that the world is big and flat and stationary and not a sphere in rapid rotation, both by spinning around its axis (at around 1000 mph at the Equator) and moving around the Sun (at around 66,000 mph). The real situation with regard to the motion of the Earth was discovered only recently in human history, and took over a century to be generally accepted.

Similarly, science tells us that the Universe behaves very differently from our common-sense experience at very small scales or very high speeds or energies. It even tells us that the local geometry of space can be altered by gravity. It therefore makes sense to ask if the Universe has a different local geometry (curvature) or global geometry (topology) at very large scales from that which we can measure locally.

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

then:

a flat space (zero curvature) is one for which this is true
an hyperbolic space (negative curvature) is one for which
a spherical space (positive curvature) is one for which

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.

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.

global geometry (topology); requires java (Jeff Weeks' pages)

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