What is a finite and bounded universe and how do scientists envision it?

[Alltimegreat1]

A number of scientists subscribe to this theory. I read up on it, but none of the explanations I found really answered my questions. How should one attempt to envision a universe that is finite and bounded?

 

[mathman]

Use torus analogy (surface) - hard to envision in three dimensions.

 

[mgkii]

It goes something like this

Start with one dimension - say a line. Imagine a creature that lives on that line and can only perceive in one dimension - i.e. backwards and forward, nothing else. If you bend that line, the creature would not be able to perceive the bend as it still travels the line backwards and forwards. If you then bend the line right the back on itself - for example, make a circle; then that creature can keep going forever, but will come back to where it started from each time. The creature can't perceive the changes you've made, as you made them in a higher dimension.

Next do it in two dimensions - e.g. a piece of paper. The creature "living" on the paper can only perceive 2 dimensions of forward, backward, left and right, so again can't perceive the changes you're about to make in a higher dimension. If you want to go off one edge an reappear on the other, then roll the paper into a cylinder. If you want to do the same for the other two edges, then roll that cylinder into a donut shape (a torus).

Now for the hard bit... try to imagine it for three dimensions! You can probably see immediately "how" you need to imagine it... but you'll have a better mind than me if you can actually build a mental image!

 

[Chalnoth]

A number of scientists subscribe to this theory. I read up on it, but none of the explanations I found really answered my questions. How should one attempt to envision a universe that is finite and bounded?

Finite and bounded?

One possible way this might happen is if our universe is a "bubble" of a different vacuum energy from its surroundings. In this model, somewhere beyond the cosmological horizon is a boundary between our value of the vacuum energy and the vacuum energy value outside. This boundary will move, at very nearly the speed of light, in the direction of lower vacuum energy (destroying everything in its path). We won't ever be able to observe it, however, as the boundary lies beyond our cosmological horizon.

But the usual thing that people talk about, and was described earlier in this thread, is a finite but unbounded universe. The surface of a sphere is like this: the area of the surface is finite, but no matter what direction you move along the surface, you'll never run into an edge. There's also the torus (doughnut-shape). A universe of this shape acts rather like the classic video game Asteroids, where moving off of one side of the screen causes your ship to appear on the opposite side of the screen.

 

[Alltimegreat1]

It goes something like this

Start with one dimension - say a line. Imagine a creature that lives on that line and can only perceive in one dimension - i.e. backwards and forward, nothing else. If you bend that line, the creature would not be able to perceive the bend as it still travels the line backwards and forwards. If you then bend the line right the back on itself - for example, make a circle; then that creature can keep going forever, but will come back to where it started from each time. The creature can't perceive the changes you've made, as you made them in a higher dimension.

Next do it in two dimensions - e.g. a piece of paper. The creature "living" on the paper can only perceive 2 dimensions of forward, backward, left and right, so again can't perceive the changes you're about to make in a higher dimension. If you want to go off one edge an reappear on the other, then roll the paper into a cylinder. If you want to do the same for the other two edges, then roll that cylinder into a donut shape (a torus).

Now for the hard bit... try to imagine it for three dimensions! You can probably see immediately "how" you need to imagine it... but you'll have a better mind than me if you can actually build a mental image!

Thanks for the input. Correct me if I'm wrong, but I think you're explaining a universe that is finite and unbounded. It's the bounded part that I find hard to conceptualize. From Wikipedia: "Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a disc, have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically."

I'd be interested to hear more detailed theories about what this edge could be like.

 

[mgkii]

You're not wrong... I need to learn to concentrate more!

 

[Alltimegreat1]

You're not wrong... I need to learn to concentrate more!

We all need to concentrate more in order to visualize the edge of the universe.

 

[Bandersnatch]

A number of scientists subscribe to this theory.

Personally, I've never heard of any, at least not in mainstream cosmology. Can you provide examples?

By the way, you mean finite and with a boundary. Bounded has pretty much the same mathematical meaning as finite in extent.

 

[Alltimegreat1]

Yes, I mean finite with a boundary.

 

[newjerseyrunner]

I think you are confused. Bounded does not mean that there is a boundary. The surface of a sphere is bounded, but it has no boundary.

 

[mgkii]

I don't think the OP is confused - extract from wiki below suggests the use of the term is correct (or wiki is wrong)

Bounded and Unbounded
Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a disc, have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is very difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the 3-sphere and 3-torus, which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact basically means that it is finite in extent ("bounded") and is a closed set. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are both closed manifolds.

 

[Bandersnatch]

An infinite universe (or infinite in a specific spatial direction) must be unbounded in that direction.

Use of finite and bounded or infinite and unbounded is correct but redundant in this context. On the other hand, it is clear from post #5 that the OP meant 'finite and with a boundary' - i.e. with and edge.
But the OP has acknowledged the mistake in post #9, so it's now just beating a dead horse.That aside, I'd still like to get some references from the OP for scientists building/favouring a cosmology with a boundary as a feature.

 

[phinds]

I think you are confused. Bounded does not mean that there is a boundary. The surface of a sphere is bounded, but it has no boundary.

No, the surface of a sphere is unbounded. You are, perhaps, thinking not of the surface of the sphere but of the full sphere embedded in 3D space. That IS bounded, although its surface is not.

 

[Flatland]

Would a finite but bounded Universe actually make any sense? Who are these so called scientists that supports this theory?

 

[phinds]

Would a finite but bounded Universe actually make any sense? Who are these so called scientists that supports this theory?

I don't think so. It would have rather severe consequences. Current physics could not explain what happens at the boundary and it would defy the Cosmological Principle.

 

[micromass]

No, the surface of a sphere is unbounded.

Nope, the surface of a sphere is bounded. It forms a bounded metric space.

 

[phinds]

Nope, the surface of a sphere is bounded. It forms a bounded metric space.

Hm. I thought, and seem to remember seeing on this forum several times, that the surface is unbounded because you can travel forever on it and never hit a boundary.

 

[micromass]

Hm. I thought, and seem to remember seeing on this forum several times, that the surface is unbounded because you can travel forever on it and never hit a boundary.

In mathematics, there is a huge difference between "having a boundary" and "being bounded".

 

[Cruz Martinez]

No, the surface of a sphere is unbounded. You are, perhaps, thinking not of the surface of the sphere but of the full sphere embedded in 3D space. That IS bounded, although its surface is not.

The sphere is bounded although it has empty boundary.

 

[Fervent Freyja]

I was under the assumption that the terms bounded and unbounded (finite and infinite) universe occurred in particular contexts and served as more of a conceptual aid alongside that datum, and as a way to help one recognize patterns later within different concepts. Also, the terms are used in all sorts of other places like topology. I cannot imagine a person wading very far in physics while trying subscribe to just one of those. Are these scientists credible, have you checked their prior work, or reputations?

My visualization of what the universe could look like has become increasingly elusive and it is difficult to maintain a focus for very long without encountering thought issues or the most unpleasant combination of running both. These terms imply much more than just topology...

There is no consensus on the shape of the universe. There is good reason that millions of scientists aren’t drawing pictures to help the public envision it… Wonder what would happen there? Would there be a war?

 

[Chronos]

The 'no boundary' proposition, which IIRC, was first proposed by Hawking, then evolved into the finite but unbounded idea [whatever that means]. I do not recall any serious proposal suggesting a finite, bounded universe - which would face obvious and serious logical and mathematical difficulties. So I share Bandersnatch's interest in a credible reference. The only generally agreed upon bounday condition that applies to the universe is temporal.

 

[Chalnoth]

The 'no boundary' proposition, which IIRC, was first proposed by Hawking, then evolved into the finite but unbounded idea [whatever that means]. I do not recall any serious proposal suggesting a finite, bounded universe - which would face obvious and serious logical and mathematical difficulties. So I share Bandersnatch's interest in a credible reference. The only generally agreed upon bounday condition that applies to the universe is temporal.

I think this is only ill-defined if you take the term "universe" literally to include everything. A finite, bounded universe is perfectly fine within some multiverse models.

For example, a finite, bounded universe is one potential consequence of a universe that starts in a high-energy false vacuum state which subsequently decays to a lower-energy vacuum state. The region with lower vacuum energy expands outward at very near the speed of light, but because there is a positive vacuum energy in this model, there's a horizon that allows this boundary to move perpetually.

Some have proposed that we could see evidence of such things by looking for collisions of expanding vacuum bubbles in the distant past (they would appear as sort of rings on the CMB). So far the statistical evidence is quite weak, so we'll probably never be able to positively detect them (at least not in that way).

 

[Bandersnatch]

So I share Bandersnatch's interest in a credible reference.

You don't share my interest in using (un)bounded and with(out) a boundary correctly, though.

 

[fresh_42]

You don't share my interest in using (un)bounded and with(out) a boundary correctly, though.

A boundary implies the existence of an outside which makes it difficult and unpopular for the universe to have one.
The boundary of Earth ($B^3$) in space ($ℝ^3$) is its surface ($S^2 = \overline{B^3}∩\overline{ℝ^3 \backslash B^3} ⊂ℝ^3$).
A sphere alone ($S^2$) has no boundary because you don't need an outside to de defined.
A bounded (metric space) means any two points can't be further away than a certain number, a maximal distance or radius. If the universe is more or less $ℝ^{3,1}$ it would be unbounded.
The universe to be finite means in this context basically that it is bounded (by some maximal expansion or radius).
So finite and bounded concerning the universe are redundant adjectives.

 

[Alltimegreat1]

That aside, I'd still like to get some references from the OP for scientists building/favouring a cosmology with a boundary as a feature.

Aside from the Wikipedia article I posted I couldn't find anything specific either. I met up with an astronomy student last week who said that bordered-universe theories should not be discredited. The main theory he talked about involves a sphere in 3D space with no curvature around the 4th or any other dimensions. The entire edge of the universe is a black hole state.