Does a finite universe require 4 spatial dimensions?

[dilletante]

A 2-dimensional creature living on the surface of a 3-dimensional sphere could conclude he lives in a finite, unbounded universe. Is it necessary for a 3-dimensional creature to assume there is a 4th spatial dimension in order to conclude the universe is finite and unbounded?

I have seen a 3-sphere posited as one possible shape of a finite universe, which Wikipedia describes as requiring 4 dimensions. I assume that only 3 dimensions are required if the universe is infinite.

Is there another choice between assuming either infinity, or an additional dimension we cannot perceive?

 

[Buzz Bloom]

Is there another choice between assuming either infinity, or an additional dimension we cannot perceive?

Hi dilletante:

My understanding of the interpretation of curvature in the universe is that there is no actual 4th spatial dimension in the universe. Whether the universe is finite or infinite, there is no 4th spatial dimension in the universe.

If the universe is finite, it will have a positive curvature, and it may be conceptually convenient to think about the geometry metaphorically as a three space curved around a center in a 4th dimension. An alternative visualization is a three space with every 2D flat-as-possible-almost-plane to actually be the surface of an ordinary sphere.

If the universe is infinite and flat, i.e, having zero curvature, no 4D metaphor is needed to assist visualization of the geometry.

If the universe is infinite with positive curvature, it may be conceptually convenient to think about the geometry metaphorically as a three space such that every 2D flat-as-possible-almost-plane to actually be (relative to a local point) a surface having a saddle shape.

I hope this is helpful.

Regards,
Buzz

 

[Hornbein]

A 2-dimensional creature living on the surface of a 3-dimensional sphere could conclude he lives in a finite, unbounded universe. Is it necessary for a 3-dimensional creature to assume there is a 4th spatial dimension in order to conclude the universe is finite and unbounded?

I have seen a 3-sphere posited as one possible shape of a finite universe, which Wikipedia describes as requiring 4 dimensions. I assume that only 3 dimensions are required if the universe is infinite.

Is there another choice between assuming either infinity, or an additional dimension we cannot perceive?

As best as I understand it, it goes like this.

Our Universe could be finite and unbounded if it has positive curvature. The curvature is due to the mass it contains. That is, the gravity of the mass bends space so that light travels in a curved path. Eventually the path has curved so much that the light is not getting any further from the point that emitted it. Indeed the distance between the emission point and the actual location of the light may decrease as the light propagates. So the Universe is bounded in this sense. Light could only get so far away from its point of emission.

This is however confounded by the expansion of the universe. It might be that light always grows further in distance from its point of emission. Then the universe is infinite in a practical sense, but not in a theoretical sense. But it is the theoretical sense that is always discussed.

 

[Chalnoth]

A 2-dimensional creature living on the surface of a 3-dimensional sphere could conclude he lives in a finite, unbounded universe. Is it necessary for a 3-dimensional creature to assume there is a 4th spatial dimension in order to conclude the universe is finite and unbounded?

The key insight in General Relativity is that it describes curvature without referring to higher dimensions. In General Relativity, it's perfectly possible to fully-describe the surface of a sphere by only describing the 2D surface, without reference to a third spatial dimension.

Furthermore, when you run through the numbers to see how many extra dimensions you would need to describe the various kinds of curvature GR permits, it turns out you need a lot more than one extra dimension (I forget how many exactly). And if we actually had a bunch of large extra dimensions, those would most likely be observable.

 

[Johninch]

Our Universe could be finite and unbounded if it has positive curvature. The curvature is due to the mass it contains. That is, the gravity of the mass bends space so that light travels in a curved path. Eventually the path has curved so much that the light is not getting any further from the point that emitted it. Indeed the distance between the emission point and the actual location of the light may decrease as the light propagates. So the Universe is bounded in this sense. Light could only get so far away from its point of emission.

I appreciate this very interesting interpretation. Could you please clarify:

You are talking about a single emission of light, but light is being emitted by galaxies everywhere in the whole universe. Are you assuming that there are no galaxies very far away from us, so that in that region no light is emitted? I understand why you are saying that “light could only get so far away from its point of emission”, but I am not understanding why you are concluding from this that it causes the universe to be bounded. Why should there be a region without emissions of light?

 

[Hornbein]

I appreciate this very interesting interpretation. Could you please clarify:

You are talking about a single emission of light, but light is being emitted by galaxies everywhere in the whole universe. Are you assuming that there are no galaxies very far away from us, so that in that region no light is emitted? I understand why you are saying that “light could only get so far away from its point of emission”, but I am not understanding why you are concluding from this that it causes the universe to be bounded. Why should there be a region without emissions of light?

First let me warn you that I don't have all that firm of a grip on this. I've always read that in a finite unbounded universe that "light always comes back to the point it was emitted." Surely this is not true, so I guess that this is a very rough approximation. I was trying to come up with something more realistic. Perhaps not a good idea.

I wrote about the unbounded case. I'm trying to show how a Universe could be finite and unbounded. The Universe we observe is almost perfectly homogenous and it is believed that all of it is homogenous. No one thinks that there is a region without emission of light.

Our Universe is observed to have very little curvature, perhaps none at all. But in its early moments it was extremely curved.

 

[PeterDonis]

Our Universe could be finite and unbounded if it has positive curvature.

It can actually be finite and unbounded even if it has zero curvature, but that case is harder to visualize (the universe would have the spatial topology of a 3-torus instead of a 3-sphere).

I've always read that in a finite unbounded universe that "light always comes back to the point it was emitted." Surely this is not true

No, it is true. Consider the simpler case of a 2-sphere, like the Earth's surface. Anything that sets off on a geodesic path (i.e., traveling on a great circle) will eventually circumnavigate the globe and return to its starting point. In a finite unbounded universe, light works the same way--if it travels long enough, it will circumnavigate the universe and return to its starting point. But it takes a long time for it to do that--how long depends on how large the universe is spatially (if it's finite), just as how long it takes something to circumnavigate the Earth depends on how large the Earth is. Since the universe has a finite age, it could be finite but unbounded even though we don't see any light that has gone all the way around it--yet. The finite size would just have to be large enough that no light has had time to circumnavigate it yet.

[Edit: it's important to note, in view of what Jorrie posted in post #9, that in order for light to circumnavigate a finite but unbounded universe, the expansion of the universe can't be accelerating. If it's accelerating, light will never make it all the way around, because the size of the universe increases faster than the light can cover it, so to speak.]

 

[PeterDonis]

Our Universe is observed to have very little curvature, perhaps none at all. But in its early moments it was extremely curved.

It's very important to distinguish spacetime curvature from space curvature. Our universe's spacetime curvature was larger in early times than it is now, because it was expanding faster. But its spatial curvature in early times could still have been zero (as it is now, according to our best current estimates).

 

[Jorrie]

This is however confounded by the expansion of the universe. It might be that light always grows further in distance from its point of emission. Then the universe is infinite in a practical sense, but not in a theoretical sense. But it is the theoretical sense that is always discussed.

The accelerated expansion that causes a finite cosmological horizon, means that distant light (outside the horizon) can never reach us. Whether one calls that "infinite in a practical sense" is a matter of choice of words - geometrically it can be spatially finite or infinite.

 

[Buzz Bloom]

First let me warn you that I don't have all that firm of a grip on this. I've always read that in a finite unbounded universe that "light always comes back to the point it was emitted." Surely this is not true, so I guess that this is a very rough approximation.

Hi Hornbein:

If the cosmological constant were positive (attracting, as originally conceived by Einstein) rather than negative (repelling, as it has been measured to be) then all of a finite universe would be the same as some future observable universe. Then in principle a future observer could "see" the spot in the universe where we are now via light that has circumnavigated the curved space to return back to where we are.

In the universe we seem to be in, as described by the "standard model", there is a possibility that the universe is finite. However the future observable universe is always much smaller than this finite universe, as both expand forever. Therefore the point where we are now cannot be observed at some future time via light that has circumnavigated the curved space to return back to where we are.

Regards,
Buzz

 

[PeterDonis]

If the cosmological constant were positive (attracting, as originally conceived by Einstein)

No, Einstein's original conception of the cosmological constant was "repelling"; he wanted a static universe, and he knew that ordinary matter produced attractive gravity, so he added a cosmological constant to provide just enough repulsion to cancel the attraction of ordinary matter.

(The sign of a "repelling" cosmological constant is positive, btw--that is, positive when it appears on the LHS of the Einstein Field Equation, where Einstein originally put it. Also, a negative cosmological constant would not really be "attracting"--it still would not work the way ordinary matter works.)

all of a finite universe would be the same as the observable universe. Then in principle a future observer could "see" the spot in the universe where we are now via light that has circumnavigated the curved space to return back to where we are.

I also don't understand what model you have in mind here.

 

[Buzz Bloom]

No, Einstein's original conception of the cosmological constant was "repelling"

Hi Peter:

I apologize for another senior moment. I got myself mixed up about the sign. Thanks for catching it.

I also don't understand what model you have in mind here.

I am NOW thinking of a finite universe with any cosmological constant other than one which is so strongly repelling it leads to a universe that eventually accelerates the expansion of the universe. This includes any value less than some small positive value so that the universe eventually reaches a maximum size and then begins to contract. At this maximum size the repelling force would be less than the matter gravitational force at that size. In such a universe a photon could in principle leave some point in space and return to the same point at some future time.

Regards,
Buzz

 

[PeterDonis]

In such a universe a photon could in principle leave some point in space and return to the same point at some future time.

In principle, yes, but in the case of zero cosmological constant, it turns out that a photon can't actually do this in a closed universe that recollapses; by the time the photon is able to make it all the way around the universe and return to its starting point, the Big Crunch has happened.

In the case of a small positive cosmological constant (small enough to not prevent recollapse, as you say), I haven't calculated whether the above is still true; i.e., whether recollapse would still happen before the photon could make it all the way around).

 

[Buzz Bloom]

in the case of zero cosmological constant, ... by the time the photon is able to make it all the way around the universe and return to its starting point, the Big Crunch has happened.

Hi Peter:

Thanks for your post.

It is always a pleasure to learn something unexpected about how both the real universe and its related mathematical models work.

In the case of a small positive cosmological constant ..., I haven't calculated whether the above is still true;

If you do sometime do this calculation, I would be very interested in seeing the result, as well as the steps you use. It would be a valuable education for me about this subject.

Regards,
Buzz