Can the universe be flat, yet finite?

[epovo]

I keep on reading that cosmologists contemplate two possibilities: either the universe is closed, unbounded and therefore finite, or else it is open (possibly flat) and infinite. I hear that a flat, finite universe "introduces many problems" and is discarded. My question would be "what problems?". I understand the the observable universe is smaller than the whole thing, so I cannot see a problem with this notion. In such a universe, there would be some observers who would certainly see anysotropy, but that would not be the case for the majority of the observers, would it? What other problems exist?

 

[Mordred]

In order for the universe to be flat and finite would require geometries such as Klien bottle,
Mobius strip etc. While these shapes are geometrically possible in terms of spatial coordinates in regards to cosmology causes other problems
How cosmology determines shape is a ratio between actual density and critical density.(conditions needed to be flat and static ).

More detail can be found in this article.
https://www.physicsforums.com/showthread.php?t=694499

much like the saddle shape in the article alternate shapes would present problems in that
different regions of the klien bottle as one example would need a curvature constant variation. As well as inhomogenous and anistrophic regions that cannot be described as homogenous and isotrophic on the same scale as other regions.
So some form of energy density would need to vary to cause a flat topography to start to loop back.
Needless to say there is insufficient evidence to support complex shapes.

 

[WannabeNewton]

In addition to Mordred's comments, take a look at this classic paper: http://xxx.lanl.gov/pdf/gr-qc/9812046v5.pdf

 

[epovo]

This is not what I meant. I actually meant a flat universe with a boundary, the admittedly naive (or maybe heretical) view of a universe starting from a singularity and expanding into some kind of nothingness. Some galaxies would actually be at the boundary. I don't have a clue what the universe would look like to observers in those galaxies, but it is clearly not our case, because our observable universe is isotropic. My question is 'why not?'

 

[Mordred]

The universe has no outside or boundary as per an edge

 

[epovo]

I know. I was asked this question by a friend and found that I could not give a convincing answer. The question remains 'how do we know that?' or, if you prefer 'what's the evidence'?

 

[WannabeNewton]

Finite translates over to compact (a topological space is compact if every open cover of the space has a finite subcover). This is not the same thing as having a boundary (manifold boundary to be exact). Compact manifolds can easily have empty manifold boundary; such is the case with the n-sphere for example. Such manifolds are called closed, lending to why you might see the $S^{3}$ FRW cosmological model being called a closed universe.

See here for a light summary: https://my.vanderbilt.edu/stacyfonstad/files/2011/10/ShapeOfSpace.pdf

I'm not sure what the theoretical justification is for choosing a model with empty manifold boundary.

 

[martinbn]

I suppose that the universe could be finite, flat with a boundary, but since there isn't any observational evidence nor any theoretical reason for such a model, there has to be something interesting about it in order to study it. Do you have anything in mind?

 

[Tzikin]

Can anybody help me to reconcile these models with the P-Brane hypothesis of theoretical physics?

 

[epovo]

I am not sure either why the universe might have an empty manifold boundary. But again I haven't heard of any observations which contradict this option. If the universe is flat, as observations suggest, why everybody seems to prefer a universe with an infinite amount of space, matter and energy as opposed to one with an empty boundary? Is this based on some theoretical reason, or maybe a philosophical bias such as the mediocrity principle or a (dubious) application of Occam's razor?

 

[Mordred]

In addition to Mordred's comments, take a look at this classic paper: http://xxx.lanl.gov/pdf/gr-qc/9812046v5.pdf

I finally had a chance to read this paper thanks WannabeNewton for pointing it out. It was an excellent read and one that goes into my collection.

I am not sure either why the universe might have an empty manifold boundary. But again I haven't heard of any observations which contradict this option. If the universe is flat, as observations suggest, why everybody seems to prefer a universe with an infinite amount of space, matter and energy as opposed to one with an empty boundary? Is this based on some theoretical reason, or maybe a philosophical bias such as the mediocrity principle or a (dubious) application of Occam's razor?

Lets try a thought experiment, The universe is defined as "everything that is", in general terms in scientific terms one can describe the universe as "everything we can see, measure or interacts with our local space time at any point in time"

Space is simply volume

So ask yourself this question "According to the above definitions, how would one describe outside the universe?" to describe such would mean a region of zero space ie no volume. b) does not interact with our universe c) what would separate our universe from a region of zero volume?

In regards to barrier "how would light interact with said barrier? it would either reflect or get absorbed by the barrier, those are the only two possibilities.

When we look further out in distance we see further into the past. so why is it we never detected any barrier?

now in regards to flat being normally described as infinite, this is shown in the articles we posted. to have a finite and flat topography we would need a more complex shape I already posted the problems with that.
Also shown in the links provided what determines our shape. the key points is how it affects expansion and light paths.

I should also point out that we don't state the universe is flat. we state that it is close to flat, this is important. a 100% flat with no unusual change in any of the values used to determine flat would be infinite. However in our case being close to flat the universe could simply be so large that our observable portion only accounts for a small portion of the entire universe. So its like a fly sitting on a ball. It looks flat but isn't.
In other words although we measure the universe to be flat because its so large it could still be a sphere or hyperbola.