general relativity - If the Universe is Flat, has Finite Mass/Energy, and is Simply Connected, Then there MUST be an Edge, Mustn't there?

Assumptions:

The universe is flat (currently supported)

The universe is simply connected (the edges aren't glued together as in a torus)

The universe contains finite mass and energy

Conclusion:

The universe must have an edge.

Yes, there is a similar question here: How can the universe be flat and have no center if universal mass-energy content is finite?

But my question is not answered. In fact, people are neatly dodging the notion of an "edge" by suggesting "unusual topologies"

This is a purely hypothetical question, but since everyone says the universe has no edge and is flat, I am forced to ask the obvious: space or some form of truly empty vacuum might go on forever, but if matter/energy are finite in the universe, then eventually, if we travel far enough past the cosmic horizon, we'll find that there are no more stars, no more galaxies, no more photons... and no more anything. Unless the universe is actually a sphere, in which case eventually we'll end up back where we started.

Is there a flaw in my reasoning? I must have read 100 articles today to get to the bottom of this.

Answer

Yes, if the universe is:

• 
  

  flat (zero spatial curvature)

  
  
  


• 
  

  has finite mass energy (since we know it is uniform this also means it is bounded. If you drop the bounded es because you don't want to admit uniformity or otherwise, i.e., if it is unbounded, then the answer is clearly no)

  
  


• 
  

  is simply connected (has what is called a trivial topology)

  
  

Then it does have to have an edge.

See the zero curvature and other sections of the wiki article on the shape of the universe, it's fairly complete, at https://en.wikipedia.org/wiki/Shape_of_the_universe

The simply connected condition is critical also. If you allow other topologies then both the torus and the Klein bottle topologies are bounded, flat and have no edges.

There are a total of 17 possible different topologies for multiply connected spaces that are flat, in 3D (our spatial dimensions, which is what is referred to when one talks about curvature of the universe) Riemannian space. See fig. 4 in the arXiv paper at https://arxiv.org/abs/0802.2236 for all of them. There are others if the space is not flat.

As far as space being unbounded but mass energy finite, that would violate what we know of the homogeneity and isotropy of the universe. From the CMB we see the (large) scale homogeneity and isotropy. Now, we only see back to 380,000 years after the Big Bang, but no sign of large inhomogeneities. It could theoretically still be true that out inflation bubble is homogeneous, and thus the part of the universe beyond our particle horizon might not be, but there is no theoretical reason to think so. The more prevalent view is that it was as uniform more or less, and the same inflation that created our bubble might have created others. If we ever fully understand our inflation (which at this point looks pretty consistent with observations but those don't rule out various versions, or other unknown mechanisms from an unknown theory of quantum gravity), we might find out better or differently. But presently, a large scale homogeneity with possible bubbles is consistent with all observations.