# 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.

• Why do you assume that the universe contains finite mass and energy? Certainly, the observable universe does, and of course it's rather difficult to make observations of the conditions outside the observable universe. ;) – PM 2Ring Feb 12 '17 at 8:19
• Are you conflating "the universe" with "the observable universe"? The universe may be infinite, with infinite mass energy. But the observable universe is finite. – user12029 Feb 12 '17 at 8:32
• The Universe may be finite, but it can still be very large, and not necessarily fixed. Say it did start from a specific point a la Big Bang and expanded outward from there. The edge may have spread much faster than the matter. In this case, there could be an edge that meets your criteria, but because we also have finite energy it could be impossible to ever catch up to and observe this edge. It could be an edge that nothing can ever get close to because it would take more energy than is available in the known and unknown to push any matter that far away. – Joel Coehoorn Feb 12 '17 at 21:31
• I assume a finite universe because in an infinite one, absurdity is the norm, and anything that can happen does, which I reject simply because it makes fundamentalist Christian views look wise. In other words, if I refuse to believe that a whale swallowed a man and he lived to tell about it, then I also refuse to believe there's a version of me that's pink, rules the world, and eats nothing but cheese. Little stuff like that makes a universe of infinite matter/energy fly out the window of serious discussion--but I'll admit there's no way to disprove it. – Jasper T. Scott Feb 13 '17 at 1:18

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.

• "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." I'm not sure if I get this. If it formed multiple "bubbles" then there must be boundaries between them right? And so then the "part beyond the horizon" would not be homogeneous because of the changes between bubbles, yet you say (cont'd) – The_Sympathizer Feb 12 '17 at 11:44
• (cont'd) there is "no theoretical reason to think so". This seems contradictory. What is going on? – The_Sympathizer Feb 12 '17 at 11:45
• Homogeneous before possibly (and it is not known if multiple bubbles, just a possibility called eternal inflation) multiple bubbles. After a bubble, like our universe, homogeneous inside and then other sprinkled bubbles. And.bubbles are just areas where there was inflation, no edges. Remember bubbles is just an unproven theory. – Bob Bee Feb 12 '17 at 12:34
• @BobBee Being simply connected does not mean that the topological space has the trivial topology; it means that the space is path-connected and the fundamental group of the space is (the) trivial (group). I now see that you attach the indefinite article to "trivial topology", so perhaps you mean something very specific by that phrase. – Eli Bashwinger Mar 17 '19 at 16:33

First, it is not known that the Universe is finite. The WMAP experiment makes it more probable, that it is infinite. It was a satellite to measure the curvature of the Universe by finding tiny disturbances in the Cosmical Microwave Background. The result was zero (in the measurement precision).

Second, it is not known that the Universe is simply connected. For example, if the Universe has a constant positive curvature (which is smaller as the precision of the WMAP experiment), the Universe can have a spherical geometry. In essence it would mean that we live in the surface of a 4-dimensional sphere. In this case, the Universe doesn't have an edge, although the 4-sphere is simply connected and finite.

Third, it is not known if the Universe is finite. We can see only structures with our telescopes, whose light started at the Big Bang (they seem 13.7billion light year far, although they are currently around 30-40billion ly away because of the expansion). On such very high scales, there are no significant visible disturbances, thus we can think, the Universe out of this horizon looks probably the same (although it is not an experimental validation).

There is no data, what is out of this horizon, but the Universe is infinite and looks similar as in our direct environment, then there is infinite mass in it.

Furthermore, even if it has a planar geometry, it can have some cyclic topology. Imagine old computer games, as the player leaved the screen on the right and then appeared on the left. In this case it again doesn't have an edge (and it is not simply connected).

We can see the Universe not only in space, but also in time. The spacetime of the Universe has a singularity in the Big Bang. That we can see as an edge - but not in a far point, but in the far past.

• The short and direct answer to your question is, in my opinion, yes (we can see the most far particle as an edge). This essay is more from the wider context (the current experimental results suggest quite different possibilities). – peterh - Reinstate Monica Feb 11 '17 at 23:20
• Both your first and third points talk about the universe being finite, but I think from the WMAP reference that your first point is supposed to be about the universe's (global) curvature. – PM 2Ring Feb 12 '17 at 8:16
• @PM2ring The WMAP results show a Universe which has at least 300billion ly radius in the case of spherical geometry. My layman intuition see this the most probable. But maybe it is really planar. – peterh - Reinstate Monica Feb 12 '17 at 8:42
• Infinite mass of the universe is eliminated due to bad interactions near (in time) to the big bang. However, the method of proof does not set an upper bound as only a true infinity fails. – Joshua Feb 12 '17 at 17:17

Yes, your conclusion would be correct, if your assumptions are correct. However, there is a lot of evidence that contradicts one of your assumptions that "... the universe is... flat."
A lot of effort has gone into trying to determine if there is just the right amount of matter-energy in the universe to make it flat. And so far, we are way off from having enough matter-energy to do it.

The universe is expanding at the speed of light. Therefore you cannot ever reach the edge without exceeding the speed of light.

This is due to the fact that you are currently not at the edge of the universe and simultaneously moving at the speed of light or faster than the speed of light.

How did I arrive at this conclusion?

Simple.

The big bang caused expansion to occur at light speed according to some researchers.

Since there was no matter in the universe at the time where the big bang took place, the expansion was able to continue outwards without anything slowing it down.

This is still happening today.

Therefore, since there's nothing to slow down the Universe's expansion, then it must still be moving at its original speed (the speed of light).

With that being said, the edge of the Universe does not exist due to the fact that you can never reach the edge of the Universe without traveling faster than light speed, which is supposedly impossible. Also, there was no matter outside of the Big Bang supposedly, which means that nothing will ever exist at the very edge of the Universe.

• The Big Bang was not some kind of explosion in a pre-existing void. According to Big Bang theory, the space-time of our universe originates at the Big Bang. – PM 2Ring Feb 12 '17 at 8:21
• Furthermore, the geometry of the spacetime has a singularity at the Big Bang, thus it is meaningless to say, what was at the Big Bang. It is like to say, what is at $\frac{1}{x}$ if $x=0$. If you nears the Big Bang, you have a continuously growing energy density (and an unknown, but constant total energy). – peterh - Reinstate Monica Feb 12 '17 at 8:46
• Gravity can slow stuff down, but matter can't move at light speed without having infinite kinetic energy (which is impossible). – wizzwizz4 Feb 12 '17 at 12:25
• At the time of the Big Bang all the mass in the universe existed. What didn't exist was the volume which that mass now occupies. – Bob Jarvis - Reinstate Monica Feb 12 '17 at 16:35
• @bob jarvis: depends what you mean by "existed". Inflation theory says that almost all the mass of the current universe condensed out of the potential energy of the inflation field at the end of the inflationary epoch, diluting the matter/energy that existed before inflation by a factor of at least 10e78. – Chappo Hasn't Forgotten Monica Feb 17 '17 at 7:52