Expert: Philip A. Stahl - 7/24/2010

Question
QUESTION: As I understand it, current evidence indicates the universe is essentially Euclidean and "flat", as opposed to extra-dimensional, spherical, or hyperbolic; and that it may be either finite or infinite in volume. I also understand that it has no "edge" nor any preferred location such as a "center."  Two questions:

1) If the universe is flat, finite, and essentially three-dimensional, how can it have no edge? (I do understand how it could be edgeless if it were, for instance, Klein bottle shaped.) Specifically, if you could travel in a straight line for an arbitrary amount of time, FTL or otherwise (or instantly teleport to any arbitrary location), wouldn't you eventually reach a boundary beyond which no structure or matter existed?  What would happen if you tried to continue past that point -- would you see nothing but black and your spacecraft just sit in place even though you were expending propulsion energy?  If not, why not?

2)If the universe is Euclidean and "flat" but infinite in size, how can it be expanding? Isn't infinite just about as "expanded" as it could possibly get? For that matter, how could it be infinite now if it were a singularity 14 billion years ago?

Thanks :)

ANSWER: Hello,

Your question poses a number of problems because: 1) I'm not sure how much background you have in topology or cosmology, and 2) I'm not sure how much mathematics you've had. In past answers such as this, I've been burned (more accurately scorched!) by assuming too much and then being waylaid with very low ratings! So, all I'd ask is that if you're unclear about any part of the answer to come back and ask additional questions, rather than just give a '6' for "clarity".  I will also ask that in future, you stick to one (or two) questions at most, the reason being is that if I answer multiple "bound" questions I still only receive credit for one! Fair enough? (Btw, you say "two questions" but please note you have 5 bound up just in #1!)

Anyway, let's look at (1) first.

The danger of terms like "edge" is that often they haven't the same meaning in the cosmological context. That's why they're dangerous to use, or overuse. When most of us use the term "edge" what we really mean (in terms of the existent cosmos we know - leaving out anything to do with a "multiverse" or "Brane space" cosmos) is the limit defined by the light "horizon" at a given time (age). The diagram I've appended should help to make this more clear. (Though it still has limitations!)

Thus, as T increases, more of the cosmos "enters" the horizon as you can see from the diagram (comparing say the enclosed area under the graph for 7 billion years age, with that now at 14 billion years age).  Thus, the "edge" is real but not so much geometrically as in terms of whether distant objects can be accessed within our "light horizon". In effect, one has to discriminate between the *observable* universe we can actually detect (within our light cone or horizon) and the putative universe that exists - which includes the subset of the observable, plus what hasn't yet entered our light horizon.

Thus, mixing terms like "flat" and "edge" gets us nowhere (Apart from the fact that the topology in cosmology and topology in mathematics don't have the same meanings). Thus, the three classes of curvature (k =0 ("flat"), k = +1, positive (spherical) and k= -1 (hyperbolic)) do not mean the same thing as the terms, e.g. flat, closed, hyperbolic, in standard topology.)

Is this complicated? Yes, you can believe it! Which is why it also makes it extremely difficult to explain. (Which is also why your Klein bottle example, analogy is useless, since a Klein bottle, reduced via algebraic topology -homology, can be "resolved" (triangulated) into triangles (normal Euclidean ones) in such a way that every 1-cycle is homologous to a cycle of form: ra + sb where r,s are integers.

This is in no way applicable to the cosmos since it is not simply a topological artifact but a dynamic one which also employs a totally distinct geometric manifold(see (2) below). For example, in the strict topology sense, the Gauss-Bonnet theorem forces the sign of curvature k to match the Euler characteristic, E. (Recall E = n(e) + n(2) - n(3) where n(e) denotes number of edges, n(2) number of 2-simplexes, and n(3) number of 3 simplexes. ) This is not required in terms of cosmology, curvature of the cosmos.

So this is why the remainder sub-questions embedded in your question (1) are simply inapplicable

As for (2):


Again, bear in mind the specialized differences in terminology noted above. In addition, you must disabuse yourself of any images of standard (orthodox) Euclidean space (say retained from high school geometry) when considering the meaning of "flat" applied to the cosmos. This will only lead you astray conceptually!  What we actually are referencing is a Euclidean 4-space (since we invoke space -time, not just space) which is a vector space R(4) possessing an inner product: (u,v) = [uo*vo + u1*v1 + u2*v2 + u3*v3].  Given this E(4) <-> R(4) combination we can easily have the universe expanding as an infinite entity and without expanding "into" any thing else. (When we use the term "infinite" here we essentially mean its geometry is self-contained. Since there is nothing "outside it".)

The singularity to which you refer (for the inception of the cosmos at the Big Bang) was not "infinite" in this sense- say applied to a vector space of any type- because its infinite density disallowed any approach for any known geometry!  Hence, it is essentially inaccessible to all intents and purposes.

I'm not sure if any of this will make much sense, but it's about the best I can do without having to write a 200-page detailed book, explaining in 100 x more detail each of the preceding concepts introduced.

However, if you're seriously interested(and you appear to be), may I recommend getting hold of Roger Penrose's terrific book, 'The Road To Reality'?  This magnificent work not only features all the math (including hyperbolic and Euclidean 4-space geometry) to understand what you're inquiring about, but also the related cosmology. Further, Penrose explains each nuance and step patiently and takes his time (and space) to do so.

If you do get the book, pay special attention to Chapters 16-19, which deal with and elaborate such issues as whether an "infinite geometry" can be applicable in physics, as well as "different sizes of infinity", Light cones, space-time for general relativity, Euclidean and Minkowskian 4-space, hyperbolic geometry in 4-space.

As you can easily see, merely because one can think of questions (that *appear* to have relatively straightforward answers) doesn't mean they must!



---------- FOLLOW-UP ----------

QUESTION: This is just to tell you that I intended to give you a high rating, but when I clicked on the Rating button it never gave me a chance to rate.  This is probably because I missed the rating section on the previous screen. In any case, I don't know what the default rating is that I assume was submitted, but if it's below 9 I would like the chance to give you my actual rating, if that can be arranged.  If it's 9 or higher then I will stick with it. :) I do still wonder about what would happen if you teleported or flew in a straight line to an arbitrarily remote distance in a closed, flat universe...

Answer
Hello,

Thanks for your ratings, which I not long ago received via e-mail from All Experts.

As to this last question, unfortunately, a "closed flat universe" is not possible. A flat cosmos has a zero or critical curvature (k =0) so we'd not define it to be "closed". It would have to have a curvature of k= +1, in which case the geometry would be spherical, to be "closed". (The "open" universe applied to k = -1, and the geometry is Lobachevskian or "horse saddle" shape).

So the conceptual basis of the question is an impossibility!