Infinate Universe... kind of

[beagrie]

Hi,

I'll just preface this post by saying my interest in space is... well it's just that, an interest not a profession, so I apologize if any terms or phrases are wrong or mis-used.

I had this thought after reading a sci-fi book. If we take space to have some kind of edge or boundry, can it still be considered infinate in a mathematical sense? What I mean by this is that when you say space is infinate, a lot people assume you mean it goes on forever. People also assume infinate is the largest number you can count to when in actual fact it's not a number at all and the very premise means you can't count to it.

The Universe has been discovered to be expanding at a rapid rate, and has been expanding atvarious rates since the big bang. As light from near the beginning of the universe is only just reaching us, it's not a great leap to accep that the universe is expanding faster than light can travel.

As nothing in this universe can move faster than light, is it fair to say that the universe is infinate to us? For no matter how far or fast we travel to the outer limits of the universe, we will never reach an edge or boundry (even I the edge of the universe is a solid wall of cheese) because it will always be moving away from us too fast to catch.

I'd like to hear any thoughts on this, even if it's to point me to an article where someone has already said what I've just said.

 

[beagrie]

I'd also liketo apologize in advanced for my spelling of "infinite"

 

[DrRocket]

I had this thought after reading a sci-fi book. If we take space to have some kind of edge or boundry, can it still be considered infinate in a mathematical sense? What I mean by this is that when you say space is infinate, a lot people assume you mean it goes on forever. People also assume infinate is the largest number you can count to when in actual fact it's not a number at all and the very premise means you can't count to it.

Some of the terminology in sci-fi and the popular literature is unfortunate. In general relativity the universe is taken to be a 4-dimensional manifold with a Lorentzian metric, space-time. What are called "infinite" and "finite" are properly termed "open" and "closed" manifolds, respectively. In either case the manifold is without boundary. A closed manifold is a compact manifold without boundary.

If there were a boundary, there would in fact be something of a problem. The boundary of a 4-manifold with boundary is a 3-manifold with boundary. So a boundary for space-time would be a portion of the universe that would have only three dimensions, which does make good sense physically. What would those three dimensions be? Two of time and one of space? Three of space and no time ? A new set of physics would be required to describe what is going on in the boundary.

This does not make it impossible. But it is quite a bit more complicated that you might initially imagine.

Infinity is also a bit more complicated than you might imagine. There are mathematical theories of infinite numbers, called cardinal or ordinal numbers. There are actually different sizes of infinity -- in fact an infinite number of them. In any case you cannot count to infinity. The first infinite number is actually the cardinality of the natural numbers, which are called countable by definition, but you cannot count "to" that number. It is simply a reflection that the natural numbers are also called the "counting numbers". Since there is no largest natural number the counting process never stops.

The Universe has been discovered to be expanding at a rapid rate, and has been expanding atvarious rates since the big bang. As light from near the beginning of the universe is only just reaching us, it's not a great leap to accep that the universe is expanding faster than light can travel.

As nothing in this universe can move faster than light, is it fair to say that the universe is infinate to us? For no matter how far or fast we travel to the outer limits of the universe, we will never reach an edge or boundry (even I the edge of the universe is a solid wall of cheese) because it will always be moving away from us too fast to catch.

I'd like to hear any thoughts on this, even if it's to point me to an article where someone has already said what I've just said.

No it is not fair to say that the universe is "faster than us". It rather depends on what your particular question and perspective might be.

There is evidence that the universe is not only expanding but that the rate of expansion is accelerating. That is a relatively recent development, and it is not understood, It is not inconceivable that this is not so or that it may not be so in the future (I am not saying that such a conclusion is warranted at this time or that it is likely, just conceivable). Tenor twelve years ago it was believed that the rate of expansion was slowing due to gravity. In any case there are legitimate questions regarding the entire space-time manifold. Such questions include the topology of "space-like slices which is related to curvature. It is not the case that simply considering them to be "infinite", or more properly open, is at all helpful. In fact it makes moot some of the deeper questions of cosmology.

From a practical matter the universe, and for that matter the galaxy are so large that travel there is totally impractical given current technology or any technology that we can even imagine at this time. But those constraints have nothing to do with our ability to understand through the application of physics the universe in which we live. To apply physics an mathematics it is necessary to consider questions involving the entire space-time manifold, no matter how large it may be.

 

[beagrie]

Hi DrRocket

I don't know if it's just me or the way you wrote your reply but it seemed you wrote as though trying to spare my feelings over how far off the truth I was. If that's the case I appreciate it. I never thought I'd be bang on the money. I'm 26 and only really just starting to take an active interest in space (eg Reading books rather than just being interested in space news stories) so I'm working my way through things with what Terry Pratchett calls "Lies to children."

Every correction helpful people like you make on what I think I know helps me learn a little more.

Thank you for your reply.

 

[derekmcd]

Hi DrRocket

I don't know if it's just me or the way you wrote your reply but it seemed you wrote as though trying to spare my feelings over how far off the truth I was. If that's the case I appreciate it. I never thought I'd be bang on the money. I'm 26 and only really just starting to take an active interest in space (eg Reading books rather than just being interested in space news stories) so I'm working my way through things with what Terry Pratchett calls "Lies to children."

Every correction helpful people like you make on what I think I know helps me learn a little more.

Thank you for your reply.

One thing I can assure you is that DrRocket spares no ones feelings... He's our Dr. House :lol:

One thing we are pretty sure of is that the observable universe has no boundaries... there is no "edge". Whether it is infinite or not is another ball of wax. Like the good Dr mentioned, if the manifold is closed, similar to the surface of a sphere, then it is considered finite. If you travel far enough, you will end up where you started.

If the curvature of the manifold is considered "open", you travel along a geodesic that never quite reaches the beginning point. If the curvature is "flat"... well, this is infinite in extent.

Our most current measurements have the observable universe very nearly flat with a possible closed curvature. Given the accelerated expansion, even with a closed curvature, it is still considered infinite in extent as you could still never reach your starting point.

Now, to change gears a bit...

There's a difference between "the observable universe" and "The Universe". The observable universe only describes what we have causal contact with. The Universe describes everything beyond...

Even if the observable universe is considered finite without boundaries, the rest of the Universe may still be infinite in extend... we just don't know. We simply don't have the technology to deal with what is beyond the observable universe.

 

[JKMurphy]

Hi DrRocket

I don't know if it's just me or the way you wrote your reply but it seemed you wrote as though trying to spare my feelings over how far off the truth I was. If that's the case I appreciate it. I never thought I'd be bang on the money. I'm 26 and only really just starting to take an active interest in space (eg Reading books rather than just being interested in space news stories) so I'm working my way through things with what Terry Pratchett calls "Lies to children."

Every correction helpful people like you make on what I think I know helps me learn a little more.

Thank you for your reply.

One thing I can assure you is that DrRocket spares no ones feelings... He's our Dr. House :lol:

One thing we are pretty sure of is that the observable universe has no boundaries... there is no "edge". Whether it is infinite or not is another ball of wax. Like the good Dr mentioned, if the manifold is closed, similar to the surface of a sphere, then it is considered finite. If you travel far enough, you will end up where you started.

If the curvature of the manifold is considered "open", you travel along a geodesic that never quite reaches the beginning point. If the curvature is "flat"... well, this is infinite in extent.

Our most current measurements have the observable universe very nearly flat with a possible closed curvature. Given the accelerated expansion, even with a closed curvature, it is still considered infinite in extent as you could still never reach your starting point.

Now, to change gears a bit...

There's a difference between "the observable universe" and "The Universe". The observable universe only describes what we have causal contact with. The Universe describes everything beyond...

Even if the observable universe is considered finite without boundaries, the rest of the Universe may still be infinite in extend... we just don't know. We simply don't have the technology to deal with what is beyond the observable universe.

[color="green]Surprisingly I understood most of what you said lol

when they clalculate the size of the universe and how fast it is traveling do they take into account how long it take for the light from the edge of what we can see takes to reach us?[/color]

 

[DrRocket]

Our most current measurements have the observable universe very nearly flat with a possible closed curvature. Given the accelerated expansion, even with a closed curvature, it is still considered infinite in extent as you could still never reach your starting point.

I know this is what you read in the literature, and it is consistent with what is found in many physics texts. But I have a problem with it.

Generally physicists relate curvature with openness and closedness of the space-time manifold. In fact, what they do is this. First it is assumed that the universe is homogeneous and isotropic (which it is not on small scales). That is defined to mean that, among other things there exists a foliation of space-time into + a one-parameter family of "space-like slices". The parameter then plays the role of a global notion of time. The space-like slices turn out to have constant curvature and a Riemannina metric inherited from the Lorentzian metric of space-time. Then they appeal to a general theorem of differential geometry that classifies spaces of constant sectional curvature. They then classify the possibilities as open (negative curvature or zero curvature) or closed (positive curvature). In doing this they ignore some potential manifolds for reasons that are not very clear to me(Wald seems to dismiss them because some are realized by means of identifications or as mathematician would say as a quotient space). That seems to me to be a rather weak argument since the means by which one constructs and example of a manifold is irrelevant. For instance they ignore the possibility of a flat toruss (a flat manifold that is closed). Maybe they have good reasons, but the reasons cannot be simply due to mathematics and so I don't see the clear connection between curvature and openness or closedness.

 

[DrRocket]

One thing I can assure you is that DrRocket spares no ones feelings... He's our Dr. House :lol:

Legitimate questions from people who really want to know and are willing to work a little to understand receive answers that I hope are helpful.

Serious reasonable inquiries receive serious reasonable responses.

Deep and thougtful issues provoke deep and thoughtful dialogue. These are the most fun.

Fools who spout nonsense as though they know the "one true thing"get what they deserve.

 

[SpeedFreek]

For instance they ignore the possibility of a flat toruss (a flat manifold that is closed). Maybe they have good reasons, but the reasons cannot be simply due to mathematics and so I don't see the clear connection between curvature and openness or closedness.

The Infinite Cosmos - Joseph Silk

Have a look through pages 190-192 (the link takes you there) - I thought this was the mainstream view, but perhaps I misunderstand what you mean by a flat torus.

If the universe is finite, this means that in a two-dimensional geometry it would be a torus.

If the universe were finite, flat and had a fundamental domain smaller than the observable universe, all possible topologies would leave a signature pattern on the surface of last scattering, so they should be detectable. Unfortunately we have so far been unable to detect any of these signatures, so the current view is that the fundamental domain of the universe is larger than our observable portion of it.

 

[CommonMan]

I got it! I think. The universe is so big we can not see all of it, so we really don’t know how big it is. We can only see what we can see, SEE. We may only be in a small portion of it for all we know. It stretches much further than what we call the observable universe.
Don’t hit me if I’m wrong here Dr. Rocket. I’m just trying to put it in words us non sciencetist can understand.

 

[Mee_n_Mac]

I got it! I think. The universe is so big we can not see all of it, so we really don’t know how big it is. We can only see what we can see, SEE. We may only be in a small portion of it for all we know. It stretches much further than what we call the observable universe.
Don’t hit me if I’m wrong here Dr. Rocket. I’m just trying to put it in words us non sciencetist can understand.

That's the big picture so far as we can tell.

 

[DrRocket]

For instance they ignore the possibility of a flat toruss (a flat manifold that is closed). Maybe they have good reasons, but the reasons cannot be simply due to mathematics and so I don't see the clear connection between curvature and openness or closedness.

The Infinite Cosmos - Joseph Silk

Have a look through pages 190-192 (the link takes you there) - I thought this was the mainstream view, but perhaps I misunderstand what you mean by a flat torus.

If the universe is finite, this means that in a two-dimensional geometry it would be a torus.

If the universe were finite, flat and had a fundamental domain smaller than the observable universe, all possible topologies would leave a signature pattern on the surface of last scattering, so they should be detectable. Unfortunately we have so far been unable to detect any of these signatures, so the current view is that the fundamental domain of the universe is larger than our observable portion of it.

Interesting. In Wald's book General Relativity he states " One could construct closed universes with flat or hyperboloid geometries by making topological identifications but it does not appear natural to do so." With that he effectively dismisses the possibility of a flat torus, among other things.

Your examples indicate that the physicists are not in agreement on the issue. Perhaps they have learned some geometry and now realize that curvature alone is not the issue. I don't understand what is mainstream now either.

Thanks for those references.

A flat torus is a torus with zero curvature. Not all tori are flat. The term torus is a topological term while "flat" is a geometric term. Geometry is more restrictive than topology.

I recall that in another forum you provided a link to the construction of a flat torus as a quotient space of a rectangle and helped me out with some problems in trying to illustrate it graphically using the tools available for that board. If you recall the link you might look at it and there you will see how a flat torus can be realized. I'm not even going to try here with the problems I am having with posting while using IE8 -- I can't even see what I am typing.

 

[SpeedFreek]

I recall that in another forum you provided a link to the construction of a flat torus as a quotient space of a rectangle and helped me out with some problems in trying to illustrate it graphically using the tools available for that board. If you recall the link you might look at it and there you will see how a flat torus can be realized. I'm not even going to try here with the problems I am having with posting while using IE8 -- I can't even see what I am typing.

I think that link was probably this one:

http://www.geom.uiuc.edu/video/sos/materials/overview/

But for anyone who wants an even more simplified view as an introduction to the subject, have a look at:

http://www.etsu.edu/physics/etsuobs/sta ... tartit.htm


I think I understand what a flat torus is, if it ends up working something like the 3-Torus shown on page 6 of that second website. But I have heard it referred to as the embedding of a 2-Torus into 4-Space (which I also think I understand at some level!) but these things start turning my brain inside out! It is obvious that my lack of education in higher mathematics is the problem here, and I feel I don't have the commitment to start down that road nowadays, but I am confident I grasp some of these principles at a conceptual level. Enough to pass some basic form of laymen's understanding on to others, at any rate, as long as I am careful how I do it!

 

[SpeedFreek]

I got it! I think. The universe is so big we can not see all of it, so we really don’t know how big it is. We can only see what we can see, SEE. We may only be in a small portion of it for all we know. It stretches much further than what we call the observable universe.

Yup, you most definitely got it! Everything we can see at the larger distances was moving away from us, when it emitted the light we now see, so the observable universe is now larger than we see it to be, due to the time it takes light to travel. But... the whole universe seems to be larger than that!

What a lot of people don't get, is how, even when our observable universe was at its smallest, the very first instant after it all started - at that time the whole universe might have been any size larger! And that's before we think inflation took a whole lot of our observable universe (as it was) and put it forever out of our reach. However much larger the whole universe was to begin with when compared to our observable universe, it should still be that much larger today, if the same sort of things are going on elsewhere as are going on in the parts we can see.

 

[quantumnumber]

"Infinity is also a bit more complicated than you might imagine. There are mathematical theories of infinite numbers, called cardinal or ordinal numbers. There are actually different sizes of infinity -- in fact an infinite number of them. In any case you cannot count to infinity. The first infinite number is actually the cardinality of the natural numbers, which are called countable by definition, but you cannot count "to" that number. It is simply a reflection that the natural numbers are also called the "counting numbers". Since there is no largest natural number the counting process never stops."

How can there possibly be different sizes of infinity if infinity goes on forever? That makes no sense to me. How can there be different sizes of something that goes on forever?

 

[DrRocket]

"Infinity is also a bit more complicated than you might imagine. There are mathematical theories of infinite numbers, called cardinal or ordinal numbers. There are actually different sizes of infinity -- in fact an infinite number of them. In any case you cannot count to infinity. The first infinite number is actually the cardinality of the natural numbers, which are called countable by definition, but you cannot count "to" that number. It is simply a reflection that the natural numbers are also called the "counting numbers". Since there is no largest natural number the counting process never stops."

How can there possibly be different sizes of infinity if infinity goes on forever? That makes no sense to me. How can there be different sizes of something that goes on forever?

Infinity does not "go on" to forever or anywhere else. Cardinal numbers, like the counting numbers are associated with the number of objects in certain sets. 3 for instance has a clear association with the set A={chicken, pig, dog}, or any other set with 3 elements (You can make this very precise in axiomatic mathematics, but we'll keep this on a more casual level).

The first infinite cardinal number, called aleph naught" is associated with the size of the natural numbers, which is the set {0,1,2,3,4,5,6,....}. It is proved in introductory classes on mathematics that aleph naught is also the cardinality of the integer ({ ...-4,-3,-2,-1,0,1,2,3,4...} and the rational numbers (all fractions with integer numerators and denominators). The real numbers are the numbers associated with positions on a line. Cantor showed using a proof technique that has come to be called "Cantor diagonalization" that the cardinality of the real numbers is strictly larger than aleph naught. The cardinality of the real numbers is usually called "c"(no relation to the speed of light which is a finite number). c is an infinite cardinal and c is larger than aleph naught. This is a quick example of two infinite cardinals of different size.

It is also proved in introductory set theory that given a set A, the set of all subsets of A has a cardinality strictly larger than A. If the cardinality of A is X then the cardinality of the set of all subsets of A is called 2^X. So 2^(aleph naught) is larger than aleph naught. You might ask then whether 2^(aleph naught) is the same as c, the cardinality of the real numbers. The answer is that one cannot determine the answer to that question within the bounds of the axioms of set theory. Paul Cohen proved that it is a question that cannot be proved in that framework. It is what is called formally "undecidable".

http://en.wikipedia.org/wiki/Cardinal_numbers

http://en.wikipedia.org/wiki/Continuum_hypothesis

http://en.wikipedia.org/wiki/Godel_inco ... ss_theorem