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Drakkith

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Have you read the FAQ about this?

Here's the link: https://www.physicsforums.com/showthread.php?t=506991 [Broken]

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Thanks. It's an interesting explanation. And another example that at both very small and very large scales, common sense cannot wrap itself around or sometimes cannot even approach what is being discussed. At this rate within fifty years no one will understand physics on any rational level at all except physicists. Possibly we are there already. As someone without training or math aptitude, I have lived for many years via popularizations. Many times now I can't even understand the popularizations anymore.Have you read the FAQ about this?

Here's the link: https://www.physicsforums.com/showthread.php?t=506991 [Broken]

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marcus

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Have you read the FAQ about this?

Here's the link: https://www.physicsforums.com/showthread.php?t=506991 [Broken]

Oh, I wouldn't sell common sense short Common sense evolves with time. It becomes more capable of understanding nature. And physical description does not necessarily become more fragmented---occasionally physics becomes more coherent (a single law is seen to explain several behaviors, or one of several competing models is validated so we get to toss the others out).Thanks. It's an interesting explanation. And another example that at both very small and very large scales, common sense cannot wrap itself around or sometimes cannot even approach what is being discussed. At this rate within fifty years no one will understand physics on any rational level at all except physicists. Possibly we are there already. As someone without training or math aptitude, I have lived for many years via popularizations. Many times now I can't even understand the popularizations anymore.

There is reason to hope that 50 years from now common sense will actually be more adequate to the job of understanding than it is today. We can't see the future, but it's possible.

=====================

If I may ask you a question---this would help me know you---can you imagine these things?

1. an infinite flat plane with no existence above or below. Just the plane, with all existence concentrated in that 2D plane.

2. a very large 2D sphere surface, with no existence either inside or outside. There is no inside and there is no outside. It takes concentration to imagine this.

3. The triangles are of secondary importance, the main thing is to imagine all existence concentrated on a large sphere surface with no inside or outside. No space inside or outside. Imagine the 2D creatures exploring their world. What would their experience of it be like?

Now imagine the 3D analog of this.

====================

Chaszz, please let me know if these things are easy or hard, familiar ("old already--been there done that") or unfamiliar. And where you encounter difficulty, if you do.

These imagination exercises are part of our inheritance from the minds of Carl Gauss and friends around 1825-1850. They realized that a creature trapped in a world of fixed dimensionality can measure geometric features (like curvature) of that world purely internally, even if outside extra dimensions do not even exist. They realized those creatures could be us. I read somewhere that Gauss even wanted to measure a very large flat triangle using mountain peaks, just to see if it would add up. I don't think he ever got around to it, but the idea was great. Like Galileo and his friends trying to measure the speed of light with blinking lanterns in the hills outside Florence. These people had fundamental curiosity! They are how our common sense grows to encompass more and more of nature.

Let me know. Are these exercises old or new for you, hard or easy?

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marcus

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If geometry gets started bending it can want to keep on bending until some restoring force stops it. So it can ripple. If geometry starts expanding it can have a kind of momentum that makes it want to keep on doing that. Until some other force slows it down.

This is not motion. With motion, somebody gets somewhere. It is dynamic geometry. And there is an equation or law governing it. So there is something analogous to momentum in the picture, which is not associated with something moving in space.

Just a two cents observation, might help.

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Thanks for your interest. I've been interested in these things since the 1950's so of course have encountered these models before many times. I think of myself as a person fascinated by physics whose aptitude for it is the inverse square of his interest. The problem with flatland (or flat-sphere if you will) etc. is that these are models of things that cannot exist in our ordinary space or space-time. They are great at modeling analogies of situations that we cannot conceive as actually existing. So their meaning is always in a way virtual to me. I understand what they mean but I do not internalize it. I do not need help, thank you, with these commonly illustrated things because as I said I've read them many times.Oh, I wouldn't sell common sense short Common sense evolves with time. It becomes more capable of understanding nature. And physical description does not necessarily become more fragmented---occasionally physics becomes more coherent (a single law is seen to explain several behaviors, or one of several competing models is validated so we get to toss the others out).

There is reason to hope that 50 years from now common sense will actually be more adequate to the job of understanding than it is today. We can't see the future, but it's possible.

=====================

If I may ask you a question---this would help me know you---can you imagine these things?

1. an infinite flat plane with no existence above or below. Just the plane, with all existence concentrated in that 2D plane.No space above or below. 2D creatures with no thickness sliding around like amoebas in that 2D universe. They know it's flat because when they measure the angles of triangles they always add up to 180 degrees.

2. a very large 2D sphere surface, with no existence either inside or outside. There is no inside and there is no outside. It takes concentration to imagine this.No space except the sphere. 2D creatures slither around in that universe. When they measure large triangles they find the angles add up to slightly more than 180 degrees, but with smaller triangles the excess is barely noticeable. Perhaps too slight for them to measure.

3. The triangles are of secondary importance, the main thing is to imagine all existence concentrated on a large sphere surface with no inside or outside. No space inside or outside. Imagine the 2D creatures exploring their world. What would their experience of it be like?

Now imagine the 3D analog of this.

====================

Chaszz, please let me know if these things are easy or hard, familiar ("old already--been there done that") or unfamiliar. And where you encounter difficulty, if you do.

These imagination exercises are part of our inheritance from the minds of Carl Gauss and friends around 1825-1850. They realized that a creature trapped in a world of fixed dimensionality can measure geometric features (like curvature) of that world purely internally, even if outside extra dimensions do not even exist. They realized those creatures could be us. I read somewhere that Gauss even wanted to measure a very large flat triangle using mountain peaks, just to see if it would add up. I don't think he ever got around to it, but the idea was great. Like Galileo and his friends trying to measure the speed of light with blinking lanterns in the hills outside Florence. These people had fundamental curiosity! They are how our common sense grows to encompass more and more of nature.

Let me know. Are these exercises old or new for you, hard or easy?

To further illustrate what I mean, the big bang is supposed to have started with a singularity smaller than an atom, or so small as to be infinitesimally small. If it then grows, even everywhere all at once, and gets bigger, well there has to be center, at least to my common sense. I understand that the sphere in your illustration has no center because its surface is only 2-dimensional and there is nothing outside or inside it. But comprehending a sphere with no outside or inside is like trying to imagine a venetian blind with minus twenty-two slats and no strings.

OK, no problem really, as I said, I've been down this road many times before. I can let that model by. I understand that these models are illustrative analogues of situations that are not like the models themselves. I understand that we have to adapt to what the math tells us and stop trying to visualize these things as being in the living room, in front of us in our everyday perceptual space. I guess what I'm really getting at is that the percentage of these non-common-sense items per subject is getting so large, even in the popularizations, that one's eyes glaze over reading them. Here is an example from Discover magazine which is a popular journal for people like me. At least it was.

http://blogs.discovermagazine.com/cosmicvariance/2011/10/26/guest-post-don-page-on-quantum-cosmology/" [Broken]

But hey, no real problem. I still find these things as fascinating as ever, am aware of the great time we live in with all the unresolved issues, and hang on every press release from CERN and wherever else. Contra to my complaint, these weirdnesses are actually why I haunt the relativity, quantum and cosmology forums and not the classical ones. It is cool to know reality is stranger than fiction.

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In case of an Euclidean space the singularity is infinite.the big bang is supposed to have started with a singularity smaller than an atom, or so small as to be infinitesimally small.

Inside is past, outside is future. If you want a 3-dimensional surface just take a hypersphere.I understand that the sphere in your illustration has no center because its surface is only 2-dimensional and there is nothing outside or inside it.

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Drakkith

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This is not accurate, and is a misunderstanding popularized to this day. The universe, to our knowledge, was NEVER a finite size that fit into a volume of space smaller than an atom. What the evidence shows us according to current models is that the DENSITY of the universe was extremely large, and that from this extremely dense state the universe "expanded". All this means is that the distance between objects increases over time, NOT that the actual universe itself is growing into some pre-existing space. When the density of the universe was 100 billion times what it is now it was still infinite in size.To further illustrate what I mean, the big bang is supposed to have started with a singularity smaller than an atom, or so small as to be infinitesimally small. If it then grows, even everywhere all at once, and gets bigger, well there has to be center, at least to my common sense. I understand that the sphere in your illustration has no center because its surface is only 2-dimensional and there is nothing outside or inside it. But comprehending a sphere with no outside or inside is like trying to imagine a venetian blind with minus twenty-two slats and no strings.

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marcus

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I think "to our knowledge" is the key qualification here. We don't really know, do we, about the overall size or whether it was finite or infinite at the start of expansion?...The universe, to our knowledge, was NEVER a finite size that fit into a volume of space smaller than an atom. What the evidence shows us according to current models is that the DENSITY of the universe was extremely large,...

Either way, according to our conventional model (the LambdaCDM) expansion started from very high density. It doesn't seem to make much difference whether you use the finite or infinite version. I've seen NASA WMAP reports where they use both versions side by side. Both fit the data well and there's hardly any difference. It seems funny that we can't say which is right but AFAIK the question is still unresolved.

Of course the observable portion is finite. And that would have occupied a very small finite volume at the start of expansion. But we haven't determined so far whether that observable portion is part of an infinite whole, or simply part of a much larger finite whole.

I admit being biased in favor of finite but try to keep my philosophical prejudices under control

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We supposedly know the age of the universe as roughly 13.7 billion years. We seemingly know the expansion rate; since it has been discovered that it's speeding up, we presumably know the rates of expansion both before and after the change in speed. How come then we don't know how big it is (assuming it is finite and leaving aside for the moment the possibility that it's infinite) as a product of the various rates pro-rated into some sort of average, multiplied by the time? (Even the early inflationary period seems like it must have a known rate).I think "to our knowledge" is the key qualification here. We don't really know, do we, about the overall size or whether it was finite or infinite at the start of expansion?

Either way, according to our conventional model (the LambdaCDM) expansion started from very high density. It doesn't seem to make much difference whether you use the finite or infinite version. I've seen NASA WMAP reports where they use both versions side by side. Both fit the data well and there's hardly any difference. It seems funny that we can't say which is right but AFAIK the question is still unresolved.

Of course the observable portion is finite. And that would have occupied a very small finite volume at the start of expansion. But we haven't determined so far whether that observable portion is part of an infinite whole, or simply part of a much larger finite whole.

I admit being biased in favor of finite but try to keep my philosophical prejudices under control

Also there were discoveries in 2011 of both a distant galaxy and a distant explosion, that were each estimated to be about 13.1 to 13.2 billion light years away. So as we get closer in these observations to the time of the Big Bang itself, aren't we establishing a finite size as well as a finite time?

This may be a good summary of the various possibilities:

http://www.desy.de/user/projects/Physics/Relativity/GR/centre.html

which leaves a finite size and even perhaps an edge in play as possibilities, simply because we just don't know?

(I know that at some point I'm going to stray into some generalization or comment that is ignorant or inappropriate in this forum, but please don't be too hard on me. I am sincerely trying to get some sort of grip on this matter and have already learned a good deal in this thread, even from postings where the posters disagree with each other).

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marcus

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Chaszz, I'm sure you understand that simply knowing a percentage expansion rate does not tell you the size today, unless you have exact knowledge of the size at some time in the past.We supposedly know the age of the universe as roughly 13.7 billion years. We seemingly know the expansion rate; since it has been discovered that it's speeding up, we presumably know the rates of expansion both before and after the change in speed. How come then we don't know how big it is (assuming it is finite and leaving aside for the moment the possibility that it's infinite) as a product of the various rates pro-rated into some sort of average, multiplied by the time? (Even the early inflationary period seems like it must have a known rate).

...

I should qualify with some clear reservation when I say "know". What I mean is estimates derived from a simple equation model of expansion that fits an immense amount of data surprisingly well, and checks with several different kinds of observations. But it's only the best we have so far and there are

1. questions it does not answer, like the overall size of the U, and

2. ongoing efforts to develop improved models to replace it if they can be shown to give an even better fit.

So this is not some kind of absolute knowledge, it is just the best so far from work in progress.

The equation model tells us the expansion history in terms of proportion. It gives a function of time called the scale-factor. The scalefactor is written a(t).

By convention it is defined so that a(now) = 1

And there are good physics reasons to estimate that there was a clearing event around year 380,000 of expansion when the hot gas cooled enough to become transparent and to estimate that

a(then) = 1/1100 ≈ 0.0009

The physics reasoning and the model does not give an estimate of the absolute overall size at that point in time. It does not give a figure in miles or feet or meters. It just tells us the history of the scalefactor as it grew from small values like 0.0009 in early days up to the present value of 1.

So it says that whatever the size was back then, say in year 380,000, it is 1100 times as big now. And other largescale distances have increase in proportion.

That includes the size of the presentday observable region, which is something we can estimate. But the observable region is not the whole thing!

======================

I think you understand (since as you say you have been following cosmology for quite some time) that what people are talking about are

That is what "acceleration" refers to as well. what is being calculated is the time derivative or slope of a(t). The first derivative a'(t) and the second derivative a''(t). You probably know the notation used in freshman calculus where they use prime and double prime. If not, please let us know.

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What marcus, the faq, and other knowledgeable posters have said. Plus here's my two cents (I'm another fascinated layman) regarding your specific question.chaszz said:

My current understanding is that the universal singularity, the point after which meaningful speculation about the expansion and properties of our universe begins, is the point beyond which meaningful backward extrapolation from the present is impossible. There's no particular size (only that the observable universe would define a much smaller volume) or preexisting conditions associated with this point.

This point might not be beginning of our universe. It's just the point at which modern physics can begin saying anything meaningful (ie., grounded in current extant observational knowledge and mainstream physical theory) about our universe.

Thus, it can't be said that our universe didn't begin as some explosive event of finite extent in, say, a preexisting medium of infinite extent. There's just no way to know. But, afaik, the largest scale observations of our universe (revealing a lacelike structure of connected filaments of radiating matter with large dark voids) don't contradict such a conjecture.

If our universe is finite (eg., a volume bounded by an expanding wave shell), then there's a region within it that we'd refer to as its center. Only if it's infinite would it have no center. Again, as marcus has pointed out, there's no way to know which it is.

Regarding, the analogies. I don't like them either. I prefer to think of the ponderable objects of our universe as occupying the interior volume of an expanding wave shell. Everything being 3D. Afaik, there's nothing ruling that picture out.

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I do not think this is correct. Even if you are modelling an LQG bounce, there was no pre-existing background. There may have been a classically contracting spacetime that led to a bounce, but I do not think this is the pre-existing background you have in mind or is even relevant to our current spacetime metric after t>planck.Thus, it can't be said that our universe didn't begin as some explosive event of finite extent in, say, a preexisting medium of infinite extent. There's just no way to know. But, afaik, the largest scale observations of our universe (revealing a lacelike structure of connected filaments of radiating matter with large dark voids) don't contradict such a conjecture.

Again I think this is incorrect. Both finite and infinite models of the U rely on the key cosmological prinicple of homogeneity. This principle is invalidated if you deposit any valuse of differentiation, so no centers, no middles, no edges - no variation on large scales other than local variation. Finite models do not assume expansion into a pre-existing background just by nature of their being finite.If our universe is finite (eg., a volume bounded by an expanding wave shell), then there's a region within it that we'd refer to as its center. Only if it's infinite would it have no center. Again, as marcus has pointed out, there's no way to know which it is.

Finite or infinite the U has no spacial center or spacial edge, which makes it a higher dimensional topology - it did not require a pre-existing space.

Cosmo

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I am currently wrestling with a bed bug invasion. I picked them up on a recent vacation in Italy. The exterminator is coming today to heat my whole apartment to 125 degrees F. to kill them. For a small fortune. The recent replies here look very interesting but I have had no time to read them, as I've spent the last few days taking everyhting out of the apt thst might be dsmaged by the heat and storing it all n a U-haul truck outside. Hopefully I'll be able to read these posts in a day or two and return comments. Do not go to a hotel.

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marcus

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How horrible! I have read something about the new breed of bug. It's a serious problem that is affecting a lot of people. Here's hoping the treatment is successful!

I live in the San Francisco Bay area. They are beginning to be a problem here:

http://articles.sfgate.com/2010-10-11/news/24130150_1_bedbugs-pests-toehold

I see that the city of SF has a map of where they have been reported. Includes some nice downtown hotels, it looks like:

http://bedbugregistry.com/metro/sf/

Here are the recent reports (for October and November 2011):

http://bedbugregistry.com/metro/sf/recent/

I hope every city has a website like this, so people can be warned of unsafe locations. Now the abbreviation BB has a new meaning, no longer Brigitte Bardot or "big bang".

I live in the San Francisco Bay area. They are beginning to be a problem here:

http://articles.sfgate.com/2010-10-11/news/24130150_1_bedbugs-pests-toehold

I see that the city of SF has a map of where they have been reported. Includes some nice downtown hotels, it looks like:

http://bedbugregistry.com/metro/sf/

Here are the recent reports (for October and November 2011):

http://bedbugregistry.com/metro/sf/recent/

I hope every city has a website like this, so people can be warned of unsafe locations. Now the abbreviation BB has a new meaning, no longer Brigitte Bardot or "big bang".

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Here is a slideshow for Cosomology I found if anyone is interested:

http://www.space.com/13320-big-bang-universe-10-steps-explainer.html

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I don't see why that would be necessary as long as the finite space wraps back on itself in some way. I mean, yeah, if there were just some kind of BORDER, but I don't think anyone seriously proposes that.... my problem with a universe which is much bigger than our observable universe, but still finite, is that at some point homogeneity and Isotropy would have to be violated?

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This shouldnt be a problem. For example if U was embedded in a higher n-sphere, this would be a finite topolgy which still adhered to the cosmological principal of homogeneity.Marcus, my problem with a universe which is much bigger than our observable universe, but still finite, is that at some point homogeneity and Isotropy would have to be violated?

The two viewpoints are not oncompatable.

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marcus

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Note that the kind of sphere we are used to is called a 2-sphere because it is a 2D surface.

==============

But we don't to get technical. One version of the standard LCDM model has space be a 3-sphere and in the NASA report on the cosmological implications of the WMAP 5-year data (Komatsu et al) they actually gave essentially what

The U circumference would have to be minimum about 600 billion LY. So if you could freeze expansion and set out at speed c in any direction then (if you don't hit something or get thrown off course) you get back to your starting point in 600 billion years.

Homog and Isotropy are built in to that model. So there is no worry about them being violated.

It is just the 3D analog of the 2D balloon model. Instead of an expanding 2 sphere and us creatures having to be flat in order to live in it, it is an expanding 3-sphere and we get to be three dimensional as we righteously should be.

It could be finite, and it could be basically on the 3-sphere model, and it could be a lot bigger than 600 billion LY circumference. All they gave us in the Komatsu et al NASA report was the MINIMUM SIZE that would be flat enough so it wouldn't be excluded with 95% confidence by the WMAP+SNe+BAO data (the three principle data sets they were combining for their report.)

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Hey marcus and chaszz, I've enjoyed reading this thread because I have totally wondered the same questions as chaszz, and with about as much physics expertise.

A question I'd like to throw in:

So far, I can't visualize the universe "wrapping back on itself" in a*3-D way* except by imagining it as a cubic volume tesselating with itself. Exit the north side = come back in on the south side (btw, would that be considered a "3-torus"?). This at least helps with understanding how the universe has no center point (because, from within the tesselating-cube universe, you could define the center of the cube to be anywhere), but setting off in different directions would result in very different distances covered before returning to your starting place.

So, question - is there a better way to visualize the 3-D universe "wrapping back on itself", in a more 3-sphere-ish way? Or is this essentially impossible because it would require us to visualize a 4-D euclidean space?

A question I'd like to throw in:

So far, I can't visualize the universe "wrapping back on itself" in a

So, question - is there a better way to visualize the 3-D universe "wrapping back on itself", in a more 3-sphere-ish way? Or is this essentially impossible because it would require us to visualize a 4-D euclidean space?

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I've had the same problem. I had to get rid of a nice couch. Then a fold out bed. Then I vacuumed extensively. Sprayed diluted bleach on possibly infected surfaces. Then wiped them down and sprayed bug spray, and wiped them down again. Got some new furniture, and washed and heat dried all clothing. And I think they're gone. Who knows? Anyway, nice thread ... great comments by marcus as usual ... the sort of stuff that gets me to wondering, thinking, speculating. Cosmology is absolutely fascinating. I hope you and others will pardon my more or less pedestrian replies.I am currently wrestling with a bed bug invasion. I picked them up on a recent vacation in Italy. The exterminator is coming today to heat my whole apartment to 125 degrees F. to kill them. For a small fortune. The recent replies here look very interesting but I have had no time to read them, as I've spent the last few days taking everyhting out of the apt thst might be dsmaged by the heat and storing it all n a U-haul truck outside. Hopefully I'll be able to read these posts in a day or two and return comments. Do not go to a hotel.

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ThomasT said:Thus, it can't be said that our universe didn't begin as some explosive event of finite extent in, say, a preexisting medium of infinite extent. There's just no way to know. But, afaik, the largest scale observations of our universe (revealing a lacelike structure of connected filaments of radiating matter with large dark voids) don't contradict such a conjecture.

How can you, or anyone, possibly know that?Cosmo Novice said:I do not think this is correct. Even if you are modelling an LQG bounce, there was no pre-existing background.

I don't have any particular preexisting background in mind. Just a preexisting medium of unknown structure, and that this is a possibility that can't be ruled out. Anyway, I don't have in mind any sort of classically contracting spacetime.Cosmo Novice said:There may have been a classically contracting spacetime that led to a bounce, but I do not think this is the pre-existing background you have in mind ...

It's obviously not relevant to t > Planck. We're talking about t < Planck and the possibility of a finite disturbance as the beginning of our universe -- which is inherently speculative, and therefore a possibiltiy.Cosmo Novice said:... or is even relevant to our current spacetime metric after t>planck.

ThomasT said:If our universe is finite (eg., a volume bounded by an expanding wave shell), then there's a region within it that we'd refer to as its center. Only if it's infinite would it have no center. Again, as marcus has pointed out, there's no way to know which it is.

Homogeneity doesn't preclude a center or edges if the homogeneous volume/medium is finite/bounded.Cosmo Novice said:Again I think this is incorrect. Both finite and infinite models of the U rely on the key cosmological prinicple of homogeneity. This principle is invalidated if you deposit any values of differentiation, so no centers, no middles, no edges ...

I don't know what you're referring to by this.Cosmo Novice said:... - no variation on large scales other than local variation.

Nor do they necessarily exclude it, even if they might obviate it.Cosmo Novice said:Finite models do not assume expansion into a pre-existing background just by nature of their being finite.

This is nonsensical. If it's finite, then, by definition, it has a spatial edge or boundary and a center. We just have no way, at least currently, of reasonably inferring that it's either finite or infinite.Cosmo Novice said:Finite or infinite the U has no spacial center or spacial edge ...

Higher dimensional topologies are, afaik, employed for calculational purposes only, and should not be taken as literal descriptions of our universe.Cosmo Novice said:... which makes it a higher dimensional topology ...

Ok, but this doesn't rule out the possibility of a preexisting medium/space, ie., the possibility that our universe is part of something quite larger, perhaps infinite.Cosmo Novice said:... - it did not require a pre-existing space.

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marcus

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Yes, go out the east side and reappear coming in thru the west.. Out the top-->in at the bottom. That's a good way to construct a 3-torus. And to imagine the experience of living in one. Actually part of visualizing is imagining the experience. Putting yourself into it helps visualize. Any object you have in there with you, you will see many copies of in the various directions you look.... imagining it as a cubic volume tesselating with itself. Exit the north side = come back in on the south side (btw, would that be considered a "3-torus"?)...

Unless the thing is expanding so fast that light can't ever make it around---cant ever reach one of the walls, even. then I guess you would not see copies.

That's the question that has to be asked. and I don't have a satisfactory answer for. Except to encourage you to try. You can familiarize yourself with 3-sphere geometry to some extent, as experienced from the inside. Imagine you lived in a small non-expanding 3-sphere with say 10-mile circumference.So, question - is there a better way to visualize the 3-D universe "wrapping back on itself", in a more 3-sphere-ish way? Or is this essentially impossible because it would require us to visualize a 4-D euclidean space?

Or maybe better, say a 32,000 km circumference. So if you flash a beam of light off in any direction it will return from the opposite direction in slightly over 1/10 of one second.

Optically it might seem a bit strange. Think of how a person placed antipodal to you would see you---your image coming in from all directions. Optics might be so distracting that you want to place screens to block the light. Or you might wish the space to be expanding so the light could'nt come full circle.

the rule that circumference of an ordinary circle is 2 pi R would not be true except for small R. After R exceeded 8000 km, the circumf would start getting smaller.

The rule that area of 2sphere is 4πR

After R got big the area wouldn't grow so fast. After a certain point it would even start to shrink with increasing R

Angles of a big triangle would add up to more than 180 degrees. I'm getting sleepy. Have to turn in. I think that you can get something out of imagining experiences in a 3sphere context.

But a very big 3sphere, especially if expanding, would be indistinguishable from the 3D space we inhabit. Just barely positive curvature is no different from zero curvature, for all practical purposes. If it's unmeasurably small it might as well be zero.

Indeed we might be living in a 3sphere. I'm getting too lazy to type the hyphen in 3-sphere, so better get to bed.

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Yeah, I have no idea what wrapping back on itself in a 3D way might mean. Why not just visualize it as an expanding, and bounded, 3D space that we're situated somewhere in?Hey marcus and chaszz, I've enjoyed reading this thread because I have totally wondered the same questions as chaszz, and with about as much physics expertise.

A question I'd like to throw in:

So far, I can't visualize the universe "wrapping back on itself" in a3-D wayexcept by imagining it as a cubic volume tesselating with itself. Exit the north side = come back in on the south side (btw, would that be considered a "3-torus"?). This at least helps with understanding how the universe has no center point (because, from within the tesselating-cube universe, you could define the center of the cube to be anywhere), but setting off in different directions would result in very different distances covered before returning to your starting place.

So, question - is there a better way to visualize the 3-D universe "wrapping back on itself", in a more 3-sphere-ish way? Or is this essentially impossible because it would require us to visualize a 4-D euclidean space?

Regarding the 4D view of geometric GR. Why not think of this as a sort of index, and calculational tool, but not necessarily a literal description of our universe?

It seems to me that this is compatible with the view that our universe is actually continually evolving, changing, 3 dimensional configurations -- possibly bounded, possibly not.

But what about the apparent warping, bending of spatial entities? I think that this might ultimately be explained via 3D wave mechanics. But I don't currently have a model to illustrate this. Just an idea.

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Chronos

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I think it is easier to think of the universe as a surface, not a volume, on a 4d hypersphere.

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