Where Is The Center Of The Big Bang Universe?

[Fredrik]

Am I using the terminolgy incorrectly? I would have thought that a family of 3-spheres with a radius that is a function of time could also be described as a 4 sphere where 1 of the 4 dimensions is a function of time.

A "one-parameter family of 3-spheres with a radius that's a function of time" would be something 5-dimensional. The family (or "set" or "class" if you prefer) doesn't have a radius, so "with a radius that.." must refer to a property of every member of the family. A 3-sphere with a time-dependent radius is a one-parameter family of 3-spheres, and that family is a 4-dimensional "thing" (technically a "manifold"), and a one-parameter family of 4-dimensional things must be 5-dimensional.

"4-sphere where 1 of the 4 dimensions is a function of time" doesn't quite make sense to me. I guess I would interpret it as meaning something like...that we're considering a 4-sphere and instead of using the coordinates (t,x,y,z) we're using (t',x,y,z) where t'=f(t) where f is some function.

You may have misunderstood how the word "dimension" is used in this context. It means (roughly) this: If there's a bunch of functions that map subsets of a set M into the set Rⁿ, then we say that M is n-dimensional.

Maybe you wanted to say that you were under the impression that a one-parameter family of 3-spheres is a 4-sphere? Let's examine a verision of that that we can visualize. Is a one-parameter framily of 1-spheres (circles) a 2-sphere (a regular sphere)? If you take a sphere and remove two points, say the north pole and the south pole, then what we have left is a one-parameter family of circles. So almost the entire sphere can be described as a one-parameter family of circles. But there are plenty of one-parameter families of circles that look nothing like a sphere, e.g. a cylinder.

If we have a circle with a time-dependent radius, and the function describing the time-dependence grows from 0 to a maximum radius and then decreases back to 0, then we can make a coordinate change that makes this one-parameter family of circles look like a sphere. But unless that radius function has a very specific form, it wouldn't really be a sphere since the formula for the length of a curve on the surface would come out looking very strange. (In this context it's appropriate to include the standard version of that formula as a part of the definition of a sphere).

OK. Where?

I'm talking about the fact that the metric of space-time is Lorentzian, not Riemannian. Unfortunately it would take much too long to explain what that means and why it's important.

 

[Cuddles]

Where Is The Center Of The Big Bang Universe?

Here I am.

Btw, Sol is right about our Hubble Limit not being the edge of the universe. It's best to say that is the edge of our observable universe. So long as the rate of expansion of spacetime doesn't out pace light-speed, as time goes on we should be able to observe more and more of the universe.

The really creepy thing is that if the expansion really is accelerating, as time goes on we'll actually see less and less of the universe. An even more fun consequence of that is that as the size of the observable universe decreases, the amount of information that can be stored in it also decreases. If dark energy is real, the universe will in fact go senile before it dies.

 

[sol invictus]

I'm talking about the fact that the metric of space-time is Lorentzian, not Riemannian. Unfortunately it would take much too long to explain what that means and why it's important.

A short explanation is that time is fundamentally different from space, and the minus sign takes care of that.

The really creepy thing is that if the expansion really is accelerating, as time goes on we'll actually see less and less of the universe. An even more fun consequence of that is that as the size of the observable universe decreases, the amount of information that can be stored in it also decreases. If dark energy is real, the universe will in fact go senile before it dies.

That's not quite accurate. The Hubble length doesn't decrease - it just asymptotes to a fixed length. In the end you're surrounded by a spherical event horizon a Hubble distance away; nothing beyond that is visible. Galaxies, and eventually everything else, fall away from you and smear themselves (as it looks to you) on that horizon. But the information capacity of that horizon is far larger than the current entropy of the observable universe today, so there is no actual decrease in information or information capacity. It does become much harder (probably impossible) to access that information.

One bizarre implication is that we seem to live at a very special time during the evolution of the universe, the time when we can see the most structure and get the most information about cosmology.

 

[quarky]

For the lurkers on the thread, a great example of a one-dimensional situation which is finite but not bound (in the sense that it has no 1D "edge" - I hope I got that right) is the Moebius strip.

Try making one and trace it - there's only one side

Try bisecting one, lengthwize.

 

[RecoveringYuppy]

A 4-sphere is a 4-dimensional Riemannian manifold. It has 4 space directions and 0 time directions.

OK. I thought that 4 space directions and 0 time was referred to as a 4-space and that 4-sphere didn't imply anything about what the 4 dimensions might be.

A spacetime is a 4 dimensional Lorentzian manifold with 3 space directions and 1 time direction.

Got that. For purposes of a laymans discussion can "manifold" be replaced with: "series of hyperspheres that change in size over time"?

"Manifold" brings in complications like "do all 3 dimensions experience the same curvature?"

it's a motion, which is something else entirely.

Yes, I was making that mistake.

A "one-parameter family of 3-spheres with a radius that's a function of time" would be something 5-dimensional. The family (or "set" or "class" if you prefer) doesn't have a radius, so "with a radius that.." must refer to a property of every member of the family. A 3-sphere with a time-dependent radius is a one-parameter family of 3-spheres, and that family is a 4-dimensional "thing" (technically a "manifold"), and a one-parameter family of 4-dimensional things must be 5-dimensional.

Thanks. I followed that. I realized some time after posting that I was at times not dinstinguishing between the universe now and a spacetime. Yes, the set of current universe does not have a radius, it's the individual universes in the set that have a radius.

"4-sphere where 1 of the 4 dimensions is a function of time" doesn't quite make sense to me. I guess I would interpret it as meaning something like...that we're considering a 4-sphere and instead of using the coordinates (t,x,y,z) we're using (t',x,y,z) where t'=f(t) where f is some function.

Yes, that sounds about like what I was thinking. In layman's terms: The current volume of the universe is a function of time since the big bang. And that function is nonlinear.

You may have misunderstood how the word "dimension" is used in this context. It means (roughly) this: If there's a bunch of functions that map subsets of a set M into the set Rⁿ, then we say that M is n-dimensional.

No. I get that.

I'm talking about the fact that the metric of space-time is Lorentzian, not Riemannian. Unfortunately it would take much too long to explain what that means and why it's important.

Hmm. Pursuing that will have to wait til this weekend when playing on the forum isn't making me late for work. Can't say I know what it means to be Lorentzian, but from what I remember of Riemannian geometry I would have thought that spacetime was Riemannian, or a subset of Riemannian.

 

[MattusMaximus]

Try bisecting one, lengthwize.

Good point. Then you get two interlocked loops of paper, each with two sides. I seem to recall a pattern here... can anyone recall what happens when you trisect a Moebius strip lengthwise?

Topology is such a weird beastie

 

[Dan O.]

Good point. Then you get two interlocked loops of paper, each with two sides.

No, You would have to bisect it twice to get that.

I seem to recall a pattern here... can anyone recall what happens when you trisect a Moebius strip lengthwise?

You can't since there is only 1 edge.

But you could cut off the edge with a width of 1/3rd the width of the original leaving a 1/3rd width Moebius strip interlocked with a double length strip with a full twist.

 

[MattusMaximus]

No, You would have to bisect it twice to get that.

You can't since there is only 1 edge.

But you could cut off the edge with a width of 1/3rd the width of the original leaving a 1/3rd width Moebius strip interlocked with a double length strip with a full twist.

Darn it, I knew I was screwing something up. Time to start playing with pieces of paper

 

[Dancing David]

Try bisecting one, lengthwize.

Ah yes, the one infinite loop becomes two. It is one of the ways I can sort of imagine the hidden dimensions of 'strings'.

 

[Fredrik]

OK. I thought that 4 space directions and 0 time was referred to as a 4-space and that 4-sphere didn't imply anything about what the 4 dimensions might be.

What you call a 4-space is what I would call a 4-dimensional smooth manifold. A 4-sphere is a smooth manifold that's isomorphic to a manifold defined as the set

[latex]\{(v,w,x,y,z)\in\mathbb{R}^5|v^2+w^2+x^2+y^2+z^2=r^2\}[/latex]

with the standard manifold structure.

For purposes of a laymans discussion can "manifold" be replaced with: "series of hyperspheres that change in size over time"?

That sounds strange to me. I would say either "a 3-sphere with a radius that changes with time" or "a one-parameter family of 3-spheres". (Either talk about it as just one three-sphere with a time-dependent radius, or as a bunch of three-spheres, each with a fixed radius. Don't mix the two descriptions). Also, note that this is just when we're talking about one particular solution of Einstein's equation. There are other solutions that we would have to describe very differently.

I objected to the use of the term "hypersphere" earlier, but Wikipedia says it's OK to use it for n-spheres with n>2, so I withdraw that objection.

Can't say I know what it means to be Lorentzian, but from what I remember of Riemannian geometry I would have thought that spacetime was Riemannian, or a subset of Riemannian.

To calculate the "length" of the path of a massive particle through space-time you would add up contributions of the form

[latex]\sqrt{-dt^2+R(t)^2d\Omega^2}[/latex]

where R(t) is the radius of the 3-sphere corresponding to time t, and the d-Omega^2-thingy is the kind of thing you would have to add up to calculate the length of an arbitrary path on that 3-sphere. (I think that's the correct form, but I was too lazy to verify that I remember it correctly). The metric is "Lorentzian" because there's a minus sign before dt^2 instead of a plus sign.

There's no need to consider spaces with curvature to understand the difference between Lorentzian and Riemannian. You should probably think about the difference in the context of special relativity instead of general relativity, to avoid unnecessary complications.

 

[Paulhoff]

The center is everywhere.

Paul

 

[RecoveringYuppy]

A 4-sphere is a smooth manifold that's isomorphic to a manifold defined as the set

with the standard manifold structure.

Now the light dawns on this "n-sphere" terminology. It's a surface, not a volume. An average ball that you can hold in your hand is not a 3-sphere, it surface is a 2-sphere. I think everything makes sense now.

 

[sol invictus]

Now the light dawns on this "n-sphere" terminology. It's a surface, not a volume. An average ball that you can hold in your hand is not a 3-sphere, it surface is a 2-sphere.

Right. That's called a 3-ball. The number is always the number of dimensions.

By the way, one way to make an n-sphere is to glue together two n-balls along their boundaries (for example a 2-sphere is two disks, or 2-balls - the two hemispheres - glued together along their boundaries).

So you can construct a 3-sphere as two 3-balls glued together at their boundaries .

 

[Olowkow]

I happened to find this astonishing (more than 4,400) collection of huge pictures of the LHC assembly, and thought some on this thread might be interested.

http://cdsweb.cern.ch/search?cc=Pho...h&c=Photos&c=&sf=&so=d&rm=&rg=25&sc=0&of=hb_p

It was set originally to show 500 results at a time. Kind of slow, so I set it for 25.
What a time sink! Enjoy.

 

[RecoveringYuppy]

Right. That's called a 3-ball. The number is always the number of dimensions.

Yes. After writing that post to Fredrik I did some googling and came across the "ball" jargon.

Physics sites on the web seem to be fairly consistent about distingushing between a sphere and a ball but a lot of geometry related sites don't seem to be. Not sure what that means.

 

[BenHad]

What Fun

As the originator of this OP, I'm glad I started it. I've scanned a lot of the links, and gotten sufficiently confused , but also gotten a lot of interesting insite. I'm beginning to suspect that I won't live long enough to see a Theory of the Universe that doesn't have a number of problems. But meanwhile I'll sit back and learn

BenHad

 

[MattusMaximus]

I happened to find this astonishing (more than 4,400) collection of huge pictures of the LHC assembly, and thought some on this thread might be interested.

http://cdsweb.cern.ch/search?cc=Pho...h&c=Photos&c=&sf=&so=d&rm=&rg=25&sc=0&of=hb_p

It was set originally to show 500 results at a time. Kind of slow, so I set it for 25.
What a time sink! Enjoy.

OMG!!! LHC slideshow... <insert Homer-esque drooling sound here>

I cannot wait for this sucker to get cranked all the way into full operation. From the Higgs Boson to dark matter, I think we're going to learn a lot of cool things...

 

[Olowkow]

OMG!!! LHC slideshow... <insert Homer-esque drooling sound here>

I cannot wait for this sucker to get cranked all the way into full operation. From the Higgs Boson to dark matter, I think we're going to learn a lot of cool things...

Glad you enjoyed it. I wonder if anyone on the forum has been there.
Sort of "geek heaven". I took a guided tour of Fermilab several years ago with one of the major scientists working there. Mind blowing! 2 story tall rectifiers, etc. As remarkable as it was, the LHC is just beyond the pale.

 

[Skeptic Ginger]

I'm sorry, but I don't know how to parse that sentence.



No, it wasn't.



In a flat or open universe, yes, there is a sense in which the singularity is infinitely large. For example, you'd have to wait an infinite amount of time for the light from the most distant parts of it to reach you. However there is also a sense in which it was a point - if you take any finite sized region of space some time after the big bang and trace it back, it will be of zero size at the bang. The real problem is that our intuition about space and time is a little bit wrong, and very wrong in this case.



I didn't ignore it. The size they're talking about is not the distance to a real edge - it's the distance to the edge of what we can see today. It's related to that fact I mentioned above, that it will take infinite time for light from all along the singularity to reach us. Someone 156 billion light years away (by the way, that number is much too big - I think it's an error) would see precisely what we see.

EDIT - that number is definitely wrong. See here, under misconceptions: http://en.wikipedia.org/wiki/Observable_universe

You can disagree with the conclusion, but as for the 156 b.l.yrs being the distance we can see, you need to re-read the article. We can see 12-14 billion light years or whatever the current estimate is for the age of the Universe. The calculation in the hypothesis is the distance the actual Universe has expanded since the singularity, its actual diameter.

As for the oxymoron, you cannot have infinity plus one because all 'ones' would be included in the infinite number.

The real problem is the usual conceptualization of the center and edge of the Universe is based on an expanding Universe where light years keep expanding but staying relative to the observer. If you freeze time, space time has a center and an edge. But we try to conceptualize it without freezing time.

How do you account for everything in 3D on a scale where time is essentially irrelevant (say the distance from here to the end of the block) having a center and an edge to having those elements just stop existing in the Universe itself? If you freeze the time dimension, you get an edge. We can't freeze the time dimension, and we can't outpace light and the expansion, so we cannot see or detect the edge. For all intents and purposes you could say there isn't one. But if expansion stopped and time stopped, and you went in a straight line, you would either return to the place you left, or reach an edge. I think reaching an edge fits the data better than returning to the place you left.

 

[sol invictus]

You can disagree with the conclusion, but as for the 156 b.l.yrs being the distance we can see, you need to re-read the article. We can see 12-14 billion light years or whatever the current estimate is for the age of the Universe. The calculation in the hypothesis is the distance the actual Universe has expanded since the singularity, its actual diameter.

I know what they did, and I don't disagree with it. They got a lower bound on the radius of curvature of the universe (but the number quoted in the newspaper article is simply wrong).

That doesn't imply the universe is that big - it implies it's at least that big.

As for the oxymoron, you cannot have infinity plus one because all 'ones' would be included in the infinite number.

Sorry, I still don't understand. There are many ways to define infinity plus 1, most of which just give back infinity. But what that has to do with the big bang, I really have no idea.

The real problem is the usual conceptualization of the center and edge of the Universe is based on an expanding Universe where light years keep expanding but staying relative to the observer. If you freeze time, space time has a center and an edge. But we try to conceptualize it without freezing time.

Sorry, but you're simply wrong. I can show you the Freedman-Robertson-Walker metric, if you'd like. The space at fixed time has neither a center nor an edge.

How do you account for everything in 3D on a scale where time is essentially irrelevant (say the distance from here to the end of the block) having a center and an edge to having those elements just stop existing in the Universe itself?

You consider the end of your block an edge? What happens if you keep walking?

If you freeze the time dimension, you get an edge. We can't freeze the time dimension, and we can't outpace light and the expansion, so we cannot see or detect the edge. For all intents and purposes you could say there isn't one. But if expansion stopped and time stopped, and you went in a straight line, you would either return to the place you left, or reach an edge. I think reaching an edge fits the data better than returning to the place you left.

You either get back to the place you left, or you keep going forever. In neither case is there a center or an edge.