Infinity and Information
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 Ixtellor
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Infinity and Information
Quick question: (feel free to insult, if its stupid)
If matter/energy = information. ( That is the impression I got with the Hawking debate about Blackholes)... then isn't there a finite amount of information in the universe? And if there is a finite amount of information, then wouldn't that be a limit to infinity?
As in, if you used all the information in the universe, you could count to X, therefore X is a finite number? (In that infinity is just a man made concept, and that it does NOT exisit in the physical universe)
If matter/energy = information. ( That is the impression I got with the Hawking debate about Blackholes)... then isn't there a finite amount of information in the universe? And if there is a finite amount of information, then wouldn't that be a limit to infinity?
As in, if you used all the information in the universe, you could count to X, therefore X is a finite number? (In that infinity is just a man made concept, and that it does NOT exisit in the physical universe)
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 gmalivuk
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Re: Infinity and Information
There is probably infinite matter, which means infinite information, yes. But most of it (an infinite amount, in fact) will be forever inaccessible due to the speed of light.
Re: Infinity and Information
Ixtellor wrote:As in, if you used all the information in the universe, you could count to X, therefore X is a finite number? (In that infinity is just a man made concept, and that it does NOT exisit in the physical universe)
Have you heard about the fact that there is more than one infinity?
Mathematically, there are many different types of infinity, the smallest of which (specifically, the smallest infinite cardinal) is defined by this exact process, it is the number (well, strictly the size (cardinality)) of natural numbers which is equivalent to saying that it is the number reached if you were to count an infinite number of objects (because the natural numbers have the property that the size of any continuous subset between 0 and n is n (by which I mean that there are no natural numbers between 0 and n not included in this set)).
Besides, need a thing exist in the physical universe for it to have some existence of its own? Whilst humans may have chosen the axioms which led to this notion of infinity, any other being which chose the same axioms would retrieve the same notion of infinity and so, clearly infinity has some existence independent of humans.
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Re: Infinity and Information
gmalivuk wrote:But most of it (an infinite amount, in fact) will be forever inaccessible due to the speed of light.
Or, the more interesting part: Only a finite amount of information is within the visible universe, and that does not change within finite time. That excludes some strange big crunch scenarios, where the definition of "finite time" is a bit tricky.
There was a thread about the maximal computing power of the visible universe some months ago, and it included numbers for that. I think it was something like 10^80 bits storage capacity in the visible universe, and 10^120 possible computing steps since the big bang.
So if "counting" is related to computing steps, you could have counted up to 10^120 (up to some small prefactors). However, it is very easy to store numbers much larger than that (this is done by computers on a daily basis). So counting speed would be more interesting than data storage.
 Ixtellor
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Re: Infinity and Information
gmalivuk wrote:There is probably infinite matter, which means infinite information, yes. But most of it (an infinite amount, in fact) will be forever inaccessible due to the speed of light.
I have never heard that there is infinite matter before. I thought that the we knew the mass of the universe(approximation), and that led me to believe that matter was finite.
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Re: Infinity and Information
According to this lecture 10^122 is the limit to the number of bits which could possibly be used in a calculation in this universe given the cosmological constant we seem to be having. So I guess we could at least as an upper limit to numbers which makes sense to talk about as the busy beaver function of 10^122. Which is of course an insanely large number.
(Yet I am talking about it, and the discussion does seem to make sense. I bet Roger Penrose would have quite a few interesting (and wrong) things to say about that.)
(Yet I am talking about it, and the discussion does seem to make sense. I bet Roger Penrose would have quite a few interesting (and wrong) things to say about that.)
 gmalivuk
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Re: Infinity and Information
In the observable universe, yes. But it doesn't just end there.Ixtellor wrote:I thought that the we knew the mass of the universe(approximation), and that led me to believe that matter was finite.
Re: Infinity and Information
Ixtellor wrote:gmalivuk wrote:There is probably infinite matter, which means infinite information, yes. But most of it (an infinite amount, in fact) will be forever inaccessible due to the speed of light.
I have never heard that there is infinite matter before. I thought that the we knew the mass of the universe(approximation), and that led me to believe that matter was finite.
Expanding from gmal's post:
When most people (particularly popular science books and tv shows) say "the universe" they strictly mean "the observable universe". AFAIK, most modern cosmologies have an infinite universe of which only a part is visible (due to c being finite) and, assuming the universe is homogeneous (which seems to be a fairly valid assumption given that the observable universe is pretty homogeneous), the fact that there is a nonzero mass density in the observable universe would require that the universe contains an infinite amount of mass.
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 Ixtellor
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Re: Infinity and Information
eSOANEM wrote:When most people (particularly popular science books and tv shows) say "the universe" they strictly mean "the observable universe". AFAIK, most modern cosmologies have an infinite universe of which only a part is visible (due to c being finite) and, assuming the universe is homogeneous (which seems to be a fairly valid assumption given that the observable universe is pretty homogeneous), the fact that there is a nonzero mass density in the observable universe would require that the universe contains an infinite amount of mass.
Correct me if I am wrong, but it sounds like your saying that there is infinite mass but its outside the known(observable) universe because any information out there (light, mass, radiation) is so far away it can't reach us because even traveling at the speed of light, there has not been enough time to reach us from the GREAT distance from which it is.. er originates.
That leaves me with lots of questions.
1) Is it due to the universe expanding faster than C, thus mass at the edge of the universe can never reach us? Or does the expanding universe only describe the observable universe and its the observable universe that is expanding faster than C?
2) I know we can look into the past due to background radiation.. is that only of the observable universe?
3) How about this... did the Big Bang singularity contain infinite mass/matter? And we just can't see it all now because its further out than C will allow the evidence of it to get back to us?
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 gmalivuk
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Re: Infinity and Information
Well strictly speaking, we don't know what's beyond the observable universe, because it's unobservable. Also, there's no "edge" to the universe, just an edge to the region of it that we can observe.Ixtellor wrote:1) Is it due to the universe expanding faster than C, thus mass at the edge of the universe can never reach us? Or does the expanding universe only describe the observable universe and its the observable universe that is expanding faster than C?
But yes, there are parts of the observable universe that are now (with "now" measured from our frame of reference) receding from us faster than light. If expansion accelerates, which it seems to be doing, most of what we now see beyond our galaxy will eventually likewise recede fast enough to go through an event horizon and redshift to invisibility.
Of course. If it's not part of the observable universe, we can't observe it. The background radiation is the remnant of what the universe looked like when it first became transparent to photons, and so it's impossible to look farther back in time than that.2) I know we can look into the past due to background radiation.. is that only of the observable universe?
Something like that, yeah. Though generally when physics comes up with infinities in quantities like density, the consensus is that we just don't understand the whole picture yet, rather than that the density there is actually infinite.3) How about this... did the Big Bang singularity contain infinite mass/matter? And we just can't see it all now because its further out than C will allow the evidence of it to get back to us?
Re: Infinity and Information
Ixtellor wrote:eSOANEM wrote:When most people (particularly popular science books and tv shows) say "the universe" they strictly mean "the observable universe". AFAIK, most modern cosmologies have an infinite universe of which only a part is visible (due to c being finite) and, assuming the universe is homogeneous (which seems to be a fairly valid assumption given that the observable universe is pretty homogeneous), the fact that there is a nonzero mass density in the observable universe would require that the universe contains an infinite amount of mass.
Correct me if I am wrong, but it sounds like your saying that there is infinite mass but its outside the known(observable) universe because any information out there (light, mass, radiation) is so far away it can't reach us because even traveling at the speed of light, there has not been enough time to reach us from the GREAT distance from which it is.. er originates.
That leaves me with lots of questions.
1) Is it due to the universe expanding faster than C, thus mass at the edge of the universe can never reach us? Or does the expanding universe only describe the observable universe and its the observable universe that is expanding faster than C?
2) I know we can look into the past due to background radiation.. is that only of the observable universe?
3) How about this... did the Big Bang singularity contain infinite mass/matter? And we just can't see it all now because its further out than C will allow the evidence of it to get back to us?
Yes, that is what I'm saying although it's a little bit more extreme, unless there are objects of infinite mass which exist in the universe (which seems very unlikely), then we would need to look at an infinite amount of the universe to see an infinite mass and, to see that far out, because c is finite, we'd have to wait, not just a very long time, but an infinite amount of time. In short, there will never have been enough time for light to reach us from an infinite amount of mass.
1) No, even if the universe wasn't expanding, or was even contracting, it would still require an infinite amount of time before an infinite amount of mass could be observed (again, assuming there are no objects of infinite mass on their own).
2) Yes. In fact, the CMBR is the best way to define the edge of the visible universe. The further back in time we look, the further away we look so, because the CMBR is the earliest light we see, it must also be the light from the furthest out. Of course, the visible universe is actually a bit smaller than the observable universe because, for the first portion of its life, the universe was so hot that it was opaque to EM radiation and so we can't see all the universe we are causally linked to.
3) It depends what you mean. In the usual flat big bang, the big bang singularity was still infinitely large (because it has to become an infinite universe in a finite time with a finite growth rate) and, by the same logic as I used to show that the universe should be infinitely massive, it must be too. If however, you mean the section of that singularity containing just the observable universe, then only a finite amount of mass (the same as in the observable universe today) would be included.
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Re: Infinity and Information
Im probably way off on this, but this topic is interesting and I would like to catch up, there are some questions that are bugging me, and if someone would take some time to read and perhaps answer I would be very grateful
For there to be infinite mass outside of the observable universe,
1. there are areas further away than light can reach us with finite volumes with infinite mass in them (as stated above, and seems highly unlikely)
2. the universe itself is infinite and contains infinite mass.
Which would imply that the universe expanded from a singularity with infinite mass into an infinitely vast volume with infinite mass in it in a finite ammount of time, right?
Which raises my actual question. Why are we assuming there is either infinite space or mass? Or both? What is wrong with a finite universe that expands at a finite and has finite mass? Sure, it may have the ability to expand infinitely or contract eventually, but why assume it is already infinite?
And the mass? I mean, the law of conservation of energy dictates that the total ammount of energy (and mass) is a constant.
And what if the density had a positive real number as a value? Wouldn't assuming an infinite universe with infinite mass *out* of our observeable part, where there are regions with clearly noninfinite mass imply that the universe is heterogeneous instead?
I mean not to bring chaos, im just a curious person
For there to be infinite mass outside of the observable universe,
1. there are areas further away than light can reach us with finite volumes with infinite mass in them (as stated above, and seems highly unlikely)
2. the universe itself is infinite and contains infinite mass.
Which would imply that the universe expanded from a singularity with infinite mass into an infinitely vast volume with infinite mass in it in a finite ammount of time, right?
Which raises my actual question. Why are we assuming there is either infinite space or mass? Or both? What is wrong with a finite universe that expands at a finite and has finite mass? Sure, it may have the ability to expand infinitely or contract eventually, but why assume it is already infinite?
And the mass? I mean, the law of conservation of energy dictates that the total ammount of energy (and mass) is a constant.
eSOANEM wrote:Expanding from gmal's post:
When most people (particularly popular science books and tv shows) say "the universe" they strictly mean "the observable universe". AFAIK, most modern cosmologies have an infinite universe of which only a part is visible (due to c being finite) and, assuming the universe is homogeneous (which seems to be a fairly valid assumption given that the observable universe is pretty homogeneous), the fact that there is a nonzero mass density in the observable universe would require that the universe contains an infinite amount of mass.
And what if the density had a positive real number as a value? Wouldn't assuming an infinite universe with infinite mass *out* of our observeable part, where there are regions with clearly noninfinite mass imply that the universe is heterogeneous instead?
I mean not to bring chaos, im just a curious person
 gmalivuk
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Re: Infinity and Information
That's what a nonzero density *is*...Scyrus wrote:And what if the density had a positive real number as a value?eSOANEM wrote:the observable universe is pretty homogeneous), the fact that there is a nonzero mass density in the observable universe would require that the universe contains an infinite amount of mass.
Infinite space and nonzero (i.e. positive) density means infinite mass.
Re: Infinity and Information
Scyrus wrote:Which would imply that the universe expanded from a singularity with infinite mass into an infinitely vast volume with infinite mass in it in a finite amount of time, right?
If the universe is infinite in spatial extent (and it very probably is), then it has always been infinite. In other words, the volume of the initial Big Bang "singularity" was infinite, it was not an infinitesimal point.
Scyrus wrote:Which raises my actual question. Why are we assuming there is either infinite space or mass? Or both? What is wrong with a finite universe that expands at a finite and has finite mass? Sure, it may have the ability to expand infinitely or contract eventually, but why assume it is already infinite?
Because it looks like the global curvature of the universe is zero, or perhaps possibly negative. The curvature would need to be positive for the volume (and matter content) of the universe to be finite. With a positively curved universe, the initial BB singularity would have zero volume under standard GR; a quantum gravity theory would permit the BB singularity to have a tiny nonzero volume.
Scyrus wrote:And the mass? I mean, the law of conservation of energy dictates that the total amount of energy (and mass) is a constant.
Energy is a torsor, so you can arbitrarily choose any convenient energy level to be the zero point. Conventionally, the gravitational potential energy of the universe is considered to be negative, and it is widely believed that under that convention that the total energy content of the universe is zero.
Re: Infinity and Information
Scyrus wrote:Which would imply that the universe expanded from a singularity with infinite mass into an infinitely vast volume with infinite mass in it in a finite ammount of time, right?
Which raises my actual question. Why are we assuming there is either infinite space or mass? Or both? What is wrong with a finite universe that expands at a finite and has finite mass? Sure, it may have the ability to expand infinitely or contract eventually, but why assume it is already infinite?
And the mass? I mean, the law of conservation of energy dictates that the total ammount of energy (and mass) is a constant.
No. A singularity has infinite density. This can either be achieved by infinite mass and/or infinitesimal volume. The big bang singularity was kindof both in that the observable universe was infinitesimal in volume, but because the entire universe is infinitely large and massive now, the entire universe was infinitely massive and so was not necessarily infinitesimally small.
In fact, as you say, because the universe as a whole appears to now (after a finite period of time) be infinite, unless we wish to introduce infinite expansion rates, we must accept that the big bang singularity was also always infinitely large.
As for why we assume that the universe is infinitely large, the reason are, I believe largely topological. We have good reason to believe that the universe is asymptotically flat. In this case, only topologies are available, either an infinite flat one without an edge or a finite one with an edge. Having an edge would introduce a lot of problems near to the boundary (for example, what happens to objects moving towards the edge?). Because a finite universe has these problems (even if they would never be observed in the probable lifespan of the human race), these problems would imply major flaws in our understanding of the universe and so an infinite universe is more consistent with other observations we have on a much smaller scale.
Scyrus wrote:And what if the density had a positive real number as a value? Wouldn't assuming an infinite universe with infinite mass *out* of our observeable part, where there are regions with clearly noninfinite mass imply that the universe is heterogeneous instead?
No, if you integrate a constant function across an infinitely large object, you get infinity regardless of what that constant is, provided it is not 0.
If we assume that the universe is homogeneous, then we can, at some scale, say that the density at any point (where each "point" is really an enormous volume of space) is equal to the mean density of the universe. If we take the volume integral of density, we get the mass so, if we take the volume integral of this density function across the entire universe then we obtain the mass of the universe however, as discussed previously, this integral must be infinite provided the density of the universe is not 0.
The assumption is not that there are regions of particularly high or infinite density outside of the observable universe, just that there is an infinite amount of it with the same (to within certain tolerances) density as the observable part.
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Re: Infinity and Information
By topology  if you have space that looks "locally flat" and ndimensional (like ours), for it to have no edge that folds back on itself requires that it be "curved" like a ball. There doesn't appear to be a way around it. Now, the larger the ball, the smaller the curvature.
We cannot detect any such curvature  so either the ball is ridiculously (and I mean ridiculously) large, or the universe is actually flat.
We cannot detect any such curvature  so either the ball is ridiculously (and I mean ridiculously) large, or the universe is actually flat.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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 Ixtellor
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Re: Infinity and Information
OK, now that my mind is blown, please help me picture something.
I thought the singularity (big bang) was small and expanded.
Now its being said that the singularity was infinite in size.
So if it exploded(expanded) from infinte size to infinite size.... I am confused.
I assume this has something to do with density, but I thought that meant that the universe prebig bang was small because it was infinitely dense.
Some clarification is needed for this layman. Also, if it was infinite in size (the singularity) what does that... er look like.
Is it like a lake of infinite air, suddenly turning into a lake of infinite water... I need an elementary school analogy people!
I thought the singularity (big bang) was small and expanded.
Now its being said that the singularity was infinite in size.
So if it exploded(expanded) from infinte size to infinite size.... I am confused.
I assume this has something to do with density, but I thought that meant that the universe prebig bang was small because it was infinitely dense.
Some clarification is needed for this layman. Also, if it was infinite in size (the singularity) what does that... er look like.
Is it like a lake of infinite air, suddenly turning into a lake of infinite water... I need an elementary school analogy people!
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Re: Infinity and Information
Actually, think more the other way around. An infinite plane of water flash boiling and becoming an infinite plane of steam, with pressure so great space expands to give it room. Then, tiny droplets form as the steam cools, but because the steam is all spread out now, it doesn't turn straight back into an infinite plane of water, it's more like a fine mist. It may eventually reform into an infinite plane of water, but not anytime soon, it might also just continue spreading out indefinitely.
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Re: Infinity and Information
I don't like this statement. It's far better to say that the universe is very, very close to being flat, but it could still be open, closed, or flat; the error bars are still too large to definitively say one way or the other, and different models give different answers. One interesting thing is that the universe had to be extremely close to flat to even get to this point; if it had substantial curvature, the expansion would have either totally run away by now, or it would have recollapsed long ago. IIRC, to get to where it is today, the universe had to be within one part in 10^12 of being perfectly flat or somesuch early in its history.Yakk wrote:We cannot detect any such curvature  so either the ball is ridiculously (and I mean ridiculously) large, or the universe is actually flat.
Re: Infinity and Information
Yakk wrote:By topology  if you have space that looks "locally flat" and ndimensional (like ours), for it to have no edge that folds back on itself requires that it be "curved" like a ball. There doesn't appear to be a way around it. Now, the larger the ball, the smaller the curvature.
We cannot detect any such curvature  so either the ball is ridiculously (and I mean ridiculously) large, or the universe is actually flat.
It could be topologically equivalent to a torus, then we wouldn't have to see any curvature, just repetition. Then it would only need to be slightly larger than the observable universe for this fact to not be empirically verifiable, barring other implications of whichever theory told us it was toroidal.
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Re: Infinity and Information
True. Or (if I remember my geometry) a flat space could also be nonorientable. But we'd probably notice the seam in that case, if it where anywhere nearby...
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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 gmalivuk
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Re: Infinity and Information
What seam?
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Re: Infinity and Information
gmalivuk wrote:What seam?
I suspect you'd get a matterantimatter seam in the 4 dimensional nonorientable universe, wouldn't you?
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Re: Infinity and Information
Why should you? CP violation can, in principle, generate matter (without antimatter) out of a symmetric matter/antimatter state.
That is independent of the shape of spacetime.
That is independent of the shape of spacetime.
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Re: Infinity and Information
In a nonorientable universe, wouldn't "going around the universe" turn matter into antimatter?
Maybe not, now that I think about it.
Maybe not, now that I think about it.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Re: Infinity and Information
Ixtellor wrote:So if it exploded(expanded) from infinte size to infinite size.... I am confused.
I assume this has something to do with density, but I thought that meant that the universe prebig bang was small because it was infinitely dense.
Some clarification is needed for this layman. Also, if it was infinite in size (the singularity) what does that... er look like.
It is a very counterintuitive concept so it is good that you are confused.
Another way to describe the expansion of the universe is that the universe actually stays the same size, but everything in it, including our measure of distance, gets smaller.
You know how when mud is baked in really hot sun it cracks? Well, this is kindof like my second description of the expansion of the universe. If we have an infinitely large field of mud which represents our initial big bang singularity. If we then draw a grid (fairly well spaced, it's not like we're short of room) and mark the vertices on the mud and then draw a short line to define our metre.
This sets up our initial universe, we have a set of points (the vertices marked in the mud) and a definition of distance. If you want, you can also draw on a few more complex structures. Say a small circle and large square (lines between four of the vertices) for example.
If we then leave the mud for a few weeks of blazing sunshine and see that the mud has cracked.
When we look at our metre we see that it has all ended up on one patch of cracked mud, but has become a lot smaller as the mud contracted. So too has the small circle but the points and the large square seem to us, outside the universe, to be in roughly the same place and the same size.
If we now measure the distances between points using our metre (the line we drew in the mud) however, we see that they have got a lot bigger. When we measure the distances between vertices, because our metre has shrunk, they seem further apart. Likewise, our large square also seems to have grown given when measured with our metre. Conversely, the circle appears to be the same size because it has contracted the same amount as our metre.
This is a rough analogy for the expansion of the universe and is a bit dodgy because it relies on being able to transform into the frame of some ring of points arbitrarily far away from earth and defining them as stationary.
It does however provide a slightly more intuitive notion of the expansion of the universe IMO and also shows why small scale structures (like a solar system) stay approximately the same size but cosmological scale structures (like collections of supperclusters) do seem to be affected.
As for the density issue. Density = mass/volume so for this to be infinite, either the mass must be infinite or the volume be infinitesimal.
If the volume is infinitesimal then the universe must be closed (or have an edge but there are very good reasons to think that is not the case) and, we have some reason to assume, this is not the case so the mass must therefore be infinite.
But, if the universe is not closed then the volume must be infinite and, given a finite age of the universe and finite expansion rate, always must have been.
So, we have that density=mass/volume therefore infinity=infinity/infinity.
This seems a bit of a problem, shouldn't infinity/infinity be 1?
Well, the thing is, infinity/infinity isn't well defined and can take any value depending on the way it's derived. In this case, it is quite correct that all of these values are infinite.
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Re: Infinity and Information
Yakk wrote:In a nonorientable universe, wouldn't "going around the universe" turn matter into antimatter?
Maybe not, now that I think about it.
Well, it could. At the very least it would reverse the parity and hence the spin of fundamental particles, but that would only be obvious with neutrinos / antineutrinos, and the observational evidence is that neutrinos only spin one way relative to their direction of travel (and of course antineutrinos have the opposite spin). If we were in some kind of Möbius strip universe that converted matter into antimatter we'd expect the proportion of matter to antimatter to be rather even, and once again the evidence says "no".
However, a Möbius strip universe would not have a seam: its nonorientableness is a global property, so there's no region within it where a seam or a twist is located.
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Re: Infinity and Information
The seam I was referring to was where matterdominated switched over to antimatter dominated, where we define both locally (as they are relative concepts). Topologically there must be a seam (or, a connected region that separates two parts of the universe, where neither matter nor antimatter is dominant) if reversing the orientation of matter involves switching it from matter to antimatter.
This seam need not be unique. Of course, in a 4d nonorientable universe, that seam could be temporal.
This seam need not be unique. Of course, in a 4d nonorientable universe, that seam could be temporal.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision  BR
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Re: Infinity and Information
Yakk wrote:By topology  if you have space that looks "locally flat" and ndimensional (like ours), for it to have no edge that folds back on itself requires that it be "curved" like a ball. There doesn't appear to be a way around it. Now, the larger the ball, the smaller the curvature.
We cannot detect any such curvature  so either the ball is ridiculously (and I mean ridiculously) large, or the universe is actually flat.
What about a "hyperbolic" universe that does have an edge, but the closer you get to the edge, the slower the speed of light is (or the more your mass increases, or some other limiting factor), so you can never actually reach the edge?
Reading posts on the xkcd forum makes me feel stupid.
Re: Infinity and Information
That looks like something you could modify to an infinite universe just by coordinate transformations. And as General Relativity teaches us, coordinates are arbitrary and without physical meaning.
If the universe changes in some way near the border region, we would live in a special place (not near the border), which is possible but may be unlikely, depending on your universe structure.
If the universe changes in some way near the border region, we would live in a special place (not near the border), which is possible but may be unlikely, depending on your universe structure.
Re: Infinity and Information
mfb wrote:That looks like something you could modify to an infinite universe just by coordinate transformations. And as General Relativity teaches us, coordinates are arbitrary and without physical meaning.
If the universe changes in some way near the border region, we would live in a special place (not near the border), which is possible but may be unlikely, depending on your universe structure.
Yep. In fact, you can convert a Euclidean space into a hyperbolic one that way: the Poincaré disk model of hyperbolic space, which Escher illustrated in his Circle Limit IV (at the suggestion of the Penroses). IIRC, Poincaré first thought of the Poincaré disk in terms of a systematic modification of lightspeed as a function of radius.
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