flat vs asymptotically flat universe

[agrays...@gmail.com]

IIUC, the difference is huge. In the former case, the universe is open, spatially infinite with infinite mass (assuming a nonzero mass distribution on large scale everywhere), whereas the latter is closed, finite in spatial extent and mass. But I notice that most cosmologists claim the universe is *flat*, as in mathematically flat. Are they just speaking loosely and really mean the universe is ASYMPTOTICALLY flat? I find it contradictory for a universe which is finite in age, to be truly mathematically FLAT. TIA.

[Brent Meeker]

Cosmologist think the universe is spatially flat.  That just means triangles have interior angles summing to 180deg.  It doesn't have anything to do with extent.  But the universe is not flat in spacetime; it's expanding and at an increasing rate.

Brent

On 11/9/2017 3:10 PM, agrays...@gmail.com wrote:

IIUC, the difference is huge. In the former case, the universe is open, spatially infinite with infinite mass (assuming a nonzero mass distribution on large scale everywhere), whereas the latter is closed, finite in spatial extent and mass. But I notice that most cosmologists claim the universe is *flat*, as in mathematically flat. Are they just speaking loosely and really mean the universe is ASYMPTOTICALLY flat? I find it contradictory for a universe which is finite in age, to be truly mathematically FLAT. TIA.

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[Alan Grayson]

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

On Thu, Nov 9, 2017 at 4:42 PM, Brent Meeker <meek...@verizon.net> wrote:

Cosmologist think the universe is spatially flat.  That just means triangles have interior angles summing to 180deg.  It doesn't have anything to do with extent.  But the universe is not flat in spacetime; it's expanding and at an increasing rate.

Brent

On 11/9/2017 3:10 PM, agrays...@gmail.com wrote:

IIUC, the difference is huge. In the former case, the universe is open, spatially infinite with infinite mass (assuming a nonzero mass distribution on large scale everywhere), whereas the latter is closed, finite in spatial extent and mass. But I notice that most cosmologists claim the universe is *flat*, as in mathematically flat. Are they just speaking loosely and really mean the universe is ASYMPTOTICALLY flat? I find it contradictory for a universe which is finite in age, to be truly mathematically FLAT. TIA.

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[Brent Meeker]

On 11/9/2017 6:23 PM, Alan Grayson wrote:

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

Not necessarily.  You could have periodic boundary conditions.  But most cosmologists do assume the universe is infinite in spatial extent.  Of course the flatness isn't measured by triangulation.  It's measured by comparing the spatial spectrum of the CMB variations to model predictions with different mass densities.  
https://arxiv.org/pdf/astro-ph/0004404

Brent

On Thu, Nov 9, 2017 at 4:42 PM, Brent Meeker <meek...@verizon.net> wrote:

Cosmologist think the universe is spatially flat.  That just means triangles have interior angles summing to 180deg.  It doesn't have anything to do with extent.  But the universe is not flat in spacetime; it's expanding and at an increasing rate.

Brent

On 11/9/2017 3:10 PM, agrays...@gmail.com wrote:

IIUC, the difference is huge. In the former case, the universe is open, spatially infinite with infinite mass (assuming a nonzero mass distribution on large scale everywhere), whereas the latter is closed, finite in spatial extent and mass. But I notice that most cosmologists claim the universe is *flat*, as in mathematically flat. Are they just speaking loosely and really mean the universe is ASYMPTOTICALLY flat? I find it contradictory for a universe which is finite in age, to be truly mathematically FLAT. TIA.

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[agrays...@gmail.com]

On Thursday, November 9, 2017 at 8:00:45 PM UTC-7, Brent wrote:

On 11/9/2017 6:23 PM, Alan Grayson wrote:

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

Not necessarily.  You could have periodic boundary conditions.  But most cosmologists do assume the universe is infinite in spatial extent.  Of course the flatness isn't measured by triangulation.  It's measured by comparing the spatial spectrum of the CMB variations to model predictions with different mass densities.  
https://arxiv.org/pdf/astro-ph/0004404

Brent

However flatness is measured, the criterion still seems Euclidean and hence infinite in extent if one believes the triangle measured has combined angles of 180 degrees. And I don't see how this is distinguishable from asymptotically flat for a huge but finite universe. Moreover, it seems contradictory that a universe which has expanded for a finite duration, could be infinite in spatial extent. TIA. 

On Thu, Nov 9, 2017 at 4:42 PM, Brent Meeker <meek...@verizon.net> wrote:

Cosmologist think the universe is spatially flat.  That just means triangles have interior angles summing to 180deg.  It doesn't have anything to do with extent.  But the universe is not flat in spacetime; it's expanding and at an increasing rate.

Brent

On 11/9/2017 3:10 PM, agrays...@gmail.com wrote:

IIUC, the difference is huge. In the former case, the universe is open, spatially infinite with infinite mass (assuming a nonzero mass distribution on large scale everywhere), whereas the latter is closed, finite in spatial extent and mass. But I notice that most cosmologists claim the universe is *flat*, as in mathematically flat. Are they just speaking loosely and really mean the universe is ASYMPTOTICALLY flat? I find it contradictory for a universe which is finite in age, to be truly mathematically FLAT. TIA.

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[Brent Meeker]

On 11/9/2017 8:55 PM, agrays...@gmail.com wrote:

On Thursday, November 9, 2017 at 8:00:45 PM UTC-7, Brent wrote:

On 11/9/2017 6:23 PM, Alan Grayson wrote:

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

Not necessarily.  You could have periodic boundary conditions.  But most cosmologists do assume the universe is infinite in spatial extent.  Of course the flatness isn't measured by triangulation.  It's measured by comparing the spatial spectrum of the CMB variations to model predictions with different mass densities.  
https://arxiv.org/pdf/astro-ph/0004404

Brent

However flatness is measured, the criterion still seems Euclidean and hence infinite in extent if one believes the triangle measured has combined angles of 180 degrees. And I don't see how this is distinguishable from asymptotically flat for a huge but finite universe.

It's not.

Moreover, it seems contradictory that a universe which has expanded for a finite duration, could be infinite in spatial extent. TIA.

It can if it starts off infinite.  Remember the expansion in Einstein's equation is just a factor, not an absolute value.

Brent

[Alan Grayson]

On Thu, Nov 9, 2017 at 10:05 PM, Brent Meeker <meek...@verizon.net> wrote:

On 11/9/2017 8:55 PM, agrays...@gmail.com wrote:

On Thursday, November 9, 2017 at 8:00:45 PM UTC-7, Brent wrote:

On 11/9/2017 6:23 PM, Alan Grayson wrote:

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

Not necessarily.  You could have periodic boundary conditions.  But most cosmologists do assume the universe is infinite in spatial extent.  Of course the flatness isn't measured by triangulation.  It's measured by comparing the spatial spectrum of the CMB variations to model predictions with different mass densities.  
https://arxiv.org/pdf/astro-ph/0004404

Brent

However flatness is measured, the criterion still seems Euclidean and hence infinite in extent if one believes the triangle measured has combined angles of 180 degrees. And I don't see how this is distinguishable from asymptotically flat for a huge but finite universe.

It's not.

That's my point. No way of distinguishing flat from asymptotically flat for a huge universe, so the assumption of infinite spatial extent by cosmologists seems unwarranted. But as you note below, the universe could have begun with infinite spatial extent. But ours didn't AFAIK. It began as astronomically tiny and expanded via inflation.

Moreover, it seems contradictory that a universe which has expanded for a finite duration, could be infinite in spatial extent. TIA.

It can if it starts off infinite.  Remember the expansion in Einstein's equation is just a factor, not an absolute value.

Brent

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[Brent Meeker]

On 11/9/2017 9:15 PM, Alan Grayson wrote:

On Thu, Nov 9, 2017 at 10:05 PM, Brent Meeker <meek...@verizon.net> wrote:

On 11/9/2017 8:55 PM, agrays...@gmail.com wrote:

On Thursday, November 9, 2017 at 8:00:45 PM UTC-7, Brent wrote:

On 11/9/2017 6:23 PM, Alan Grayson wrote:

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

Not necessarily.  You could have periodic boundary conditions.  But most cosmologists do assume the universe is infinite in spatial extent.  Of course the flatness isn't measured by triangulation.  It's measured by comparing the spatial spectrum of the CMB variations to model predictions with different mass densities.  
https://arxiv.org/pdf/astro-ph/0004404

Brent

However flatness is measured, the criterion still seems Euclidean and hence infinite in extent if one believes the triangle measured has combined angles of 180 degrees. And I don't see how this is distinguishable from asymptotically flat for a huge but finite universe.

It's not.

That's my point. No way of distinguishing flat from asymptotically flat for a huge universe, so the assumption of infinite spatial extent by cosmologists seems unwarranted. But as you note below, the universe could have begun with infinite spatial extent. But ours didn't AFAIK. It began as astronomically tiny and expanded via inflation.

But you don't know that.  According to Einstein's equations the visible part of the universe started at zero size.  Of course no one takes that entirely seriously since at very small distances quantum mechanics must invalidate Einstein's equations.

Brent

[Alan Grayson]

On Thu, Nov 9, 2017 at 10:32 PM, Brent Meeker <meek...@verizon.net> wrote:

On 11/9/2017 9:15 PM, Alan Grayson wrote:

On Thu, Nov 9, 2017 at 10:05 PM, Brent Meeker <meek...@verizon.net> wrote:

On 11/9/2017 8:55 PM, agrays...@gmail.com wrote:

On Thursday, November 9, 2017 at 8:00:45 PM UTC-7, Brent wrote:

On 11/9/2017 6:23 PM, Alan Grayson wrote:

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

Not necessarily.  You could have periodic boundary conditions.  But most cosmologists do assume the universe is infinite in spatial extent.  Of course the flatness isn't measured by triangulation.  It's measured by comparing the spatial spectrum of the CMB variations to model predictions with different mass densities.  
https://arxiv.org/pdf/astro-ph/0004404

Brent

However flatness is measured, the criterion still seems Euclidean and hence infinite in extent if one believes the triangle measured has combined angles of 180 degrees. And I don't see how this is distinguishable from asymptotically flat for a huge but finite universe.

It's not.

That's my point. No way of distinguishing flat from asymptotically flat for a huge universe, so the assumption of infinite spatial extent by cosmologists seems unwarranted. But as you note below, the universe could have begun with infinite spatial extent. But ours didn't AFAIK. It began as astronomically tiny and expanded via inflation.

But you don't know that.  According to Einstein's equations the visible part of the universe started at zero size.  Of course no one takes that entirely seriously since at very small distances quantum mechanics must invalidate Einstein's equations.

Brent

If you're invoking QM, aren't you conceding it started out very small, if not exactly zero size? So it seems more plausible to assume it started out very small, surely not infinite. But according to your previous statements and those that I have read by cosmologists, the assumption of infinite spatial extent is generally accepted and IMO unwarranted. 

Moreover, it seems contradictory that a universe which has expanded for a finite duration, could be infinite in spatial extent. TIA.

It can if it starts off infinite.  Remember the expansion in Einstein's equation is just a factor, not an absolute value.

Brent

[Brent Meeker]

On 11/10/2017 4:06 AM, Alan Grayson wrote:

On Thu, Nov 9, 2017 at 10:32 PM, Brent Meeker <meek...@verizon.net> wrote:

On 11/9/2017 9:15 PM, Alan Grayson wrote:

On Thu, Nov 9, 2017 at 10:05 PM, Brent Meeker <meek...@verizon.net> wrote:

On 11/9/2017 8:55 PM, agrays...@gmail.com wrote:

On Thursday, November 9, 2017 at 8:00:45 PM UTC-7, Brent wrote:

On 11/9/2017 6:23 PM, Alan Grayson wrote:

The difference between spatially flat and asymptotically flat for a huge universe would be virtually impossible to distinguish by measuring the sum of angles in a triangle. Moreover, I don't see how spatially flat can have nothing to do with extent, since in applying Euclidean geometry we surely seem to be dealing with an infinitely extended plane. TIA.

Not necessarily.  You could have periodic boundary conditions.  But most cosmologists do assume the universe is infinite in spatial extent.  Of course the flatness isn't measured by triangulation.  It's measured by comparing the spatial spectrum of the CMB variations to model predictions with different mass densities.  
https://arxiv.org/pdf/astro-ph/0004404

Brent

However flatness is measured, the criterion still seems Euclidean and hence infinite in extent if one believes the triangle measured has combined angles of 180 degrees. And I don't see how this is distinguishable from asymptotically flat for a huge but finite universe.

It's not.

That's my point. No way of distinguishing flat from asymptotically flat for a huge universe, so the assumption of infinite spatial extent by cosmologists seems unwarranted. But as you note below, the universe could have begun with infinite spatial extent. But ours didn't AFAIK. It began as astronomically tiny and expanded via inflation.

But you don't know that.  According to Einstein's equations the visible part of the universe started at zero size.  Of course no one takes that entirely seriously since at very small distances quantum mechanics must invalidate Einstein's equations.

Brent

If you're invoking QM, aren't you conceding it started out very small, if not exactly zero size? So it seems more plausible to assume it started out very small, surely not infinite. But according to your previous statements and those that I have read by cosmologists, the assumption of infinite spatial extent is generally accepted and IMO unwarranted.

If it's flat or has negative curvature then the equations imply it's infinite or perhaps periodic (no matter what the scale factor is).  If the curvature is positive then it's finite and closed and the scale factor can be taken to be the radius, so it indeed starts small in the absolute sense.  Atkatz and Pagels showed that only FRW universes that are closed (positive curvature) or De Sitter (flat with a positive cosmological constant) can "tunnel out of nothing". 

http://www.quantum-gravitation.de/media/99f63994b9064eb6ffff8004fffffff2.pdf

So most cosmologists liked the closed universe model, until it was found that expansion is accelerating.  So now more of them look to some modification of the De Sitter space universe. 

Brent

[agrays...@gmail.com]

Modification of De Sitter will be flat and therefore open. I find the open universe model in contradiction to the finite age of the universe. Is this unreasonable? Why is the closed universe model less favored when it was discovered that expansion is accelerating? TIA.

[Alan Grayson]

[Brent Meeker]

Well, if you have an infinite universe, and toward the past it is scaled by a factor a, and a->0 does the universes size go to zero?

Why is the closed universe model less favored when it was discovered that expansion is accelerating?

Because the De Sitter universe that can "tunnel from the vacuum" automatically has a positive cosmological constant.

Brent

TIA.

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[agrays...@gmail.com]

Unfortunately, my understanding of the scale factor and cosmological constant as they relate to the various geometries is insufficient to appreciate your comments. Maybe you could restate your above comments with that in mind. TIA. 

[Brent Meeker]

Didn't you read Vic's "Comprehensible Cosmos"?  Why are you over here on the everythinglist asking physics questions anyway, Alan?  You should try the stackechange or quora.

Brent

[agrays...@gmail.com]

I wasn't aware of those sites.  Since "everything" includes physics, I thought this group would be appropriate. And I see some topics here include physics. I don't see the harm. 

[agrays...@gmail.com]

I haven't read "Comprehensible Cosmos", but have done other reading. Vic had a way of glibly dismissing complex issues and it turned me off. Nevertheless, I basically respect his work. However, I really don't believe the Cosmos is Comprehensible. In fact, QM tells me the opposite. Ultimately, it's IRREDUCIBLY RANDOM as I think Bohr believed, and thus, IMO, INCOMPREHENSIBLE. God DOES play dice with the Universe. (Opinions can differ.)

[agrays...@gmail.com]

Fans of Sherlock Holmes are familiar with the story where Holmes solved the case by noting that the dog did not bark. In the case of Vic's "glib" solution to the enigma of non-locality, the "dog" in this case was the physics community which was totally unimpressed by his alleged solution, involving time reversal. AFAIK, there were absolutely no references to it in the literature as a candidate solution or explanation. And I recall never getting any answers to my critiques of the proposal. 

[Bruno Marchal]

Sometimes we discuss physics, but in general it is in or through relations with fundamental problems, including the mind-body problem, notably. 

You might look at the Space-Times Series (it exists on YouTube, link below). I just  "understood" (as far as a mathematician can understand physics of course) why the "tunneling from the vacuum" imposes a positive cosmological constant to some of solution of Einstein's GR equation, or others (probably the De Sitter one I guess). 

https://www.youtube.com/watch?v=X5rAGfjPSWE&t=712s

(That is not the episode which makes me get what Brent said.

I haven't read "Comprehensible Cosmos", but have done other reading. Vic had a way of glibly dismissing complex issues and it turned me off. Nevertheless, I basically respect his work. However, I really don't believe the Cosmos is Comprehensible. In fact, QM tells me the opposite. Ultimately, it's IRREDUCIBLY RANDOM as I think Bohr believed, and thus, IMO, INCOMPREHENSIBLE. God DOES play dice with the Universe. (Opinions can differ.)

You seem to take for granted that Reality is the or a Physical Reality. But where that Physical reality comes from, and what is physical about it?

Theoretical computer science, and its arithmetization, entails that if we assume the local finiteness of the observers, the observable correctly inferable of almost all finite observers emerges from a lattice of computations restrained by self-correctness self-referential relations.

God get just asleep and lost itself innumerably in a labyrinth of dreams. technically, this should (and seems already) reduce the laws of the observable to arithmetical self-reference (unfortunately only well known by logicians (self-reference is handled in the arithmetical sense explained in Gödel 1931 paper. 

The Universal machine of Turing has a theology (a science of its own limitation toward truth if truth exists) which includes physics, so it is testable/refutable/improvable.

All mystics said this: the Universe/Truth is in your head. So I suggest we compare the Universe which is in the head of the Universal (Turing) machine and what we infer from observations. 

You are right, there is something irreducibly random, if only the first person view of self-duplication i Arithmetic. But there are enumerable pieces of computations with an extreme redundancy in elementary Arithmetic (which is Turing complete). Physics is a first person plural view of arithmetic, and there must be derivation of the role of each pieces of its mathematics, like the group theory implied. Then the theology separates the true from the rationally justifiable, etc. The big discovery is the discovery of the universal machine (Emil Post, Alan Turing, Stephen Kleene, ...). It lives already in the diophantine polynomial equations.

Bruno

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[agrays...@gmail.com]

I'll watch it. TY. I could be mistaken, but I thought the CC is (a possibly IMPOSED) positive for an expanding universe. Before it was discovered that the expansion is accelerating, cosmologists preferred a closed, finite geometry; presumably a huge expanding hypersphere. But Brent wrote that the open, flat, infinite model became preferable after that discovery, presumably because "tunneling from the vacuum" automatically IMPOSES a positive cosmological constant. Is it the automatic imposition of a positive CC that makes the open, flat, infinite model preferable? This is really a huge change in perspective (and FWIW, not one I am happy with). I wonder; does the imposition of a positive CC ALSO automatically imply open, flat, and infinite? TIA, AG  

[Bruno Marchal]

That is my feeling, from what I can understand.  I work on "physics" from the other side (the possible observable of the universal turing machine): that explains the bottom symmetries, the quantum logic and alternate dream, and "duration", but it will take many generations of mathematics to derive particles, spaces and (physical) times (if those things exist). I suspect that mechanism could imply a infinite physical past, making the "block space time" infinite. The physical is the whole border of the "mind of the universal turing machine", but that does not make it infinite automatically, due to many group equivalence class. 

This is really a huge change in perspective (and FWIW, not one I am happy with).

Already with the very weak Robinson Arithmetic, shit happens.

With Peano arithmetic, shit happens, but also the guilt, and the questions.

If you don't like something, expect it to be true or plausible. 

Beware, perhaps, wishful thinking for the fundamental questions. 

(although even that is an open problem, and by some token, the "fundamental equation" of the mind, of the arithmetically sound universal machine looks like a sort of wishful thinking formula (Löb's formula, B(Bp -> p) -> Bp). If you convince Peano-Arithmetic that if ever proving the existence of Santa Klaus entails the existence  of Santa Klaus, then Peano Arithmetic will prove the existence of Santa Klaus!

I wonder; does the imposition of a positive CC ALSO automatically imply open, flat, and infinite? TIA, AG  

I don't know. 

I don't know if there is an well definite answer for all solutions of Einstein's equation, and I am not sure of the numerical approximations made.

I content myself to contemplate the spectacle as a lazy layman. I admire the field, but it misses (for methodological simplification) what I think is the main ingredient, if interested in the very fundamental question including consciousness, which is the notion of "finite observer", which leads to the invariance of both theology and physics, for the base theory. Of course, using this is not useful to make direct prediction, only to make sense of the big picture.

No doubt that there is a physical reality, but when we assume mechanism, we can no more invoke it to justify the experience of the mind, we need to extract the appearances of the observable from number self-reference. It fits up to now (at place where many have expected the collapse of the theory).

Bruno

I haven't read "Comprehensible Cosmos", but have done other reading. Vic had a way of glibly dismissing complex issues and it turned me off. Nevertheless, I basically respect his work. However, I really don't believe the Cosmos is Comprehensible. In fact, QM tells me the opposite. Ultimately, it's IRREDUCIBLY RANDOM as I think Bohr believed, and thus, IMO, INCOMPREHENSIBLE. God DOES play dice with the Universe. (Opinions can differ.)

You seem to take for granted that Reality is the or a Physical Reality. But where that Physical reality comes from, and what is physical about it?

Theoretical computer science, and its arithmetization, entails that if we assume the local finiteness of the observers, the observable correctly inferable of almost all finite observers emerges from a lattice of computations restrained by self-correctness self-referential relations.

God get just asleep and lost itself innumerably in a labyrinth of dreams. technically, this should (and seems already) reduce the laws of the observable to arithmetical self-reference (unfortunately only well known by logicians (self-reference is handled in the arithmetical sense explained in Gödel 1931 paper. 

The Universal machine of Turing has a theology (a science of its own limitation toward truth if truth exists) which includes physics, so it is testable/refutable/improvable.

All mystics said this: the Universe/Truth is in your head. So I suggest we compare the Universe which is in the head of the Universal (Turing) machine and what we infer from observations. 

You are right, there is something irreducibly random, if only the first person view of self-duplication i Arithmetic. But there are enumerable pieces of computations with an extreme redundancy in elementary Arithmetic (which is Turing complete). Physics is a first person plural view of arithmetic, and there must be derivation of the role of each pieces of its mathematics, like the group theory implied. Then the theology separates the true from the rationally justifiable, etc. The big discovery is the discovery of the universal machine (Emil Post, Alan Turing, Stephen Kleene, ...). It lives already in the diophantine polynomial equations.

Bruno

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[agrays...@gmail.com]

On Thursday, November 9, 2017 at 4:42:45 PM UTC-7, Brent wrote:

Cosmologist think the universe is spatially flat.  That just means triangles have interior angles summing to 180deg.  It doesn't have anything to do with extent.  But the universe is not flat in spacetime; it's expanding and at an increasing rate.

Brent

I don't understand how a flat universe has nothing to do with extent.

I don't understand the distinction; spatially flat (and therefore presumably infinite in extent?), but not flat in spacetime. 

Also, when it was discovered that the expansion is accelerating, did the solutions of the field equations which allowed tunneling from a vacuum and presumably automatically came with a positive CC (indicating expansion), also automatically come with a spatially flat (and presumably infinite in extent) geometry? Hard to grasp how a universe emerging from a vacuum could start out spatially flat and therefore IMO infinite in extent.  

TIA.

[agrays...@gmail.com]

On Sunday, November 12, 2017 at 1:42:13 PM UTC-7, agrays...@gmail.com wrote:

On Thursday, November 9, 2017 at 4:42:45 PM UTC-7, Brent wrote:

Cosmologist think the universe is spatially flat.  That just means triangles have interior angles summing to 180deg.  It doesn't have anything to do with extent.  But the universe is not flat in spacetime; it's expanding and at an increasing rate.

Brent

I don't understand how a flat universe has nothing to do with extent.

I don't understand the distinction; spatially flat (and therefore presumably infinite in extent?), but not flat in spacetime. 

Also, when it was discovered that the expansion is accelerating, did the solutions of the field equations which allowed tunneling from a vacuum and presumably automatically came with a positive CC (indicating expansion), also automatically come with a spatially flat (and presumably infinite in extent) geometry? Hard to grasp how a universe emerging from a vacuum could start out spatially flat and therefore IMO infinite in extent.  

TIA.

I must have committed some unforgivable crime. I am just not sure what, exactly, it was. I wrote to Ned Wright and Lawrence Krauss, asking how a universe of finite age could be infinite in spatial extent. Including you, Brent, I am 0 for 3. I suppose the answer could be that it tunneled out of the Multiverse, and was infinite in spatial extent at that initial time. Hard to imagine such an event.