Curvature

[John Clark]

I counted, the post I am now responding to had 10 iterated quotes, that's quotes of quotes of quotes of quotes of quotes of quotes of quotes of quotes of quotes. People are too lazy to trim anything and that's why in long threads the signal to noise ratio declines exponentially and soon becomes almost unreadable. And that's why I'm starting a new one.

On Mon, Jan 20, 2020 at 7:49 AM Alan Grayson <agrays...@gmail.com> wrote:

> What I am established is that flatness is incompatible with a universe which had a beginning. So if it's flat, it never had a beginning; or else it did, and is closed, hyper-spherical in shape. AG

Are you talking about spatial curvature or spacetime curvature? By "curvature" do you mean the angles of a triangle add up to something other than 180 degrees, or do you mean if you keep going in one direction you will eventually end up where you started? They are not necessarily the same thing. 

If the universe once expanded faster than the speed of light (as inflation hypothesizes) then it's conceivable the angles of a triangle could add up to be more than 180 degrees, so the universe would have a positive spatial curvature like a sphere does, and yet you'd be going further and further from your starting point into infinity and never return. And as long as there had been a faster than light expansion at some point in the universe's history if the angles of a triangle added up to less than 180 degrees then the universe would have negative spatial curvature, like the shape of a saddle does, and you'd still never return to your starting place.

John K Clark

[Alan Grayson]

On Mon, Jan 20, 2020 at 7:49 AM Alan Grayson <agrays...@gmail.com> wrote:

> What I  HAVE established is that flatness is incompatible with a universe which had a beginning. So if it's flat, it never had a beginning; or else it did, and is closed, hyper-spherical in shape. AG

Are you talking about spatial curvature or spacetime curvature?

I think the former, but I am unclear about the difference between the cases. There might not be a difference. AG

 

By "curvature" do you mean the angles of a triangle add up to something other than 180 degrees, or do you mean if you keep going in one direction you will eventually end up where you started? They are not necessarily the same thing. 

One would eventually end up where one started IF the universe were perfectly homogeneous, but since it isn't this is an idealization for discussion purposes. And since the curvature is slightly positive, if the expansion were frozen, the sum of angles of a triangle would be larger than 180 deg. AG

If the universe once expanded faster than the speed of light (as inflation hypothesizes) then it's conceivable the angles of a triangle could add up to be more than 180 degrees, so the universe would have a positive spatial curvature like a sphere does,

This is what I imagine, but I don't see the necessity of expansion faster than light. AG

 

and yet you'd be going further and further from your starting point into infinity and never return.

You might never return due to the rate of expansion being temporarily faster than light, but I think the universe would still be closed. AG

[John Clark]

On Mon, Jan 20, 2020 at 12:58 PM Alan Grayson <agrays...@gmail.com> wrote:

>> Are you talking about spatial curvature or spacetime curvature?

> I think the former, but I am unclear about the difference between the cases. There might not be a difference. AG

Of course there is a difference! People had worked out the mathematics for non-Euclidian spatial curvature by the early 18th century, but nobody knew anything about spacetime curvature and how it relates to physics before Einstein.

> You might never return due to the rate of expansion being temporarily faster than light, but I think the universe would still be closed. AG

Then Universe, Closed Universe, and Open Universe would all mean the same thing. But they don't.

John K Clark

[Alan Grayson]

On Monday, January 20, 2020 at 1:47:49 PM UTC-7, John Clark wrote:

On Mon, Jan 20, 2020 at 12:58 PM Alan Grayson <agrays...@gmail.com> wrote:

>> Are you talking about spatial curvature or spacetime curvature?

> I think the former, but I am unclear about the difference between the cases. There might not be a difference. AG

Of course there is a difference! People had worked out the mathematics for non-Euclidian spatial curvature by the early 18th century, but nobody knew anything about spacetime curvature and how it relates to physics before Einstein.

I meant the definition of curvature for space-time is certainly different, but I'm not sure how this difference relates to what I've conjectured. AG 

> You might never return due to the rate of expansion being temporarily faster than light, but I think the universe would still be closed. AG

Then Universe, Closed Universe, and Open Universe would all mean the same thing. But they don't.

 Not the same. There would still be a difference between a sphere and a plane. AG 

John K Clark

[John Clark]

On Tue, Jan 21, 2020 at 1:09 AM Alan Grayson <agrays...@gmail.com> wrote:

> if it started as finite, it must remain finite

Not necessarily. If the universe started off finite but then expanded, not infinitely fast just faster than light and just for a short time (as in inflation), then the universe would be open regardless of how many degrees the angles of a triangle add up to because spatial curvature and spacetime curvature are not the same thing.

 John K Clark

[Alan Grayson]

I agree they're different and concede that I don't know the exact distinction. However, I strongly disagree that finite rates of expansion will result in an open universe. I believe it will be a closed hyper-sphere, but I am open to being wrong. AG 

[Alan Grayson]

If one has an expanding hyper-sphere which is closed, why would expansion faster than light at some point in its history, make it open? AG 

[Lawrence Crowell]

In the de Sitter cosmology the curvature of the spatial surface is determined by how it is embedded in the hyperboloid. Below is a diagram of such a hyperboloid of 2 dimensions where the spatial surface is the green curve. In this situation the spatial surface is infinite. If the spatial surface is embedded as a circle, such as the black circle around the coordinates q_1 and q_4, the spatial surface is a sphere S^3. Finally the hyperbolic spatial surface occurs if it is embedded along a more vertical direction. 

This is based on the idea the de Sitter manifold is embedded in 5 dimensions with metric s = t^2 - u^2 - x^2 - y^2 - z^2, and the constraint s = 1 results in the dS metric. The metric reduced to 4 dimensions is

ds^2 = dt^2 - cosh(t√{3/Λ})dS^2

for dS^2 the spatial metric. The FLRW metric approximates this for large time with exp(t√{3/Λ}) ≈ cosh(t√{3/Λ}).

LC

[John Clark]

On Wed, Jan 22, 2020 at 12:48 AM Alan Grayson <agrays...@gmail.com> wrote:

> I strongly disagree that finite rates of expansion will result in an open universe. I believe it will be a closed hyper-sphere, but I am open to being wrong. AG 

If empty space has a residual intrinsic energy of any value greater than zero, which General Relativity allows for and Quantum Mechanics demands, then the expansion of the universe will accelerate. If the universe is accelerating then it is open regardless of what its spatial shape is, regardless of how many degrees the angles of a triangle add up to (please remember the term "spatial shape" is not equivalent with the term "spacetime shape").

And when you ask "How big is the universe?" you need to know exactly what you're really asking. You have nothing outside of the universe to compare it to so one answer would be "The universe is as big as the universe", but you may find that unsatisfying. What you really want to know is if the universe is open or closed, you want to know "If you keep going in one direction will you head out for infinity and keep getting further and further from your starting point for eternity, OR will you eventually hit some sort of wall or eventually start getting closer to your starting point?" 

Today we have very good evidence the universe is accelerating and thus open. Independently we have moderately good evidence that Inflation occured. And we have pretty good evidence the universe is spatially pretty flat. We'll never be able to observationally prove it has exactly zero spatial curvature but if all you want to know is how big the universe is, that is to say if all you want to know is if it's open or closed, then how many degrees the angles of a triangle add up to is not important. 

John K Clark  

[Lawrence Crowell]

The accelerated expansion does not imply an open universe. The dS spacetime permits closed S^3 spherical spatial worlds. As I said above this depends on how the hyperboloid is spatially sliced. With the FLRW metric 

ds^2 = dt^2 - [exp(r√{3/Λ}}/(1 - kr^2]dS^2

is accelerated and closed for k = 1, flat and open for k = 0 and saddle-hyperbolic shaped for k = -1.

LC

[John Clark]

On Wed, Jan 22, 2020 at 8:43 AM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> The accelerated expansion does not imply an open universe. The dS spacetime permits closed S^3 spherical spatial worlds. 

But a De Sitter universe contains no matter, normal or dark, that's not the sort of universe we live in.

 John K Clark

[Lawrence Crowell]

It is approximately the case, and maybe closer is the FLRW. As the universe expands it will asymptote to these configurations. It is then possible to have an expanding accelerated cosmos that is spherically closed.

LC 

[John Clark]

On Wed, Jan 22, 2020 at 12:06 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> It is then possible to have an expanding accelerated cosmos that is spherically closed.

So if I keep going I will eventually return to where I started even though everything is constantly getting more distant from me and is doing so at an accelerating rate?

 John K Clark

[Lawrence Crowell]

For an accelerated expansion of the sphere there is a cosmological horizon that one can't cross. in other words, the sphere will keep expanding faster than you can ever go. Think of the scene in the movie "The Shining" with Jack Nicholson where the hotel hallway telescoped away faster than he could run.

LC

[John Clark]

On Wed, Jan 22, 2020 at 3:34 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> For an accelerated expansion of the sphere there is a cosmological horizon that one can't cross. in other words, the sphere will keep expanding faster than you can ever go. Think of the scene in the movie "The Shining" with Jack Nicholson where the hotel hallway telescoped away faster than he could run.

OK. But in that case in what sense could it be said that such a universe is "closed"? It seems to me if the expansion is accelerating I'll never get back to where I started no matter how far I go even if it's spherically curved as you say.

John K Clark

 

[Alan Grayson]

I don't think it depends on acceleration. As long as the universe expands, even at a constant rate, at some distance, the distance between, say, an Earth observer, and some terminal point along a line of sight, will exceed 300,000 km (the distance light travels in one second) and points beyond that will keep increasing the increment every second, creating a cosmological horizon that light cannot cross. This is because the creation of the horizon is purely a geometric effect of the expansion, and the rate of expansion is irrelevant. AG

[Alan Grayson]

If the universe is a hyper-sphere, it's finite in volume regardless of whether light returns or not. So maybe your definition of "closed" is not right. AG

 

[Alan Grayson]

On Wednesday, January 22, 2020 at 6:12:35 AM UTC-7, John Clark wrote:

On Wed, Jan 22, 2020 at 12:48 AM Alan Grayson <agrays...@gmail.com> wrote:

> I strongly disagree that finite rates of expansion will result in an open universe. I believe it will be a closed hyper-sphere, but I am open to being wrong. AG 

If empty space has a residual intrinsic energy of any value greater than zero, which General Relativity allows for and Quantum Mechanics demands,

WRT QM, are you depending on the HUP to make this statement? AG

 

then the expansion of the universe will accelerate.

Why? Can't there be a balance between gravity and residual intrinsic repulsive energy of the vacuum, that would create UN-accelerated expansion? AG

 

If the universe is accelerating then it is open regardless of what its spatial shape is, regardless of how many degrees the angles of a triangle add up to (please remember the term "spatial shape" is not equivalent with the term "spacetime shape").

Maybe the issue is whether the universe has infinite or finite volume, and "closure" is irrelevant? AG 

And when you ask "How big is the universe?" you need to know exactly what you're really asking. You have nothing outside of the universe to compare it to so one answer would be "The universe is as big as the universe", but you may find that unsatisfying. What you really want to know is if the universe is open or closed, you want to know "If you keep going in one direction will you head out for infinity and keep getting further and further from your starting point for eternity, OR will you eventually hit some sort of wall or eventually start getting closer to your starting point?" 

A hyper-sphere has no edge or boundary, and if it is expanding, you might never return to your starting point even though it is finite in spatial volume. Same for a torus. AG 

[Brent Meeker]

That's not quite right.  Light can cross it just fine.  But a photon crossing it toward us, can never reach us.  This is how the Hubble boundary differs from a black hole event horizon.

Brent

This is because the creation of the horizon is purely a geometric effect of the expansion, and the rate of expansion is irrelevant. AG

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[Alan Grayson]

On Wednesday, January 22, 2020 at 8:54:37 PM UTC-7, Brent wrote:

On 1/22/2020 6:38 PM, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 1:34:00 PM UTC-7, Lawrence Crowell wrote:

On Wednesday, January 22, 2020 at 11:33:04 AM UTC-6, John Clark wrote:

On Wed, Jan 22, 2020 at 12:06 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> It is then possible to have an expanding accelerated cosmos that is spherically closed.

So if I keep going I will eventually return to where I started even though everything is constantly getting more distant from me and is doing so at an accelerating rate?

 John K Clark

For an accelerated expansion of the sphere there is a cosmological horizon that one can't cross. in other words, the sphere will keep expanding faster than you can ever go. Think of the scene in the movie "The Shining" with Jack Nicholson where the hotel hallway telescoped away faster than he could run.

LC

I don't think it depends on acceleration. As long as the universe expands, even at a constant rate, at some distance, the distance between, say, an Earth observer, and some terminal point along a line of sight, will exceed 300,000 km (the distance light travels in one second) and points beyond that will keep increasing the increment every second, creating a cosmological horizon that light cannot cross.

That's not quite right.  Light can cross it just fine.  But a photon crossing it toward us, can never reach us.  This is how the Hubble boundary differs from a black hole event horizon.

Brent

Good point. TY. AG 

This is because the creation of the horizon is purely a geometric effect of the expansion, and the rate of expansion is irrelevant. AG

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[Alan Grayson]

On Wednesday, January 22, 2020 at 9:03:55 PM UTC-7, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 8:54:37 PM UTC-7, Brent wrote:

On 1/22/2020 6:38 PM, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 1:34:00 PM UTC-7, Lawrence Crowell wrote:

On Wednesday, January 22, 2020 at 11:33:04 AM UTC-6, John Clark wrote:

On Wed, Jan 22, 2020 at 12:06 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> It is then possible to have an expanding accelerated cosmos that is spherically closed.

So if I keep going I will eventually return to where I started even though everything is constantly getting more distant from me and is doing so at an accelerating rate?

 John K Clark

For an accelerated expansion of the sphere there is a cosmological horizon that one can't cross. in other words, the sphere will keep expanding faster than you can ever go. Think of the scene in the movie "The Shining" with Jack Nicholson where the hotel hallway telescoped away faster than he could run.

LC

I don't think it depends on acceleration. As long as the universe expands, even at a constant rate, at some distance, the distance between, say, an Earth observer, and some terminal point along a line of sight, will exceed 300,000 km (the distance light travels in one second) and points beyond that will keep increasing the increment every second, creating a cosmological horizon that light cannot cross.

That's not quite right.  Light can cross it just fine.  But a photon crossing it toward us, can never reach us.  This is how the Hubble boundary differs from a black hole event horizon.

Brent

Good point. TY. AG 

Now I'm not so sure. ISTM, the photons that never reach us, never cross the event horizon. They're emitted in a region receding faster than the SoL, so they can never cross it. AG 

[John Clark]

On Wed, Jan 22, 2020 at 9:53 PM Alan Grayson <agrays...@gmail.com> wrote:

>>If empty space has a residual intrinsic energy of any value greater than zero, which General Relativity allows for and Quantum Mechanics demands,

> WRT QM, are you depending on the HUP to make this statement? AG

Quantum Mechanics demands that virtual particles give empty space an intrinsic energy, although the number it came up with was 10^120 times larger than what the observed value turned out to be. And IHA.

>> then the expansion of the universe will accelerate.

> Why?

Because that's what the 4D tensor equations of General Relativity say.

 

>> If the universe is accelerating then it is open regardless of what its spatial shape is, regardless of how many degrees the angles of a triangle add up to (please remember the term "spatial shape" is not equivalent with the term "spacetime shape").

> Maybe the issue is whether the universe has infinite or finite volume, and "closure" is irrelevant? AG 

When you ask "is the universe infinite?" if you don't mean can you keep getting further from your starting point forever then I don't know what you mean by the question.

> A hyper-sphere has no edge or boundary, and if it is expanding, you might never return to your starting point

Exactly.

> even though it is finite in spatial volume.

If the Universe is finite then you should be able to visit every cubic meter of it, at least in principle. But in a expanding and accelerating universe you can't.

John K Clark

[Lawrence Crowell]

I would say the spatial surface is topologically closed, but not causally closed. 

LC

 

[John Clark]

On Thu, Jan 23, 2020 at 5:40 AM Alan Grayson <agrays...@gmail.com> wrote:

> Now I'm not so sure. ISTM, the photons that never reach us, never cross the event horizon. They're emitted in a region receding faster than the SoL, so they can never cross it. AG

If even a photon is not fast enough to reach all parts of the universe then the Universe must be infinite. And IHA.

John K Clark

[Alan Grayson]

On Thursday, January 23, 2020 at 3:54:31 AM UTC-7, John Clark wrote:

On Wed, Jan 22, 2020 at 9:53 PM Alan Grayson <agrays...@gmail.com> wrote:

>>If empty space has a residual intrinsic energy of any value greater than zero, which General Relativity allows for and Quantum Mechanics demands,

> WRT QM, are you depending on the HUP to make this statement? AG

Quantum Mechanics demands that virtual particles give empty space an intrinsic energy, although the number it came up with was 10^120 times larger than what the observed value turned out to be.

That's what QED gives for the summed vacuum state energy of every frequency mode, assuming some cutoff. Since it's clearly wrong, no point in using it in an argument. And nothing to do with virtual particles, at least as far as QED is concerned. AG 

And IHA.

>> then the expansion of the universe will accelerate.

> Why?

Because that's what the 4D tensor equations of General Relativity say.

 

>> If the universe is accelerating then it is open regardless of what its spatial shape is, regardless of how many degrees the angles of a triangle add up to (please remember the term "spatial shape" is not equivalent with the term "spacetime shape").

> Maybe the issue is whether the universe has infinite or finite volume, and "closure" is irrelevant? AG 

When you ask "is the universe infinite?" if you don't mean can you keep getting further from your starting point forever then I don't know what you mean by the question.

 I mean the total volume is finite at any moment in time, even though its volume is increasing without limit, while the matter content remains constant. AG

> A hyper-sphere has no edge or boundary, and if it is expanding, you might never return to your starting point

Exactly.

> even though it is finite in spatial volume.

If the Universe is finite then you should be able to visit every cubic meter of it, at least in principle.

There's no such principle, and in fact it's wrong. Just imagine an expanding hyper-sphere, even one expanding at less than light speed, and you'll see your conjecture isn't true. AG

[Alan Grayson]

I'm pretty sure your claim is false. I explained earlier that for any expansion rate, even lower than the SoL, for an observer there will regions which eventually become non-observable. This is because the winking out, say of galaxies, is purely a geometric effect of the expansion. AG 

[Alan Grayson]

On Thursday, January 23, 2020 at 4:03:24 AM UTC-7, Lawrence Crowell wrote:

On Wednesday, January 22, 2020 at 2:46:35 PM UTC-6, John Clark wrote:

On Wed, Jan 22, 2020 at 3:34 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> For an accelerated expansion of the sphere there is a cosmological horizon that one can't cross. in other words, the sphere will keep expanding faster than you can ever go. Think of the scene in the movie "The Shining" with Jack Nicholson where the hotel hallway telescoped away faster than he could run.

Yes, you nailed it, and Brent was incorrect to say that photons beyond the horizon can cross it. AG 

OK. But in that case in what sense could it be said that such a universe is "closed"? It seems to me if the expansion is accelerating I'll never get back to where I started no matter how far I go even if it's spherically curved as you say.

John K Clark

I would say the spatial surface is topologically closed, but not causally closed. 

Yes, you nailed it again. You're hired!  AG

LC

 

[Alan Grayson]

On Thursday, January 23, 2020 at 4:03:24 AM UTC-7, Lawrence Crowell wrote:

As I just posted, this is correct, but can you give a precise mathematical meaning to "topologically closed"? TIA, AG 

LC

 

[John Clark]

On Thu, Jan 23, 2020 at 6:40 AM Alan Grayson <agrays...@gmail.com> wrote:

>>Lawrence Crowell wrote: I would say the spatial surface is topologically closed, but not causally closed.

> As I just posted, this is correct, but can you give a precise mathematical meaning to "topologically closed"? TIA, AG 

The Universe is topologically closed if you can give me any point in the universe I can give you a number greater than zero that you can use as a radius to draw a sphere centered on that point such that every point within that sphere is also in the universe. Or to put it more succinctly, if the universe is topologically closed then it would contain all its limit points. But you want to know if it's finite or infinite and this would tell you nothing about that.

 John K Clark

[John Clark]

On Thu, Jan 23, 2020 at 6:20 AM Alan Grayson <agrays...@gmail.com> wrote:

>> If the Universe is finite then you should be able to visit every cubic meter of it, at least in principle.

 

>There's no such principle,

There had better be, otherwise the terms "finite universe" and "infinite universe" would be indistinguishable and so could be replace by simply saying "universe".

 > and in fact it's wrong. Just imagine an expanding hyper-sphere, even one expanding at less than light speed, and you'll see your conjecture isn't true. AG

You're imagining yourself standing outside the universe viewing something that looks like an inflating balloon and looking at all the space the balloon hasn't expanded into yet. Do I really have to point out all the flaws in this picture?

>  I mean the total volume is finite at any moment in time,

And you're imagining there is a universal time and everyone at every place agrees on exactly what a "moment in time" means. Do I have to explain what's wrong with that idea too?

 John K Clark

[Lawrence Crowell]

AKA Heine-Borel theorem.

LC 

[Brent Meeker]

On 1/23/2020 2:40 AM, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 9:03:55 PM UTC-7, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 8:54:37 PM UTC-7, Brent wrote:

On 1/22/2020 6:38 PM, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 1:34:00 PM UTC-7, Lawrence Crowell wrote:

On Wednesday, January 22, 2020 at 11:33:04 AM UTC-6, John Clark wrote:

On Wed, Jan 22, 2020 at 12:06 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> It is then possible to have an expanding accelerated cosmos that is spherically closed.

So if I keep going I will eventually return to where I started even though everything is constantly getting more distant from me and is doing so at an accelerating rate?

 John K Clark

For an accelerated expansion of the sphere there is a cosmological horizon that one can't cross. in other words, the sphere will keep expanding faster than you can ever go. Think of the scene in the movie "The Shining" with Jack Nicholson where the hotel hallway telescoped away faster than he could run.

LC

I don't think it depends on acceleration. As long as the universe expands, even at a constant rate, at some distance, the distance between, say, an Earth observer, and some terminal point along a line of sight, will exceed 300,000 km (the distance light travels in one second) and points beyond that will keep increasing the increment every second, creating a cosmological horizon that light cannot cross.

That's not quite right.  Light can cross it just fine.  But a photon crossing it toward us, can never reach us.  This is how the Hubble boundary differs from a black hole event horizon.

Brent

Good point. TY. AG 

Now I'm not so sure. ISTM, the photons that never reach us, never cross the event horizon. They're emitted in a region receding faster than the SoL, so they can never cross it. AG

Sure they do.  If galaxy Z is at our Hubble boundary, we're at galaxy Z's Hubble boundary.  Does that mean we can't emit a photon toward galaxy Z?

Brent

[Brent Meeker]

On 1/23/2020 2:53 AM, John Clark wrote:

On Wed, Jan 22, 2020 at 9:53 PM Alan Grayson <agrays...@gmail.com> wrote:

>>If empty space has a residual intrinsic energy of any value greater than zero, which General Relativity allows for and Quantum Mechanics demands,

> WRT QM, are you depending on the HUP to make this statement? AG

Quantum Mechanics demands that virtual particles give empty space an intrinsic energy, although the number it came up with was 10^120 times larger than what the observed value turned out to be. And IHA.

>> then the expansion of the universe will accelerate.

> Why?

Because that's what the 4D tensor equations of General Relativity say.

 

>> If the universe is accelerating then it is open regardless of what its spatial shape is, regardless of how many degrees the angles of a triangle add up to (please remember the term "spatial shape" is not equivalent with the term "spacetime shape").

> Maybe the issue is whether the universe has infinite or finite volume, and "closure" is irrelevant? AG 

When you ask "is the universe infinite?" if you don't mean can you keep getting further from your starting point forever then I don't know what you mean by the question.

But that happens in an arbitrarily small space that is expanding.  The proper definition is you need infinite coordinate values to label all the distinct points.

> A hyper-sphere has no edge or boundary, and if it is expanding, you might never return to your starting point

Exactly.

> even though it is finite in spatial volume.

If the Universe is finite then you should be able to visit every cubic meter of it, at least in principle.

Only on the principle that you can go faster than the expansion rate between any two points of the universe...which appears to be false even for a small part of the universe.

But in a expanding and accelerating universe you can't.

Even if the universe has the topology of a sphere and hence is closed and finite.  I don't see why you have a problem with a finite, expanding space in which you can't reach every point of it because you speed is limited.  Having limited speed and space being infinite are different things.

Brent

[Alan Grayson]

On Thursday, January 23, 2020 at 12:53:58 PM UTC-7, Brent wrote:

On 1/23/2020 2:53 AM, John Clark wrote:

On Wed, Jan 22, 2020 at 9:53 PM Alan Grayson <agrays...@gmail.com> wrote:

>>If empty space has a residual intrinsic energy of any value greater than zero, which General Relativity allows for and Quantum Mechanics demands,

> WRT QM, are you depending on the HUP to make this statement? AG

Quantum Mechanics demands that virtual particles give empty space an intrinsic energy, although the number it came up with was 10^120 times larger than what the observed value turned out to be. And IHA.

>> then the expansion of the universe will accelerate.

> Why?

Because that's what the 4D tensor equations of General Relativity say.

 

>> If the universe is accelerating then it is open regardless of what its spatial shape is, regardless of how many degrees the angles of a triangle add up to (please remember the term "spatial shape" is not equivalent with the term "spacetime shape").

> Maybe the issue is whether the universe has infinite or finite volume, and "closure" is irrelevant? AG 

When you ask "is the universe infinite?" if you don't mean can you keep getting further from your starting point forever then I don't know what you mean by the question.

But that happens in an arbitrarily small space that is expanding.  The proper definition is you need infinite coordinate values to label all the distinct points.

How do you define "topologically closed"? I refreshed my memory last night and I don't find this concept helpful in distinguishing a sphere from a plane. Nor does the concept of connectedness. TIA, AG 

[Alan Grayson]

But how does this definition of "topologically closed" distinguish a sphere from a plane? Nor does connectedness help. AG 

LC 

[John Clark]

On Thu, Jan 23, 2020 at 2:53 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

>  I don't see why you have a problem with a finite, expanding space in which you can't reach every point of it because you speed is limited. 

If the Universe is not just expanding but accelerating then you couldn't visit all of it, not even it you were a photon of light and moving at the speed of causality, not even if you had a infinite amount of time at your disposal. So in what sense would such a universe be "finite" and how would it differ from a universe that was "infinite"?  

John K Clark

[Lawrence Crowell]

That is a toughy. A closed spherical space with a huge radius of curvature may be virtually indistinguishable from an infinite universe. The only prospect for distinguishing between the two cases might be with the quantum field implications of the two.

[Brent Meeker]

I already answered that.  It would take an infinite range of coordinate values to label all the points in an infinite universe, but not in a finite one.

Brent

John K Clark

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[John Clark]

On Thu, Jan 23, 2020 at 5:46 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

> It would take an infinite range of coordinate values to label all the points in an infinite universe, but not in a finite one.

So you're saying in a finite universe you'd only need a finite number of distinguishable labels for each point in space. OK I understand that, but if new points are constantly being created in this "finite" universe, even worse new points are being created at an accelerated rate in this "finite" universe, then you're going to run out of labels for every point in this "finite" universe if you only have a finite number of labels.

John K Clark

[Bruce Kellett]

You seem to have missed an important little word in Brent's post: Brent talked about needing an infinite RANGE of coordinate values for an infinite universe -- nothing whatsoever about having only a finite set of distinguishable labels......

Bruce

[John Clark]

On Thu, Jan 23, 2020 at 6:35 PM Bruce Kellett <bhkel...@gmail.com> wrote:

 

> You seem to have missed an important little word in Brent's post: Brent talked about needing an infinite RANGE of coordinate values for an infinite universe

OK fine, so in a finite universe you'd only need a finite RANGE of coordinate values printed on a finite number of labels for all the finite number of points in that finite universe. But as I said, if new points are constantly being made at an accelerating rate in that "finite" universe then you're going to run out of those finite labels.

> nothing whatsoever about having only a finite set of distinguishable labels......

Nothing whatsoever? He specifically said a "range of coordinate values to label all the points". And if a label isn't distinguishable then it isn't a label. 

John K Clark

[Bruce Kellett]

How many real numbers are there? Would that not be enough labels?....

Bruce

[Alan Grayson]

But they are distinguishable. Moreover, accelerating expansion is irrelevant. A constant rate of expansion produces non-observable regions for every observer. As for using a single time label for the whole universe, doesn't GR do just that in its scale factor "a", which is a function of one variable, t, time. Admittedly, this seems to violate the concept in relativity that each observer has its own clock.  No doubt someone here can rationalize this apparent anomaly in relativity. AG

[Alan Grayson]

On Thursday, January 23, 2020 at 2:52:50 PM UTC-7, Lawrence Crowell wrote:

That is a toughy. A closed spherical space with a huge radius of curvature may be virtually indistinguishable from an infinite universe. The only prospect for distinguishing between the two cases might be with the quantum field implications of the two.

How does topology distinguish them? ISTM, that a sphere and a plane are both closed, since each contains its accumulation points. No? AG 

[Brent Meeker]

I carefully wrote "range of coordinate values".  I'm assuming a continuum spacetime.  So even a 1cm interval takes an infinite number of labels.

Brent

John K Clark

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[Lawrence Crowell]

 If you have a sphere that is expanding the coordinate grid comoves with that. The spacing between coordinate points increases. The number of points needed to specify things does not need to change. The points on a space are not physical information. In some ways they are just mathematical fantasies of sorts that happen to satisfy requirements of a self-consistent axiomatic system called point-set topology. If the sphere has constant curvature the only intrinsic piece of information you need then is just one point, which you define as your coordinate. All other coordinates can be derived.

If the 3-sphere has lots of hills and valleys then you do need to specify more points. In this situation there is more real information. If these hills and valleys becomes infinitely craggy in a fractal then the amount of information required to specify this sphere has unbounded Kolmogoroff complexity. But for a smooth sphere, and one that is expanding so it becomes every smoother, does not at all require added information to describe it as it expands.

LC

[Bruce Kellett]

On Fri, Jan 24, 2020 at 12:22 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

 If you have a sphere that is expanding the coordinate grid comoves with that. The spacing between coordinate points increases.

If the space between two real numbers increases, you just add more real numbers to fill the gap. Coordinates are used to specify locations in the space, not the topology of the space. From the locations you can derive distances between points and the like. We are talking about a metric space here, not just a topological space.

Bruce

[Brent Meeker]

On 1/23/2020 5:22 PM, Lawrence Crowell wrote:

On Thursday, January 23, 2020 at 5:56:08 PM UTC-6, John Clark wrote:

On Thu, Jan 23, 2020 at 6:35 PM Bruce Kellett <bhkel...@gmail.com> wrote:

 

> You seem to have missed an important little word in Brent's post: Brent talked about needing an infinite RANGE of coordinate values for an infinite universe

OK fine, so in a finite universe you'd only need a finite RANGE of coordinate values printed on a finite number of labels for all the finite number of points in that finite universe. But as I said, if new points are constantly being made at an accelerating rate in that "finite" universe then you're going to run out of those finite labels.

> nothing whatsoever about having only a finite set of distinguishable labels......

Nothing whatsoever? He specifically said a "range of coordinate values to label all the points". And if a label isn't distinguishable then it isn't a label. 

John K Clark

 If you have a sphere that is expanding the coordinate grid comoves with that. The spacing between coordinate points increases. The number of points needed to specify things does not need to change.

But if "the spacing increases" means anything at all, it means the range of coordinate values to define those points must increase.

Brent

The points on a space are not physical information. In some ways they are just mathematical fantasies of sorts that happen to satisfy requirements of a self-consistent axiomatic system called point-set topology. If the sphere has constant curvature the only intrinsic piece of information you need then is just one point, which you define as your coordinate. All other coordinates can be derived.

If the 3-sphere has lots of hills and valleys then you do need to specify more points. In this situation there is more real information. If these hills and valleys becomes infinitely craggy in a fractal then the amount of information required to specify this sphere has unbounded Kolmogoroff complexity. But for a smooth sphere, and one that is expanding so it becomes every smoother, does not at all require added information to describe it as it expands.

LC

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[John Clark]

On Thu, Jan 23, 2020 at 7:23 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

 

>> if new points are constantly being created in this "finite" universe, even worse new points are being created at an accelerated rate in this "finite" universe, then you're going to run out of labels for every point in this "finite" universe if you only have a finite number of labels.

> I carefully wrote "range of coordinate values".  I'm assuming a continuum spacetime.  So even a 1cm interval takes an infinite number of labels.

This entire thing started because I have a problem with a "finite" universe that was not only expanding faster than light but accelerating too, so I asked in what sense could such a universe be called "finite". You responded with, and I quote:

 "It would take an infinite range of coordinate values to label all the points in an infinite universe, but not in a finite one."

If you're assuming that Real Numbers exist and that even a 1 cm universe would need a infinite number of labels then obviously your above answer is nonsense, so I ask my question again: 

What is the difference between a "finite" universe that is expanding and accelerating and an infinite universe that is expanding and accelerating?

John K Clark

[John Clark]

On Thu, Jan 23, 2020 at 8:22 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> If you have a sphere that is expanding the coordinate grid comoves with that. The spacing between coordinate points increases. The number of points needed to specify things does not need to change. The points on a space are not physical information.

So you're saying only the orientation of matter determines the informational content of the universe not energy such as the vacuum energy inherent in empty space, thus as everything expands and matter becomes more dilute we can start to ignore it and just think about empty space. But if I wanted to know, even approximately, the total amount of energy in the universe the number of digits needed to express that approximate value would keep increasing forever in a expanding accelerating universe.

 John K Clark

[Lawrence Crowell]

On Thursday, January 23, 2020 at 9:54:29 PM UTC-6, Brent wrote:

On 1/23/2020 5:22 PM, Lawrence Crowell wrote:

On Thursday, January 23, 2020 at 5:56:08 PM UTC-6, John Clark wrote:

On Thu, Jan 23, 2020 at 6:35 PM Bruce Kellett <bhkel...@gmail.com> wrote:

 

> You seem to have missed an important little word in Brent's post: Brent talked about needing an infinite RANGE of coordinate values for an infinite universe

OK fine, so in a finite universe you'd only need a finite RANGE of coordinate values printed on a finite number of labels for all the finite number of points in that finite universe. But as I said, if new points are constantly being made at an accelerating rate in that "finite" universe then you're going to run out of those finite labels.

> nothing whatsoever about having only a finite set of distinguishable labels......

Nothing whatsoever? He specifically said a "range of coordinate values to label all the points". And if a label isn't distinguishable then it isn't a label. 

John K Clark

 If you have a sphere that is expanding the coordinate grid comoves with that. The spacing between coordinate points increases. The number of points needed to specify things does not need to change.

But if "the spacing increases" means anything at all, it means the range of coordinate values to define those points must increase.

Brent

If the sphere has constant curvature then to specify a point between two comoving or expanding coordinate points one can calculate it. While there is a continuum of space in point-set topology this does not really mean a space requires an infinite, indeed uncountable infinite, amount of information to describe it.

LC

 

The points on a space are not physical information. In some ways they are just mathematical fantasies of sorts that happen to satisfy requirements of a self-consistent axiomatic system called point-set topology. If the sphere has constant curvature the only intrinsic piece of information you need then is just one point, which you define as your coordinate. All other coordinates can be derived.

If the 3-sphere has lots of hills and valleys then you do need to specify more points. In this situation there is more real information. If these hills and valleys becomes infinitely craggy in a fractal then the amount of information required to specify this sphere has unbounded Kolmogoroff complexity. But for a smooth sphere, and one that is expanding so it becomes every smoother, does not at all require added information to describe it as it expands.

LC

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[Brent Meeker]

On 1/24/2020 4:14 AM, John Clark wrote:

On Thu, Jan 23, 2020 at 7:23 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

 

>> if new points are constantly being created in this "finite" universe, even worse new points are being created at an accelerated rate in this "finite" universe, then you're going to run out of labels for every point in this "finite" universe if you only have a finite number of labels.

> I carefully wrote "range of coordinate values".  I'm assuming a continuum spacetime.  So even a 1cm interval takes an infinite number of labels.

This entire thing started because I have a problem with a "finite" universe that was not only expanding faster than light but accelerating too, so I asked in what sense could such a universe be called "finite". You responded with, and I quote:

 "It would take an infinite range of coordinate values to label all the points in an infinite universe, but not in a finite one."

If you're assuming that Real Numbers exist and that even a 1 cm universe would need a infinite number of labels

But not an infinite range of labels.

Brent

then obviously your above answer is nonsense, so I ask my question again: 

What is the difference between a "finite" universe that is expanding and accelerating and an infinite universe that is expanding and accelerating?

Imagine the Earth is expanding like a balloon and at an accelerating pace.  You can't go fast enough to circumnavigate it because there's a speed limit.  In your imagination is it finite or infinite?  Are there locations on it which are finite distances apart?  Is there a set of such locations connecting any two points?  Is the sum of the distances between locations of such a set finite?

Brent

[Brent Meeker]

On 1/24/2020 5:06 AM, Lawrence Crowell wrote:

On Thursday, January 23, 2020 at 9:54:29 PM UTC-6, Brent wrote:

On 1/23/2020 5:22 PM, Lawrence Crowell wrote:

On Thursday, January 23, 2020 at 5:56:08 PM UTC-6, John Clark wrote:

On Thu, Jan 23, 2020 at 6:35 PM Bruce Kellett <bhkel...@gmail.com> wrote:

 

> You seem to have missed an important little word in Brent's post: Brent talked about needing an infinite RANGE of coordinate values for an infinite universe

OK fine, so in a finite universe you'd only need a finite RANGE of coordinate values printed on a finite number of labels for all the finite number of points in that finite universe. But as I said, if new points are constantly being made at an accelerating rate in that "finite" universe then you're going to run out of those finite labels.

> nothing whatsoever about having only a finite set of distinguishable labels......

Nothing whatsoever? He specifically said a "range of coordinate values to label all the points". And if a label isn't distinguishable then it isn't a label. 

John K Clark

 If you have a sphere that is expanding the coordinate grid comoves with that. The spacing between coordinate points increases. The number of points needed to specify things does not need to change.

But if "the spacing increases" means anything at all, it means the range of coordinate values to define those points must increase.

Brent

If the sphere has constant curvature then to specify a point between two comoving or expanding coordinate points one can calculate it. While there is a continuum of space in point-set topology this does not really mean a space requires an infinite, indeed uncountable infinite, amount of information to describe it.

I don't think the discussion of what makes a negative curvature or flat universe infinite, referred to the amount of information needed to describe it.  It referred to distance in the metric space.

Brent

[John Clark]

On Fri, Jan 24, 2020 at 3:06 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

 

>> If you're assuming that Real Numbers exist and that even a 1 cm universe would need a infinite number of labels

> But not an infinite range of labels.

OK now it's official, I have no idea what you're talking about.

>> I ask my question again:
What is the difference between a "finite" universe that is expanding and accelerating and an infinite universe that is expanding and accelerating?

> Imagine the Earth is expanding like a balloon and at an accelerating pace. 

A balloon is a terrible analogy for the Earth and a inflating balloon is an even worse analogy for a universe that will expand and accelerate forever. With the balloon you're standing outside of it watching the balloon expand into something that's already there, but you can't stand outside of the universe and the universe is not expanding into anything that's already there.

 

> You can't go fast enough to circumnavigate it because there's a speed limit. 

And to call that speed limit the speed of light would be true but tends to trivialize it, really it's something far more fundamental and profound, it's the very speed of causality.

 

> In your imagination is it finite or infinite? Are there locations on it which are finite distances apart? Is there a set of such locations connecting any two points?  Is the sum of the distances between locations of such a set finite?

I would say a infinite amount of information would be needed to adequately describe the evolution of the phase space (all possible values of the position and momentum of the particles in the universe) of such a expanding accelerating universe. It's infinite because no amount of approximation would be good enough for prediction, due to the accelerated creation of new space there will always be more values of position and momentum that particles can be in tomorrow than they can be in today. By the way, all this talk about the distance between particles in a expanding accelerating universe is rather ambiguous if you don't specify when, and "now" has no meaning everybody agrees with.

  

And I've heard a bunch of bad analogies but I still haven't heard a direct answer to my question:
What is the difference between a "finite" universe that is expanding and accelerating forever and an infinite universe that is expanding and accelerating forever?

John K Clark

[Bruce Kellett]

On Sat, Jan 25, 2020 at 9:09 AM John Clark <johnk...@gmail.com> wrote:

And I've heard a bunch of bad analogies but I still haven't heard a direct answer to my question:
What is the difference between a "finite" universe that is expanding and accelerating forever and an infinite universe that is expanding and accelerating forever?

If you don't understand Brent's answer in terms of the range of values in coordinate maps, then you will never understand the difference. A finite universe has a finite range of coordinate values. Even if it is expanding exponentially, the range of coordinate values only ever increases at the same exponential rate -- i.e., never becomes infinite. In the case of an initially infinite universe, the coordinate range is necessarily infinite, and remains infinite even with expansion.

Bruce

[Brent Meeker]

On 1/24/2020 2:08 PM, John Clark wrote:

On Fri, Jan 24, 2020 at 3:06 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

 

>> If you're assuming that Real Numbers exist and that even a 1 cm universe would need a infinite number of labels

> But not an infinite range of labels.

OK now it's official, I have no idea what you're talking about.

>> I ask my question again:
What is the difference between a "finite" universe that is expanding and accelerating and an infinite universe that is expanding and accelerating?

> Imagine the Earth is expanding like a balloon and at an accelerating pace. 

A balloon is a terrible analogy for the Earth and a inflating balloon is an even worse analogy for a universe that will expand and accelerate forever. With the balloon you're standing outside of it watching the balloon expand into something that's already there,

But you don't have to stand outside of it.  Everything in the analogy is observable for a Flatland creature living on the sphere.

but you can't stand outside of the universe and the universe is not expanding into anything that's already there.

 

> You can't go fast enough to circumnavigate it because there's a speed limit. 

And to call that speed limit the speed of light would be true but tends to trivialize it, really it's something far more fundamental and profound, it's the very speed of causality.

So what.  I'm making an analogy, not a model.

 

> In your imagination is it finite or infinite? Are there locations on it which are finite distances apart? Is there a set of such locations connecting any two points?  Is the sum of the distances between locations of such a set finite?

I would say a infinite amount of information would be needed to adequately

Nobody asked about the amount of information.  That's a red herring that LC threw in.  The question was about the expansion and size of the universe.

describe the evolution of the phase space (all possible values of the position and momentum of the particles in the universe) of such a expanding accelerating universe. It's infinite because no amount of approximation would be good enough for prediction, due to the accelerated creation of new space there will always be more values of position and momentum that particles can be in tomorrow than they can be in today. By the way, all this talk about the distance between particles in a expanding accelerating universe is rather ambiguous if you don't specify when, and "now" has no meaning everybody agrees with.

  

And I've heard a bunch of bad analogies but I still haven't heard a direct answer to my question:
What is the difference between a "finite" universe that is expanding

As in my analogy, in a finite universe there are a finite number of intervals of finite distance that can link any two points in the universe.  Of course this refers to it being finite at a given time, and you raised the problem of defining what counts as "at the same time".  The answer is that it is at the same time if it is at the same degree of expansion...operationally it means that two distant events are "at the same time" if the isotropic temperature of the CMB looks the same to them.

Brent

and accelerating forever and an infinite universe that is expanding and accelerating forever?

John K Clark

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[Lawrence Crowell]

The distinction between an accelerated universe that is open and one that is closed is not hard to understand theoretically. I think the real question is how can an observer know which they inhabit. A spherical universe that is accelerated has a cosmological horizon that makes galaxies accelerate away faster than any signal or particle from Earth could ever catch up with. The same of course for the flat accelerated universe. If the radius of curvature of the spherical universe is large enough it might be impossible to determine which one inhabits by astronomical or astrometric means.

The difference between these two is can be seen as topological, and according to why a timelike curve can't be transformed into a null ray. I attach my figure below of the dS spacetime

The green diagonal lines are surfaces for the final states of an open cosmology and the dotted line between the two green lines is the initial state of the two cosmologies in a spatial surface. The two green lines may be identified with the I^+ in a conformal diagram. This is referred to by some as a biverse. The red curves are time evolved spatial surfaces and the blue curves are time directions. The closed spatial universe is more intuitive as analogous to the circle at the waist and a foliation of circles would be the time evolved cosmos. There is then no way to transform one type of cosmology into the other. 

The difference is also topological, and the different topological numbers (Betti numbers, Chern numbers etc) are associated with different quantum numbers on the open and closed spatial surfaces. In that way the two have the same physical information. This would then mean there is some possible way to ascertain whether the universe is open or closed by some quantum field information. 

LC

[Alan Grayson]

On Friday, January 24, 2020 at 3:50:54 PM UTC-7, Brent wrote:

On 1/24/2020 2:08 PM, John Clark wrote:

On Fri, Jan 24, 2020 at 3:06 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

 

>> If you're assuming that Real Numbers exist and that even a 1 cm universe would need a infinite number of labels

> But not an infinite range of labels.

OK now it's official, I have no idea what you're talking about.

>> I ask my question again:
What is the difference between a "finite" universe that is expanding and accelerating and an infinite universe that is expanding and accelerating?

> Imagine the Earth is expanding like a balloon and at an accelerating pace. 

A balloon is a terrible analogy for the Earth and a inflating balloon is an even worse analogy for a universe that will expand and accelerate forever. With the balloon you're standing outside of it watching the balloon expand into something that's already there,

But you don't have to stand outside of it.  Everything in the analogy is observable for a Flatland creature living on the sphere.

but you can't stand outside of the universe and the universe is not expanding into anything that's already there.

 

> You can't go fast enough to circumnavigate it because there's a speed limit. 

And to call that speed limit the speed of light would be true but tends to trivialize it, really it's something far more fundamental and profound, it's the very speed of causality.

So what.  I'm making an analogy, not a model.

 

> In your imagination is it finite or infinite? Are there locations on it which are finite distances apart? Is there a set of such locations connecting any two points?  Is the sum of the distances between locations of such a set finite?

I would say a infinite amount of information would be needed to adequately

Nobody asked about the amount of information.  That's a red herring that LC threw in.  The question was about the expansion and size of the universe.

describe the evolution of the phase space (all possible values of the position and momentum of the particles in the universe) of such a expanding accelerating universe. It's infinite because no amount of approximation would be good enough for prediction, due to the accelerated creation of new space there will always be more values of position and momentum that particles can be in tomorrow than they can be in today. By the way, all this talk about the distance between particles in a expanding accelerating universe is rather ambiguous if you don't specify when, and "now" has no meaning everybody agrees with.

  

And I've heard a bunch of bad analogies but I still haven't heard a direct answer to my question:
What is the difference between a "finite" universe that is expanding

As in my analogy, in a finite universe there are a finite number of intervals of finite distance that can link any two points in the universe.  Of course this refers to it being finite at a given time, and you raised the problem of defining what counts as "at the same time".  The answer is that it is at the same time if it is at the same degree of expansion...operationally it means that two distant events are "at the same time" if the isotropic temperature of the CMB looks the same to them.

Brent

You sometimes refer to the scale factor in GR being a function of time, namely a(t). But in relativity each observer has a clock, and time is what the observer reads on his clock. So what time are you referring to; the clock of a bird's eye observer outside the universe? TIA, AG 

and accelerating forever and an infinite universe that is expanding and accelerating forever?

John K Clark

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[Brent Meeker]

In the equation "t" is just a parameter.  The physical clock is the expansion itself, which is most conveniently measured by the CMB temperature.  But if you read Ned Wright's tutorial he points out that there are different ways to assign a size to the observable universe because of the frame dependence of simultaneity.

Brent

and accelerating forever and an infinite universe that is expanding and accelerating forever?

John K Clark

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[Alan Grayson]

OK, but then, as I suggested many moons ago, there IS such a thing as absolute motion, and it's with respect to the CMB. AG

 

But if you read Ned Wright's tutorial he points out that there are different ways to assign a size to the observable universe because of the frame dependence of simultaneity.

Brent

and accelerating forever and an infinite universe that is expanding and accelerating forever?

John K Clark

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[Brent Meeker]

On 1/24/2020 7:57 PM, Alan Grayson wrote:

In the equation "t" is just a parameter.  The physical clock is the expansion itself, which is most conveniently measured by the CMB temperature. 

OK, but then, as I suggested many moons ago, there IS such a thing as absolute motion, and it's with respect to the CMB. AG

It's no more absolute motion than my motion relative to the Moon is absolute.  It's motion defined relative to a physically meaningful frame, particularly for cosmological calculation.  But the dynamic equations of physics are independent of motion relative to it.  The Earth is moving at 368 km/sec relative to the CMB, but we don't have to put that into any of our equations.

Brent

[Alan Grayson]

On Thursday, January 23, 2020 at 12:37:21 PM UTC-7, Brent wrote:

On 1/23/2020 2:40 AM, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 9:03:55 PM UTC-7, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 8:54:37 PM UTC-7, Brent wrote:

On 1/22/2020 6:38 PM, Alan Grayson wrote:

On Wednesday, January 22, 2020 at 1:34:00 PM UTC-7, Lawrence Crowell wrote:

On Wednesday, January 22, 2020 at 11:33:04 AM UTC-6, John Clark wrote:

On Wed, Jan 22, 2020 at 12:06 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> It is then possible to have an expanding accelerated cosmos that is spherically closed.

So if I keep going I will eventually return to where I started even though everything is constantly getting more distant from me and is doing so at an accelerating rate?

 John K Clark

For an accelerated expansion of the sphere there is a cosmological horizon that one can't cross. in other words, the sphere will keep expanding faster than you can ever go. Think of the scene in the movie "The Shining" with Jack Nicholson where the hotel hallway telescoped away faster than he could run.

LC

I don't think it depends on acceleration. As long as the universe expands, even at a constant rate, at some distance, the distance between, say, an Earth observer, and some terminal point along a line of sight, will exceed 300,000 km (the distance light travels in one second) and points beyond that will keep increasing the increment every second, creating a cosmological horizon that light cannot cross.

That's not quite right.  Light can cross it just fine.  But a photon crossing it toward us, can never reach us.  This is how the Hubble boundary differs from a black hole event horizon.

Brent

Good point. TY. AG 

Now I'm not so sure. ISTM, the photons that never reach us, never cross the event horizon. They're emitted in a region receding faster than the SoL, so they can never cross it. AG

Sure they do.  If galaxy Z is at our Hubble boundary, we're at galaxy Z's Hubble boundary.  Does that mean we can't emit a photon toward galaxy Z?

Brent

It's still not clear. Let's say galaxy Z is in our non-observable region and emits a photon in our direction. Since space is expanding faster than light at that distance, the photon isn't traveling fast enough to enter our observable region, presumably where the event horizon is located.  AG

[John Clark]

On Fri, Jan 24, 2020 at 5:21 PM Bruce Kellett <bhkel...@gmail.com> wrote:

>> And I've heard a bunch of bad analogies but I still haven't heard a direct answer to my question:
What is the difference between a "finite" universe that is expanding and accelerating forever and an infinite universe that is expanding and accelerating forever?

 

> If you don't understand Brent's answer in terms of the range of values in coordinate maps, then you will never understand the difference.

Then I guess I'll never understand the difference.

 

> A finite universe has a finite range of coordinate values.

NOPE! Brent specifically said "I'm assuming a continuum spacetime. So even a 1cm interval takes an infinite number of labels".  Thus even if the universe is not expanding at all and even if it's only 1cm across a infinite number of labels with a infinite rage of coordinate values printed on them would be needed.

 

> Even if it is expanding exponentially, the range of coordinate values only ever increases at the same exponential rate -- i.e., never becomes infinite.

For the sake of argument let's take the opposite of Brent's assumption and say that spacetime is not a continuum but is composed of discrete chunks. It doesn't help. Any finite range of values you name regardless of how huge it is will soon (very soon, the universe is after all accelerating) be shown to be insufficiently large.

So I have to ask my question yet again:
What exactly is so finite about this finite universe of yours? How can I tell the difference between a "finite" universe that is expanding and accelerating forever and an infinite universe that is expanding and accelerating forever, and if I can't tell the difference is there any difference?

 John K Clark

[John Clark]

On Fri, Jan 24, 2020 at 5:50 PM 'Brent Meeker'  Everything List <everyth...@googlegroups.com> wrote:

>> I would say a infinite amount of information would be needed to adequately

> Nobody asked about the amount of information. 

Well you certainly didn't ask about it, you ignored information entirely and I would say that was the fundamental reason your analysis failed.

 

> That's a red herring that LC threw in. 

A red herring?! Lawrence is wise enough to know that if you're developing a cosmological model while pretending information does not exist then you're heading for trouble.

> The question was about the expansion and size of the universe.

No, the question was if the universe was infinite or finite. Yes if the position space (aka plain ordinary space) is infinite then it would be safe to say the universe is infinite, but that's just one attribute the universe can have, there is also momentum space and informational content; if either of those was infinite I would say that regardless of whether position space was infinite or not it would be misleading at best and dead wrong at worse to say the universe was finite.

 

> As in my analogy, in a finite universe there are a finite number of intervals of finite distance that can link any two points in the universe.  Of course this refers to it being finite at a given time, and you raised the problem of defining what counts as "at the same time".  The answer is that it is at the same time if it is at the same degree of expansion...operationally it means that two distant events are "at the same time" if the isotropic temperature of the CMB looks the same to them.

Even if we ignore Quantum Mechanics any finite level of precision used to measure the current position and momentum of those two particles or of the temperature of the CMB will soon (very soon because the universe is accelerating) prove to be insufficiently precise to predict their future position and momentum because phase space keeps getting larger at an accelerating rate. The fundamental reason you can't make a good prediction is you don't have enough information, an infinite amount is required and you don't have that. 

John K Clark

[Alan Grayson]

I don't think coordinate arguments will solve your problem. Suppose we have an observer at the origin of a one-dimensional space. If you consider a 1 meter line starting at the origin and increasing in length at an accelerating rate at its end point, then obviously, at any observer time t, the length of the line is finite and can be easily calculated. Now consider an expanding 4 dimensional space-time continuum. For any observer, and therefore for all observers, the volume of this space is obviously finite regardless of how fast it's expanding. So it's conceptually easy to distinguish finite from infinite volume when comparing an expanding hyper-sphere (finite) from an expanding plane (infinite). Your doubt seems to depend on the inability of observers to make the measurements when, due to expansion, both geometries have non-observable regions. So, does the distinction between finite and infinite volumes really depend upon what observers can measure? IMO, this is where the rubber hits the road. AG

[Alan Grayson]

You keep referring to accelerating expansion. I explained several times that acceleration has nothing to do with this issue. A constant rate of expansion will ALSO create non-observable regions. Do you accept this or not; and if not, why? AG 

[Lawrence Crowell]

On Saturday, January 25, 2020 at 6:23:54 AM UTC-6, John Clark wrote:

On Fri, Jan 24, 2020 at 5:21 PM Bruce Kellett <bhkel...@gmail.com> wrote:

>> And I've heard a bunch of bad analogies but I still haven't heard a direct answer to my question:
What is the difference between a "finite" universe that is expanding and accelerating forever and an infinite universe that is expanding and accelerating forever?

 

> If you don't understand Brent's answer in terms of the range of values in coordinate maps, then you will never understand the difference.

Then I guess I'll never understand the difference.

 

> A finite universe has a finite range of coordinate values.

NOPE! Brent specifically said "I'm assuming a continuum spacetime. So even a 1cm interval takes an infinite number of labels".  Thus even if the universe is not expanding at all and even if it's only 1cm across a infinite number of labels with a infinite rage of coordinate values printed on them would be needed.

Nope. Space and spacetime are an epiphenomenology. They are mental perceptual models that result from large N-entanglements of quantum states. There are no infinite sets of points and labels, that would in fact be uncountably infinite. These things only exist in our mathematical representations or axiomatic systems. Now, what information we can get about space from the IR domain of energy at extreme distances, such as with burstars etc,, is the representation of what we call space being smooth fits the data. This does not mean that fundamentally there is an actual smooth continuum of space.

LC

[Brent Meeker]

On 1/25/2020 4:32 PM, Lawrence Crowell wrote:

On Saturday, January 25, 2020 at 6:23:54 AM UTC-6, John Clark wrote:

On Fri, Jan 24, 2020 at 5:21 PM Bruce Kellett <bhkel...@gmail.com> wrote:

>> And I've heard a bunch of bad analogies but I still haven't heard a direct answer to my question:
What is the difference between a "finite" universe that is expanding and accelerating forever and an infinite universe that is expanding and accelerating forever?

 

> If you don't understand Brent's answer in terms of the range of values in coordinate maps, then you will never understand the difference.

Then I guess I'll never understand the difference.

 

> A finite universe has a finite range of coordinate values.

NOPE! Brent specifically said "I'm assuming a continuum spacetime. So even a 1cm interval takes an infinite number of labels".  Thus even if the universe is not expanding at all and even if it's only 1cm across a infinite number of labels with a infinite rage of coordinate values printed on them would be needed.

Nope. Space and spacetime are an epiphenomenology. They are mental perceptual models that result from large N-entanglements of quantum states. There are no infinite sets of points and labels, that would in fact be uncountably infinite. These things only exist in our mathematical representations or axiomatic systems. Now, what information we can get about space from the IR domain of energy at extreme distances, such as with burstars etc,, is the representation of what we call space being smooth fits the data. This does not mean that fundamentally there is an actual smooth continuum of space.

I don't disagree, but you're getting further and further from saying what it means for spacetime to be finite versus infinite.  Since it's our mathematical model, that should have a simple mathematical answer.

Brent

[Lawrence Crowell]

There seems to be some sort of issue with the idea of continuum or space having an infinite number of points. I see this as a modern day version of asking how many angels can dance on a pin.

LC 

[Alan Grayson]

I've answered JC's question above. The answer has nothing to do with coordinate systems. JC wants to know how to distinguish a finite spherical universe from an infinite flat universe (finite or infinite in VOLUME) and he thinks if he can't directly MEASURE non-observational regions, which both have, he can't distinguish the cases. The difference is this: every observer in a spherical universe can calculate its radius if he knows the rate of expansion and how long it has persisted for, which is not the case for observers in a flat space which presumably has an infinite past (for otherwise it would have what is impossible, an edge). AG

 

[Brent Meeker]

I have no issue with it.  But it doesn't mean that a spherical spacetime is infinite.  The infinity of metric distance in a Riemannian space is not the same as the infinite cardinality of point in a real interval.

Brent

[Philip Thrift]

If the Universe is truly infinite, if you travel outwards from Earth, eventually you will reach a place where there's a duplicate cubic meter of space. The further you go, the more duplicates you'll find.

Ooh, big deal, you think. One hydrogen pile looks the same as the next to me. Except, you hydromattecist, you'll pass through places where the configuration of particles will begin to appear familiar, and if you proceed long enough you'll find larger and larger identical regions of space, and eventually you'll find an identical you. And finding a copy of yourself is just the start of the bananas crazy things you can do in an infinite Universe.

In fact, hopefully you'll absorb the powers of an immortal version of you, because if you keep going you'll find an infinite number of yous. You'll eventually find entire duplicate observable universes with more yous also collecting other yous. And at least one of them is going to have a beard.

So, what's out there? Possibly an infinite number of duplicate observable universes. We don't even need multiverses to find them. These are duplicate universes inside of our own infinite universe. That's what you can get when you can travel in one direction and never, ever stop.

Whether the Universe is finite or infinite is an important question, and either outcome is mindblenderingly fun. So far, astronomers have no idea what the answer is, but they're working towards it and maybe someday they'll be able to tell us.

https://phys.org/news/2015-03-universe-finite-infinite.html

@philipthrift

 

[Bruce Kellett]

On Sun, Jan 26, 2020 at 6:20 PM Philip Thrift <cloud...@gmail.com> wrote:

If the Universe is truly infinite, if you travel outwards from Earth, eventually you will reach a place where there's a duplicate cubic meter of space. The further you go, the more duplicates you'll find.

Ooh, big deal, you think. One hydrogen pile looks the same as the next to me. Except, you hydromattecist, you'll pass through places where the configuration of particles will begin to appear familiar, and if you proceed long enough you'll find larger and larger identical regions of space, and eventually you'll find an identical you. And finding a copy of yourself is just the start of the bananas crazy things you can do in an infinite Universe.

In fact, hopefully you'll absorb the powers of an immortal version of you, because if you keep going you'll find an infinite number of yous. You'll eventually find entire duplicate observable universes with more yous also collecting other yous. And at least one of them is going to have a beard.

So, what's out there? Possibly an infinite number of duplicate observable universes. We don't even need multiverses to find them. These are duplicate universes inside of our own infinite universe. That's what you can get when you can travel in one direction and never, ever stop.

Whether the Universe is finite or infinite is an important question, and either outcome is mindblenderingly fun. So far, astronomers have no idea what the answer is, but they're working towards it and maybe someday they'll be able to tell us.

Unjustifiable assumptions about initial distributions are being made here -- so it is yet another load of codswallop!

Bruce

 

https://phys.org/news/2015-03-universe-finite-infinite.html

@philipthrift

[Lawrence Crowell]

This is the case for a spatial surface that is infinite, but distance is using the idea of Poincare recurrence around 10^{10^{100}} light years away. This is far beyond the cosmological horizon and you could never get there no matter how long or extremely you try to accelerate outwards. With the spherical universe much the same also holds, but where getting around a spatial sphere with an enormous radius of curvature is impossible because it will always expand faster than you can travel. With the flat spacetime the existence of repeated versions of this local world means there is some covering space that is a torus or maybe the Poincare dodecahedral space. I tend to think this covering space is some form of quasi-crystal. For all we know we are in a cosmos with that sort of space. 

LC

 

 

[Alan Grayson]

I don't see any basis for assuming infinite repetitions in an infinite universe. It's sort-of like the claim that every thing that can happen, must happen.  What's your take? AG

 

 

[Philip Thrift]

Did you send the writer an email explaining that?

There's nothing wring with his article I can see.

Author:

https://twitter.com/fcain

Fraser Cain

@fcain

Publisher of http://universetoday.com, co-host of http://astronomycast.com. Named after Asteroid 158092.

Courtenay, British Columbiaabout.me/frasercain

@philipthrift

[Lawrence Crowell]

 

These repetitions are an aspect of what is called the level I multiverse. It is just the world beyond the horizon that if infinite is by statistical necessity going to reproduce local regions.

LC

[John Clark]

On Sat, Jan 25, 2020 at 9:10 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

> There seems to be some sort of issue with the idea of continuum or space having an infinite number of points. I see this as a modern day version of asking how many angels can dance on a pin.

I can see what you mean in a de Sitter universe that had no matter in it, then the continuum vs non-continuum argument and even the infinite universe vs finite universe argument would be pretty meaningless, just philosophers running around in circles chasing their own tails. I would even question if a universe that had no matter in it could even meaningfully be said to exist because the best definition of "nothing" I've ever heard is "infinite unbounded homogeneity", and that sure sounds a lot like a de Sitter universe to me.

But if it's not de Sitter and real particles exist not just virtual particles, and if space or time is a continuum then even if we ignore Quantum Mechanics you'd need an infinite number of digits to describe the distance between 2 particles and to state their momentum. And no amount of approximation would be good enough to make long term predictions about their position and momentum because even the smallest error could cause huge differences in outcome; and that would be even more true if the universe were accelerating.

Anyway, would you agree that there is no effective way to tell the difference between a "finite" universe that is expanding and accelerating forever and a "infinite" universe that is expanding and accelerating forever?  If there isn't then it is indeed a modern day version of asking how many angels can dance on a pin.

 John K Clark

[Alan Grayson]

There is, but obviously you're not interested. AG

[Alan Grayson]

This is consistent with my intuitions; namely, that we live in a finite universe, approximately hyper-spherical, with no repetitions. AG 

[Philip Thrift]

https://www.linkedin.com/in/frasercain/

@philipthrift

[John Clark]

On Sat, Jan 25, 2020 at 10:27 PM Alan Grayson <agrays...@gmail.com> wrote:

> JC wants to know how to distinguish a finite spherical universe from an infinite flat universe

No, JC doesn't care if the universe is spherical or flat, he just wants to know if he keeps going will he always keep getting further from his starting point, or will he start to return. or will he eventually hit some sort of wall.

> he thinks if he can't directly MEASURE non-observational regions, which both have, he can't distinguish the cases.

Correct.

 

> The difference is this: every observer in a spherical universe can calculate its radius if he knows the rate of expansion and how long it has persisted for,

Incorrect. He can't determine how big the universe is, from the date of the expansion and its rate of acceleration he can only calculate how far into the universe he can see. You're error is you ignored a fact that is fundamental and very important, the speed of causality is not infinite. That's why regardless of if the universe is infinite or finite I doubt there is a cosmologist alive who thinks that what we'll someday be able to see with even tomorrows planet sized telescopes is all of the universe that there is. There will always be part of the universe we will be unable to see even in principle because parts of it are moving away from us faster than the speed of causality. We have a good lower limit of how big that unobservable part is, it's larger than zero, but its upper limit is pure speculation and always will be.

John K Clark

[Brent Meeker]

On Sunday, January 26, 2020 at 1:30:58 AM UTC-6, Bruce wrote:

On Sun, Jan 26, 2020 at 6:20 PM Philip Thrift <cloud...@gmail.com> wrote:

 

Unjustifiable assumptions about initial distributions are being made here -- so it is yet another load of codswallop!

Bruce

Just as a point of curiosity is codswallop a delicacy you can order in Oz, like vegimite?   What is it?

Brent

[Bruce Kellett]

I refer you to an authoritative source -- which also has no explanation for the origin of the term "codswallop".

https://www.phrases.org.uk/meanings/codswallop.html

By the way, the Australian and New Zealand savoury delicacy is spelled "Vegemite". I have no idea what "vegimite" might be!

As a point of historical interest, vegemite was originally called "pawill', to avoid confusion with the English savoury delicacy called "Marmite". But the name never caught on, and it was relabelled "vegemite".

https://twistedhistory.net.au/tag/if-marmite-pawill/

Bruce

[Brent Meeker]

On 1/26/2020 1:48 PM, Bruce Kellett wrote:

On Mon, Jan 27, 2020 at 7:57 AM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On Sunday, January 26, 2020 at 1:30:58 AM UTC-6, Bruce wrote:

On Sun, Jan 26, 2020 at 6:20 PM Philip Thrift <cloud...@gmail.com> wrote:

 

Unjustifiable assumptions about initial distributions are being made here -- so it is yet another load of codswallop!

Bruce

Just as a point of curiosity is codswallop a delicacy you can order in Oz, like vegimite?   What is it?

I refer you to an authoritative source -- which also has no explanation for the origin of the term "codswallop".

https://www.phrases.org.uk/meanings/codswallop.html

By the way, the Australian and New Zealand savoury delicacy is spelled "Vegemite". I have no idea what "vegimite" might be!

Having tasted it, I have no idea what Vegemite might be either.  :-)

Brent

As a point of historical interest, vegemite was originally called "pawill', to avoid confusion with the English savoury delicacy called "Marmite". But the name never caught on, and it was relabelled "vegemite".

https://twistedhistory.net.au/tag/if-marmite-pawill/

Bruce

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[Alan Grayson]

On Sunday, January 26, 2020 at 6:39:00 AM UTC-7, John Clark wrote:

On Sat, Jan 25, 2020 at 10:27 PM Alan Grayson <agrays...@gmail.com> wrote:

> JC wants to know how to distinguish a finite spherical universe from an infinite flat universe

No, JC doesn't care if the universe is spherical or flat, he just wants to know if he keeps going will he always keep getting further from his starting point, or will he start to return.

If the universe is expanding, even lower than c, you'll never return to starting point. AG 

or will he eventually hit some sort of wall.

No walls, ever. AG 

> he thinks if he can't directly MEASURE non-observational regions, which both have, he can't distinguish the cases.

Correct.

 

> The difference is this: every observer in a spherical universe can calculate its radius if he knows the rate of expansion and how long it has persisted for,

Incorrect. He can't determine how big the universe is, from the date of the expansion and its rate of acceleration he can only calculate how far into the universe he can see.

If it's a hyper-sphere, its curvature is constant. So if you can measure OR calculate its curvature, by simple trigonometry you can calculate its total finite volume from the curvature, including the non observable region. According to the article on posted on a related thread, the authors claim that the CMB data suggests the universe is curved. I anticipate that at some time in the future. we'll be able to measure the curvature from the CMB. AG 

You're error is you ignored a fact that is fundamental and very important, the speed of causality is not infinite.

I have not. AG

 

That's why regardless of if the universe is infinite or finite I doubt there is a cosmologist alive who thinks that what we'll someday be able to see with even tomorrows planet sized telescopes is all of the universe that there is.

For an expanding universe, we'll never be able to observe the entire universe, whether hyper-spherical or flat. But for a hyper-spherical universe we don't have to. All we need is to determine its curvature, from which we can easily calculate its total finite volume, including the non-observable region. AG

 

There will always be part of the universe we will be unable to see even in principle because parts of it are moving away from us faster than the speed of causality.

Agreed. That's why I objected recently to Brent's claim that light from a galaxy in our non-observable region can cross our event horizon. Those photons are not traveling fast enough to overcome the rate of expansion of space. AG

 

We have a good lower limit of how big that unobservable part is, it's larger than zero, but its upper limit is pure speculation and always will be.

Doesn't matter if our universe is a hyper-sphere and we want to determine its finite volume, provided we can determine its curvature. AG

John K Clark

[John Clark]

On Sun, Jan 26, 2020 at 6:39 PM Alan Grayson <agrays...@gmail.com> wrote:

>  I objected recently to Brent's claim that light from a galaxy in our non-observable region can cross our event horizon.

It's just the opposite. Regardless of if space is positively curved or negatively curved or as flat as a pancake, if the universe is accelerating and not just expanding then galaxies in our OBSERVABLE region will eventually cross over our event horizon into our UNOBSERVABLE region, and there is no way to tell how much is already there, no way to know if that unobservable region is finite or infinite because it is...well... unobservable.

John K Clark

[Alan Grayson]

Did you study trigonometry in high school? AG 

[Alan Grayson]

On Sunday, January 26, 2020 at 5:24:46 PM UTC-7, John Clark wrote:

On Sun, Jan 26, 2020 at 6:39 PM Alan Grayson <agrays...@gmail.com> wrote:

>  I objected recently to Brent's claim that light from a galaxy in our non-observable region can cross our event horizon.

It's just the opposite.

You misunderstand. What I thought Brent posted, was that if a photon is in our non-observable region, it could cross our event horizon (and therefore become visible). If that's what he claimed, I think it's wrong. AG

 

Regardless of if space is positively curved or negatively curved or as flat as a pancake, if the universe is accelerating and not just expanding then galaxies in our OBSERVABLE region will eventually cross over our event horizon into our UNOBSERVABLE region,

How many times must I say this? It does NOT depend on acceleration, just expansion, because the effect is purely geometrical, which I earlier explained. AG

 

and there is no way to tell how much is already there, no way to know if that unobservable region is finite or infinite because it is...well... unobservable.

There is a physical clock for the universe, namely, the temperature of the CMBR. Moreover, the curvature at any time t, is the same everywhere, if the universe is a hyper-sphere, So if you want to calculate its radius, all you need is its curvature! -- which, I conjecture, will one day be able to be measured. And that measurement need NOT include the unobservable region. AG

John K Clark

[Philip Thrift]

On Sunday, January 26, 2020 at 5:39:31 PM UTC-6, Alan Grayson wrote:

 According to the article on posted on a related thread, the authors claim that the CMB data suggests the universe is curved. I anticipate that at some time in the future. we'll be able to measure the curvature from the CMB. AG 

What specific mathematical model are they fitting the data to and finding curvature in?

@philipthrift

 

[Alan Grayson]

Brent posted a link to the scientific paper on which the article is based. I tried to read it, but it's somewhat above my head. I suggest you try reading it and let us know if you understand the technical points and whether it's persuasive. AG 

 

[Alan Grayson]

Here's the scientific paper:  https://arxiv.org/pdf/1911.02087.pdf  .  Let us know what you make of it. AG

 

[Philip Thrift]

I copied it to my Desktop and will read it today!

@philipthrift

 

[John Clark]

On Sun, Jan 26, 2020 at 9:49 PM Alan Grayson <agrays...@gmail.com> wrote:

Note: At the end of this email I posted a picture, if anybody responds to it PLEASE don't just hit the reply button, take 2 seconds to edit the picture out so we don't get endless recursive iterations of it.

>>Regardless of if space is positively curved or negatively curved or as flat as a pancake, if the universe is accelerating and not just expanding then galaxies in our OBSERVABLE region will eventually cross over our event horizon into our UNOBSERVABLE region,

> How many times must I say this?

42.

> It does NOT depend on acceleration, just expansion, because the effect is purely geometrical, which I earlier explained. [...] So if you want to calculate its radius, all you need is its curvature! -- which, I conjecture, will one day be able to be measured.

First of all, you'll never be able to measure exactly zero curvature and prove it's flat, you might in principle be able to measure positive curvature and show that the universe is spherically shaped, but in a expanding accelerating universe that wouldn't prove the universe is finite. You can fit an infinite volume inside a expanding sphere if you take length contraction into account. Einstein tells us that if the universe is a expanding sphere then the more distant a star is from us the faster it will be moving away from us and thus the thinner it will look to us, this is even more important if it's not just expanding but accelerating. It's rather like one of M C Escher's "Circle Limit" series of 2D woodcuts blown up to 3D. They're all great but take a look at Circle Limit III:

John K Clark

[Alan Grayson]

On Monday, January 27, 2020 at 6:41:45 AM UTC-7, John Clark wrote:

On Sun, Jan 26, 2020 at 9:49 PM Alan Grayson <agrays...@gmail.com> wrote:

Note: At the end of this email I posted a picture, if anybody responds to it PLEASE don't just hit the reply button, take 2 seconds to edit the picture out so we don't get endless recursive iterations of it.

>>Regardless of if space is positively curved or negatively curved or as flat as a pancake, if the universe is accelerating and not just expanding then galaxies in our OBSERVABLE region will eventually cross over our event horizon into our UNOBSERVABLE region,

> How many times must I say this?

42.

> It does NOT depend on acceleration, just expansion, because the effect is purely geometrical, which I earlier explained. [...] So if you want to calculate its radius, all you need is its curvature! -- which, I conjecture, will one day be able to be measured.

First of all, you'll never be able to measure exactly zero curvature and prove it's flat, you might in principle be able to measure positive curvature and show that the universe is spherically shaped, but in a expanding accelerating universe that wouldn't prove the universe is finite.

It sure would. If it's expanding at a finite rate for finite time (the age of the universe), how could it be other than finite? AG

 

You can fit an infinite volume inside a expanding sphere if you take length contraction into account.

Failing to apply length contraction (and I'm not sure it is applicable in this situation), would just mean that the estimate without it would be too large, but not infinite. AG 

Einstein tells us that if the universe is a expanding sphere then the more distant a star is from us the faster it will be moving away from us and thus the thinner it will look to us, this is even more important if it's not just expanding but accelerating.

Irrelevant IMO. AG

[John Clark]

On Mon, Jan 27, 2020 at 2:18 PM Alan Grayson <agrays...@gmail.com> wrote:

>>You can fit an infinite volume inside a expanding sphere if you take length contraction into account.Einstein tells us that if the universe is a expanding sphere then the more distant a star is from us the faster it will be moving away from us and thus the thinner it will look to us, this is even more important if it's not just expanding but accelerating.

> Failing to apply length contraction (and I'm not sure it is applicable in this situation),

Interesting. Why aren't you sure? We know for a fact time runs slower relative to us for an observer in a distant galaxy because we can see the redshift, the decrease in frequency, of light that comes from there. But if clocks ran slower for them but lengths did not also contract for them then they would observe a different speed of light then we do. But we also know for a fact from other experiments that the speed of light is the one true constant for everyone everywhere, the observed speed of light does not depend on the speed of the observer or on the speed of the source producing the light. So why are you "not sure it is applicable in this situation"?

 > would just mean that the estimate without it would be too large, but not infinite. AG 

Neither Einstein's theory or anything else in physics says length contraction, time dilation, and mass increase discontinuously stops at some point short of the speed of light, they don't suddenly stop increasing, they increase continuously up to the speed of light. An expanding spherical universe that has a constant speed of causality would follow 3D hyperbolic geometry just as MC Escher's woodcut "Circle Limit III" follows 2D hyperbolic geometry.

John K Clark

[Alan Grayson]

On Monday, January 27, 2020 at 3:14:13 PM UTC-7, John Clark wrote:

On Mon, Jan 27, 2020 at 2:18 PM Alan Grayson <agrays...@gmail.com> wrote:

>>You can fit an infinite volume inside a expanding sphere if you take length contraction into account.Einstein tells us that if the universe is a expanding sphere then the more distant a star is from us the faster it will be moving away from us and thus the thinner it will look to us, this is even more important if it's not just expanding but accelerating.

> Failing to apply length contraction (and I'm not sure it is applicable in this situation),

Interesting. Why aren't you sure? We know for a fact time runs slower relative to us for an observer in a distant galaxy because we can see the redshift, the decrease in frequency, of light that comes from there. But if clocks ran slower for them but lengths did not also contract for them then they would observe a different speed of light then we do. But we also know for a fact from other experiments that the speed of light is the one true constant for everyone everywhere, the observed speed of light does not depend on the speed of the observer or on the speed of the source producing the light. So why are you "not sure it is applicable in this situation"?

Simple. Because length contraction, say of a rod, depends on comparing measurement of the rod's length as observed in two frames of reference, moving wrt each other.  In this case, we're making a measurement of the CMBR to determine curvature. AG

 > would just mean that the estimate without it would be too large, but not infinite. AG 

Neither Einstein's theory or anything else in physics says length contraction, time dilation, and mass increase discontinuously stops at some point short of the speed of light, they don't suddenly stop increasing, they increase continuously up to the speed of light.

I haven't stated anything about discontinuities. They don't exist in this situation. AG

[John Clark]

On Mon, Jan 27, 2020 at 8:54 PM Alan Grayson <agrays...@gmail.com> wrote:

>> We know for a fact time runs slower relative to us for an observer in a distant galaxy because we can see the redshift, the decrease in frequency, of light that comes from there. But if clocks ran slower for them but lengths did not also contract for them then they would observe a different speed of light then we do. But we also know for a fact from other experiments that the speed of light is the one true constant for everyone everywhere, the observed speed of light does not depend on the speed of the observer or on the speed of the source producing the light. So why are you "not sure it is applicable in this situation"?

> Simple.

Yes your answer is very simple, but that word has more than one meaning.

 > Because length contraction, say of a rod, depends on comparing measurement of the rod's length as observed in two frames of reference, moving wrt each other.  In this case, we're making a measurement of the CMBR to determine curvature. AG

I'm not talking about Euclidean curvature! I'm trying to show you the volume in a expanding sphere can be infinite. An observer in a distant galaxy using a clock and a meter stick can measure the speed of light. We know for a fact his clock runs slower than our clock (we know this from the redshift). So if his meter stick is not shorter than our meter stick (from relativistic length contraction) then he would measure a different speed for light than we do.  But we know all observers measure the same speed for light. Therefore he must experience both time dilation AND length contraction. So regardless of what the local geometry is, on a large scale the geometry of our universe must be hyperbolic; and the same would be true for any universe that was expanding and had a finite speed of causality.

 >>> would just mean that the estimate without it would be too large, but not infinite. AG 

>> Neither Einstein's theory or anything else in physics says length contraction, time dilation, and mass increase discontinuously stops at some point short of the speed of light, they don't suddenly stop increasing, they increase continuously up to the speed of light.

> I haven't stated anything about discontinuities. They don't exist in this situation. AG

OK fine, but if there are no discontinuities then as galaxies get more and more distant from us the clocks in them can run arbitrarily slower than ours from time dilation. And galaxies can be arbitrarily thin from length contraction. And so you could fit a arbitrarily large number of galaxies in a arbitrarily small volume of space. And so globally the universe must follow the rules of hyperbolic geometry not those of Euclid.  And so there is nothing to prevent the volume of a sphere from being infinite if it is expanding and does what Einstein says.

John K Clark

[Alan Grayson]

Since you can't measure anything in the NON-observable region, your argument fails. Moreover, the radius of a sphere is the same everywhere, so if we measure it via the CMBR, this is sufficient to calculate its total volume, including the NON-observable region. AG 

[Alan Grayson]

On Tuesday, January 28, 2020 at 5:44:52 AM UTC-7, John Clark wrote:

On Mon, Jan 27, 2020 at 8:54 PM Alan Grayson <agrays...@gmail.com> wrote:

>> We know for a fact time runs slower relative to us for an observer in a distant galaxy because we can see the redshift, the decrease in frequency, of light that comes from there. But if clocks ran slower for them but lengths did not also contract for them then they would observe a different speed of light then we do. But we also know for a fact from other experiments that the speed of light is the one true constant for everyone everywhere, the observed speed of light does not depend on the speed of the observer or on the speed of the source producing the light. So why are you "not sure it is applicable in this situation"?

> Simple.

Yes your answer is very simple, but that word has more than one meaning.

 > Because length contraction, say of a rod, depends on comparing measurement of the rod's length as observed in two frames of reference, moving wrt each other.  In this case, we're making a measurement of the CMBR to determine curvature. AG

I'm not talking about Euclidean curvature! I'm trying to show you the volume in a expanding sphere can be infinite. An observer in a distant galaxy using a clock and a meter stick can measure the speed of light. We know for a fact his clock runs slower than our clock (we know this from the redshift). So if his meter stick is not shorter than our meter stick (from relativistic length contraction) then he would measure a different speed for light than we do.  But we know all observers measure the same speed for light. Therefore he must experience both time dilation AND length contraction.

For the observer situated in a distant galaxy, his clock does not dilate, and his length does not contract. Rather, that's how it appears for an observer far from that galaxy, moving away with some relative speed.  Moreover, if that galaxy is in the non-observable region wrt to the distant observer "measuring" time and length, no measurements are possible. And even if the impossible measurement could be made, those galaxies would NOT shrink in length to zero, presumably allowing for infinite volume, since the expansion has been going on for finite time, 13.8 BY. AG