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Perhaps the only way to understand it, then, is by learning the math

However I saw many words and little maths, really only a few unsubstantiated formulae.

There was no development of a mathematical argument.

Are you suggesting I should learn this?

Certainly there was no mathematical answer to the straightforward mathematical question I raised, although the article did state that there was no expansion (section 2.0)

We can certainly agree that this kind of misuse of

the term “expansion of space” is fallacious and most

certainly dangerous.

Unfortunately the article introduces an aether theory in section 2.1 et seq, which substatially diminishes it in my view.

I would add that my question is linked to my comments in post#5 and represents the beginning of a critical examination of the idea of expansion.

I sympathise with the OP's dilemma and seek some solid mathematics.

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if their is a creator HE must be timeless, but i do not think there is a creator, and space can not be finite, it has to be infinite. because if there's boundaries what is past them. (MORE SPACE)

My view on a creator is the universe and it's laws

Edited by mattrsmith88
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However I saw many words and little maths, really only a few unsubstantiated formulae.

There was no development of a mathematical argument.

Are you suggesting I should learn this?

I just thought that the introductory sections were a good non-mathematical description of what the "space is expanding" analogy refers to. (The paper does get more mathematical later; those bits probably don't make sense without a good understanding of the Einstein Field Equations).

I sympathise with the OP's dilemma and seek some solid mathematics.

Then you need to get to grips with the FLRW solution to the Einstein Field Equations. (Although this seems to be what you think is an "aether theory". It isn't.)

http://en.wikipedia.org/wiki/FLRW

http://math.ucr.edu/home/baez/physics/Relativity/GR/expanding_universe.html

http://en.wikipedia.org/wiki/Metric_expansion_of_space

http://math.ucr.edu/home/baez/gr/gr.html

space can not be finite, it has to be infinite. because if there's boundaries what is past them. (MORE SPACE)

Space can be finite and unbounded. Consider the 2D surface of a sphere: it is finite but with no edges. The same can be applied to 3 dimensions.

Edited by Strange
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i do not believe the universe is a sphere, any evidence?

i picture the physical universe as a massive nebula, but i'm asking what is beyond that

and i do not need to get to grips, nothing is proven in this field.

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Let me extrapolate on what Strange said. The spacetime continuum can be curved. Einstein said gravity was a warpature of the spacetime continuum. So what about the whole universe? The universe could be warped like the surface of a globe. Light traveling in a strait line could come back to the beginning. This would mean the universe would be finite but unbounded (no end).

In the early universe it would be like the surface of a small moon (I'm only talking 2 dimensions for simplicity) you could travel in a strait line and get back to where you started.

Then the universe has been expanding and much harder to get back to your starting point by traveling in a strait line.

i do not believe the universe is a sphere, any evidence?

i picture the physical universe as a massive nebula, but i'm asking what is beyond that

and i do not need to get to grips, nothing is proven in this field.

NO NO NO the universe is not at all like a nebula in space, Filling up a certain amount of space.

The universe IS SPACE. The universe IS TIME. The universe IS the SPACETIME CONTINUUM.

But it does have matter within it.

i picture the physical universe as a massive nebula, but i'm asking what is beyond that

Stop thinking about the universe as if it was the stars and matter within the universe. When we say the universe is like a sphere we are talking about 2 dimensions of the spacetime continuum. For the sake of understanding forget about the stars and matter within the universe.

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That's all very well but that does not explain what is meant by 'expansion'.

Do you consider the universe finite or infinite?

By that I mean is there at least one longest line, of length N metres (however large) that can be traversed in a constant direction without passing the same point more than once.

Strange, once again I thank you for putting all that typing effort in replying to my questions.

Unfortunately you replied to neither, since you answered the the second as I don't know

I don't know (it doesn't make any difference to the "expanding space" model).

However I am quite certain that it makes a very big difference to the mathematics whether the universe is finite or infinite.

Further since you do not know the answer to this you cannot state that the mathematics is A or B since finite and transfinite mathematics is different at the most fundamental level.

In particular you cannot answer my first question until the second one is answered.

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Unfortunately you replied to neither, since you answered the the second as I don't know

I don't know because no one knows. There is no strong evidence either way.

However I am quite certain that it makes a very big difference to the mathematics whether the universe is finite or infinite.

Not as far as I know. But that isn't definitive. Everything I have read suggests that GR, and specifically the FLRW metric, works equally well in either case.

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Of course it makes a big difference.

Consider my first question, which might be used to stand in for 'the diameter of the universe'.

Case1) The universe is infinite.

Then there is no such N as I described. The number of metres is unbounded above.

And what of the mathematics of such an 'N' ?

It means that whatever value of N is proposed I can find a larger N+1 within the existing system.

What then is the meaning of expansion or making N larger? ie what is the 'value' of infinity plus one? Mathematically it is still infinity.

Infinity is capable of sustaining an infinite level of expansion, without appearing any different.

Case2) The universe is finite, is more interesting.

It automatically brings in the unavoidable issue of the boundary, since then N is bounded above and the question of what about an M > N arises.

Mathematically this question can be avoided by considering an open set, ie excluding the boundary, as in the frst example in my post#5.

Alternatively you can address this issue by fiddling the metric as in the Poincare disk model (which also excludes the boundary).

However if you exclude something, how can you describe the universe as including everything?

Edited by studiot
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By that I mean is there at least one longest line, of length N metres (however large) that can be traversed in a constant direction without passing the same point more than once.

You are assuming that the line itself can't expand.

Imagine walking down a 5 kilometer road and someone drops in an extra kilometer on you. At your end point you've gone 6 km without ever traversing the same point twice.

In the hypothetical external view, there is probably nothing to see. The Universe either has no external surface area or no mass-energy has reached you.

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You are assuming that the line itself can't expand.

Not at all.

Read again my last post and post #5.

For expansion to occur

Either there are more metres or each metre is 'longer' or there is some other agent also at work as with the temperature scale.

Edited by studiot
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Einstein says space time is warped. Consider 2 dimensions of the universe like the surface of a balloon and the balloon is inflating. As the balloon gets bigger and bigger it is harder and harder for a bug to walk all the way around it.

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Of course it makes a big difference.

Consider my first question, which might be used to stand in for 'the diameter of the universe'.

Case1) The universe is infinite.

Then there is no such N as I described. The number of metres is unbounded above.

And what of the mathematics of such an 'N' ?

It means that whatever value of N is proposed I can find a larger N+1 within the existing system.

What then is the meaning of expansion or making N larger? ie what is the 'value' of infinity plus one? Mathematically it is still infinity.

Infinity is capable of sustaining an infinite level of expansion, without appearing any different.

The fact that a line can be infinitely long makes no difference to some aspects of the description of the geometry of space-time. For example, imagine you have a surface and you are doing some good old fashioned (Euclidean) geometry on that plane. It makes no difference to the way you would describe or construct a triangle, whether the plane is infinite or not.

Obviously, it does make a difference that you can extend a line forever. But that doesn't change the fundamental characteristics of the geometry. In this case, the fact that the spatial metric will evolve over time - either contracting or expanding (I can't say a huge amount more because I don't fully understand the mathematics of pseduo-Riemannian manifolds. I am just parroting my limited understanding.)

Case2) The universe is finite, is more interesting.

It automatically brings in the unavoidable issue of the boundary, since then N is bounded above and the question of what about an M > N arises.

So, the answer to this in GR is that the manifold is finite but unbounded. A 2D analog is the surface of a sphere, say. This has finite area, "straight" lines (called geodesics) have a maximum length, etc. But there is not edge or boundary on the surface.

3D space can be described in the same way.

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Strange.

So, the answer to this in GR is that the manifold is finite but unbounded. A 2D analog is the surface of a sphere, say. This has finite area, "straight" lines (called geodesics) have a maximum length, etc. But there is not edge or boundary on the surface.

3D space can be described in the same way.

Einstein says space time is warped. Consider 2 dimensions of the universe like the surface of a balloon and the balloon is inflating. As the balloon gets bigger and bigger it is harder and harder for a bug to walk all the way around it.

Thank you both I am aware of this and pointed it out in post#5 and repeated it several times since in this thread.

However you miss my point.

'Finite but unbounded' is a glib phrase that applies to some aspects of the ball but not all of them.

In particular the longest line you can draw on the manifold is finite and bounded (by its length).

If the manifold expands then one of my conditions for expansion must be met. ie can the line get longer?

You need to consider carefully the implications of a boundary.It is not a question of Ricci tensors or high falutin maths, it is a question of some basic set theory.

The temperature scale is comfortably within the mathematical framework because with the temperature scale we are not claiming 'universal properties'.

We can establish a temperature scale that is bounded below by absolute zero, but choose an open set of temperature values that do not include the boundary, without penalty.

However.

Set theory, for continuous sets such as we are discussing here, identifies two types of members, interior members (or points) and boundary members (or points)

Interior points

All interior members only possess other set members as neighbours or all the neighbours of interior points are in the set.

Boundary points

Boundary points possess some neighbours that are memebrs of the set and some that are not members. That is not all the neighbouring points of a boundary point are in the set.

Thus it is no problem that -273.15 is not in the set but -273.12 is in the set of all possible temperatures, where absolute zero at -273.14 is a boundary point, -273.12 is an interior point and -273.15 is not in the set.

However it is a real problem that if N metres is the longest line that can be drawn that a line of length (N+1) metres is not in the set of everything.

Edited by studiot
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In particular the longest line you can draw on the manifold is finite and bounded (by its length).

If the manifold expands then one of my conditions for expansion must be met. ie can the line get longer?

Yes. In the 2D "surface of a sphere" example, consider that the sphere gets bigger; then the surface gets bigger and lines can be longer.

Set theory, for continuous sets such as we are discussing here, identifies two types of members, interior members (or points) and boundary members (or points)

I don't really follow this argument, I'm afraid. In the set of points making up the surface of a sphere, which are the boundary members?

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I don't really follow this argument, I'm afraid. In the set of points making up the surface of a sphere, which are the boundary members?

You need to complete your 2D analogy, not pick part of it.

You can only have the surface of a sphere as a 2D surface if you embed it in 3D space.

The boundary is then between the inside and outside of the sphere.

So all points are boundary points in 3D space, and of course you must have 3D space.

Again this is not a problem unless you are claiming 'everything'. This must then include the rest of 3D space.

In our 3D world manifold, we need to introduce 4D space to accomodate this.

And the same boundary issue immediately arises.

A further complication with this view is that it is possible for a 2D 'being' to detect the effect (existence) of 3D space by such physical phenomena as shadows.

But we have never observed these in our universe.

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You can only have the surface of a sphere as a 2D surface if you embed it in 3D space.

Ah, I see. No, mathematically we can treat the surface of the sphere as a 2D object without embedding it in a higher dimensional space. (There is a reason that mathematicans call a three-dimensional sphere a 2-sphere. )

Similarly, the mathematics of 4D space-time is not embedded in a five dimensional space.

http://mathworld.wolfram.com/IntrinsicCurvature.html

http://en.wikipedia.org/wiki/Curvature#Higher_dimensions:_Curvature_of_space

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Ah, I see. No, mathematically we can treat the surface of the sphere as a 2D object without embedding it in a higher dimensional space.

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Go on do tell.

Look up "intrinsic curvature".

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I was just starting to add to my post when the phone rang.

Yes you can restrict the properties to create an 'intrinsic geometry' see for instance O'Neill @Elementary Differential Geometry' p271, Intrinsic Geometry of Surfaces in E3.

However we have that word 'restrict' and we are back to the difficulty of 'everything'.

Take the surface of a sphere.

It is (in some ways) isomorphic to the plane.

But

It was also known to surveyors by the mid 18 century that a line of constant bearing or heading was not the shortest distance between two points on the Earth, as it would be with the plane. In fact it is an S shaped curve on the surface of the Earth called a geodesic. In this way some properties of the embedding higher dimension are required.

Edited by studiot
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Consider the set of real numbers and place them on an infinitely long yardstick studiot. The set is infinite and is spaced at 1mm intervals between numbers. Notice that this infinite set comfortably fits on the infinite yardstick.

We now take each of those numbers and double it such that 1 becomes 2, 2 becomes 4,etc. by removing all odd numbers. Note that we have a one to one correspondence between the original set and the new set ( one might think that the new set is smaller since it only includes even numbers ). Both are infinite and both comfortably fit on the infinite yardstick, however the separation between adjacent numbers has now doubled or 'expanded'. Maybe 'expand' isn't the proper term to use, but your argument doesn't negate the option of an infinite universe.

Since you are familiar with Poincare and Reimann, I would assume that you realise a geometric space does not need to be embedded in higher dimension. It may, but it doesn't have to be. So, yes, a finite universe is also an option.

In bot of these cases, however, I prefer the unbounded option. Only because I wouldn't want to explain what's on the other side of the boundary.

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Since you are familiar with Poincare and Reimann, I would assume that you realise a geometric space does not need to be embedded in higher dimension. It may, but it doesn't have to be. So, yes, a finite universe is also an option.

Taking the geometric equivalent of saying we can have a full (possibly infinite) set of numbers without using negative numbers.

This is true, but we cannot claim we include all the numbers or all their properties.

Using intrinsic geometry is the geometric equivalent.

But you cannot then declare your subset to be the universal set.

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i get that you can have a infinite universe in a bounded sphere, sort of like the pi ratio of a circle. that makes a little more sense but still doesn't answer my question on if space itself is timeless, this big static sphere must sit somewhere

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i get that you can have a infinite universe in a bounded sphere, sort of like the pi ratio of a circle. that makes a little more sense but still doesn't answer my question on if space itself is timeless, this big static sphere must sit somewhere

Okay, so you understand the sphere represents 2 dimensions of our universe. We are talking about the universe. The universe is the space-time continuum. You are asking where is this universe? Where in space is space? You must realize these questions are meaningless.

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I also feel I need to make one more clarification. In order to help you understand the dimensions of the universe we have stated that the universe may be curved like the surface of a sphere. If this is the case then cosmologist would say the universe is finite and unbounded.

The universe is finite because a line can only be so long (before it retraces itself). A plane (actually a sphere because of the curvature of space-time) can only have so much area.

The universe is unbounded because there is no boundaries. You will never get to the end of the universe. If you keep traveling in a strait line you would end up where you started. For this reason some have erroneously said this type of universe would be bounded. That's not true. An unbounded universe is one that has no boundaries. If a universe was bounded then you could reach the end of the universe and if you were to go farther then you would leave the universe. No cosmologist believes the universe is bounded. Everyone is sure there is no end to the universe.

Einstein showed, in his General Relativity that the spacetime continuum can be curved because of gravity. It is extremely unlikely that the curvature of the spacetime continuum is perfectly flat. It could have a positive curvature, which would be like the sphere we have been talking about. I hate to bring it up (at the risk of confusing your understanding) but the other possibility is that the curvature of the space-time continuum would have a negative curvature. This would be more like the shape of a Pringle's Potato Chip. I tried to find a better image of this spacetime continuum but a Pringles chip was the best I could come up with. If the dimensions of the universe were shaped like this then the universe would have negative curvature. The area would increase as you moved farther from the starting point.

If you continued the Pringle's chip forever then the area would be infinite. Hence Infinite but Unbounded.

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No cosmologist believes the universe is bounded.

It's not a question of belief, that is for religion.

it's a question of observation, followed by mathematical deduction. (not the other way round as that can lead to error).

Of course the surface of a finite sphere is finite and there are no bounds in the surface.

So what? Further this is not true of any folded plate structure like your pringle.

A being on that surface could very easily take measurements on sets of three points and deduce that the distances between them do not match a planar triangle and that the area of the figure they enclose is subject to spherical excess.

This would lead to the deduction that there must be a third dimension allowing curvature for this to happen.

Yet that same being could develop a consistent system of mathematics on his surface.

But he could not claim it to be a complete system of mathematics.

Consider the following.

The rotations of a plane triangle form a group and allow a consistent algebra.

However for the triangle to rotate, there must be at least a disk of diameter equal to length of the side of one triangle available.

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