Toy model space that is not expanding into

[Martin]

It isn't obvious to me that we need this example. And it's somewhat mathematical so may actually cause confusion. But just in case it might help, here goes.

 

We'll take an extremely simplified set S to represent space, say a unique copy of the numbers between 0 and 360, including zero but not including 360.

 

[math]0 \leq x < 360[/math]

 

If you want, to convince yourself that this interval is distinct from all other intervals of numbers, pour whisky on it or paint the numbers some color that is not like any other color in the universe. However you do it, think of S as different and not part of anything else. Imagine all existence concentrated in S, no outside.

 

For any two members of S, define a function f(x,y) = min {|x-y|, 360 - |x-y|}

 

Notice we still have no distance entering in here. I haven't mentioned inches or meters or miles or lightyears. But we do have this numerical-valued function f(x,y) defined for any two elements x and y that you might pick.

 

Now we will define a distance function on the set S. For any two members x and y, the distance between them is going to be dist(x, y) defined this way:

 

Today, namely Sunday,

dist(x,y) = f(x,y) times 6 inches

 

Tomorrow

dist(x,y) = f(x,y) times 7 inches

 

Tuesday

dist(x,y) = f(x,y) times 8 inches.

 

And so on.

 

Notice that the set S remains the same and it has no surroundings. But distances within S are changing (General Relativity, our theory of how gravity works, tells us to expect distances between points to change, that's part of the idea of curvature.) The toy model is not working according to General Relativity, but we nevertheless we have distances change to make it at little more realistic. The set S is not expanding, at least not in a simpleminded sense of expanding into somewhere else:D. Instead, all that's happening is internal distances are changing (and the word internal is redundant here since all the distances in this example are internal to S.) So let's get rid of the redundant word and say it more clearly---there is no expansion into, what's happening is that distances are changing.

 

Just to take a concrete example, let x = 3.51 and let y = 359.51.

So f(x,y) = min{356, 360-356} = min{356, 4} = 4

 

So today, Sunday, the distance between x and y is 24 inches.

Tomorrow the distance will be 28 inches.

And on Tuesday the distance will be 32 inches.

 

You can think the toy model equally well in lightyears as inches, and you can pick any other two members x and y of the set. The main thing is that the set is not in any surroundings and it doesn't expand like an object might expand.

It is even misleading to say "space expands" because then some people can't help thinking that this means space is an object:D. Maybe the answer is to erase the words "space expands" from your brain and concentrate on the regular pattern of increasing distances.

 

I don't have to mention, but will anyway, the number 360 doesn't matter, we could have used 11 or 17 just as well. And S did not have to be a set of numbers, an interval of numbers is just convenient to work with. It cut out a step or two and saved us bother.

 

Hubble law is a pattern of increasing distances. It may have been unfortunate that astronomers happened to translate Hubble law for the general public by saying "space expands" because when some people hear those words they apparently get hung up on the image of an expanding object and never get past the words to the pattern.

Edited January 25, 2009 by Martin

[ajb]

Geometrically, I liken the expansion of the universe to a cone. The cone is topologically (lets cut out the apex) [math]R^{1}\times S^{1}[/math]. Now if we take time to be [math]R^{1}[/math] and space to be [math]S^{1}[/math]. Then we see that space gets larger as time does, yet it is expanding into nothing.

 

I would not take my illustration too seriously, but it does point to the common misconception.

[NowThatWeKnow]

Geometrically, I liken the expansion of the universe to a cone. The cone is topologically (lets cut out the apex) [math]R^{1}\times S^{1}[/math]. Now if we take time to be [math]R^{1}[/math] and space to be [math]S^{1}[/math]. Then we see that space gets larger as time does, yet it is expanding into nothing.

 

Do you think it is possible that time is responsible for the expansion of space? Maybe time could be a type of ether. It seems the universe is a vacuum full of matter and electromagnetic radiation. Is it expanding into a vacuum that is void?

[ajb]

Do you think it is possible that time is responsible for the expansion of space? Maybe time could be a type of ether. It seems the universe is a vacuum full of matter and electromagnetic radiation. Is it expanding into a vacuum that is void?

 

Time is responsible in the sense that the metric depends explicitly on time.

[Martin]

Is it expanding into a vacuum that is void?

 

NowThat,

why should it be expanding into anything?

 

A thing does not need any surroundings in order to exist. Or?

Do you think that because the universe exists it must have some surroundings?

 

Indeed there are hierarchical multi-universe concepts that one can read about but so far they are mainly ornamental.

 

To discuss ordinary observation-based cosmology all one needs is the one boundaryless universe concept.

Space which has no boundary.

 

If we take that as cosmology concept A, then I guess concept B would be absolute stillness. Practical cosmology has a basic concept of absolute rest. Rest means no doppler hotspot in your microwave background sky. Hubble already got the idea before the microwave background had been detected or even thought of. He realized you are at rest (universally speaking) if the increases in distance around you are symmetric. He may not have known the exact coordinates but he knew that the solar system has an absolute motion which is roughly in the direction of constellation Leo. We now know it is about 380 km/second and the coordinates are known precisely. That individual motion is what is subtracted out to make microwave temperature maps---subtracting out the CMB dipole---the Leo hotspot. The maps you see published, those red/blue mottled ovals, are made from a viewpoint of absolute rest---with the solarsystem's (and etc.) motion subtracted out.

 

Concept C is that if two widely separated points are not moving then the distance between them is increasing by approximately 1/140 percent every million years. This is Hubble's Law. The percentage has been larger in the past and it's history is known with some degree of confidence. It agrees with our theory of dynamic geometry, i.e. gravity, namely General Relativity. Gen Rel teaches us not to expect distances between stationary objects to remain constant, and in fact they do not remain constant.

 

Dynamic geometry is what GR is about. Geometry is affected by shifting concentrations of matter. Geometry undulates. Distances between stationary objects change. The observed Hubble pattern of increasing distances is one possible solution resulting from GR.

 

The conventional solution depends on no boundary, no outside empty surround. The GR equation works, and predicts what we see, when space has no boundary and is more or less uniformly filled with matter, as we see it, with the average density that we see.

 

GR is the mother of geometry. It explains why the geometry we experience is approximately Euclidean because that is what the solution to the GR equation is at low density. Euclid geometry is not basic. GR is basic and explains why Euclid works at low density. GR teaches us not to expect geometry to remain static, not to expect distances between stationary objects to remain constant, and they don't. It teaches us to expect things like Hubble law increase in distances, and black hole collapse, and these things do seem to be observed.

 

GR fits the data most conveniently if no outside is assumed. Whether the universe is finite volume or infinite volume is unimportant as long as it is approximately uniformly filled. In other words GR works if it is uniformly filled with the observed average density and boundaryless.

 

So from a working cosmology point of view it is impractical to imagine a boundary and some outer place. Because then you have shot yourself in the foot. The dynamics become elaborate and difficult to make realistic. People have attempted things along those lines such as braneworld higher dimension scenarios, but they appear to be going out of fashion.

Edited January 26, 2009 by Martin

[NowThatWeKnow]

NowThat,

why should it be expanding into anything?

I was hoping to get ajb to elaborate on "Then we see that space gets larger as time does, yet it is expanding into nothing."

What could be closer to nothing then a vacuum that is void?

 

 

A thing does not need any surroundings in order to exist. Or?

Do you think that because the universe exists it must have some surroundings?

 

Not having a surrounding woud be easier to visualize if we observed a static universe but that is not the case. Our observations show that the distance between matter is expanding. Maybe it is not expanding into anything but is expanding through nothing.

 

None of us know for sure what is happening and it would be foolish to rule out any of the many possibilities. We all just put a different likelihood on each scenario. your knowledge allows you to grasp what you say much better then my knowledge allows me to grasp what you say. However, my respect for your knowledge tells me to give your views much consideration and I listen when you speak.

 

Now, if someone would get off their double sided end and figure out the real answer.

[ajb]

NowThatWeKnow , start thinking 4-d and not 3-d with an "external" parameter = time.

 

In doing so you see that the expansion is really an "illusion" of the 4-d nature of space-time. Thus, no need to expand into anything.

 

By geometry being dynamical one simply means the metric depends on both where and when you are. However, one can also mean by dynamical that a metric is a solution to an action principle, the Einstein-Hilbert action. The second notion is probably more important in quantum gravity than classical.

[NowThatWeKnow]

NowThatWeKnow , start thinking 4-d and not 3-d with an "external" parameter = time.

 

In doing so you see that the expansion is really an "illusion" of the 4-d nature of space-time. Thus, no need to expand into anything.

 

By geometry being dynamical one simply means the metric depends on both where and when you are...

 

I try to think of space/time as a single construct. As time passes it stretches the metric of space with it. It sounds like you are implying that maybe space is not expanding. Would that require time to continually speed up for us to observe the expansion we see? It is hard enough to understand with out illusions making it even harder.

 

It is probably just as hard to imagine a finite universe with a boundary we can not cross as a infinite universe without a boundary to cross.