Small and large extra dimension(s) of the physical space

[victorvmotti]

Trying to make sense of small and large extra dimension(s) of phyiscal space in a simple intuitive example.

Consider a two dimensional manifold like R2 and we are trying to add a small and a large extra dimension.

Do we mean by smale extra dimension in this case something like (0,1)×R (the flat case) or S1×R (the curved case)?

Do we mean by large extra dimension something like R2×R=R3?

Do we mean in the case of our three dimensional space that basically we have a base space of our phyiscal three dimensional space with a total space built by adding a fiber and thus creating a fiber bundle or a even more general an arbitrary total space?

Does the extra dimension need to be real or can we even consider the complex manifolds, in the case of adding extra dimension to the phyiscal space, for example C×R3 or (Riemann surface) ×R3

 

[robphy]

Is this a relativity question?

 

[victorvmotti]

I was not sure where exactly to put it. It is mostly a math question perhaps. But you can answer it as it relates to general relativity and cosmology. Like the shape of the cosmos if we need to think about the extra small and large dimensions. The focus here could be on how different small and large extra dimensions are related to the universe.

 

[PeterDonis]

Trying to make sense of small and large extra dimension(s) of phyiscal space in a simple intuitive example.

What you're giving isn't a simple intuitive example; you haven't constrained anything. If you don't constrain anything, your question is unanswerable since there are an infinite number of possibilities.

You would be better served by looking for an actual example in the literature. You might try looking up references for Kaluza Klein theory; I believe that is the simplest "extra dimensions" type of model that has been looked at historically.

 

[victorvmotti]

Very good suggestion, thank you and will follow up.

My key question is perhaps related to math.

Am I giving correct examples of small and large extra dimension in the lower dimensional cases?

Are they among those infinite number of possibilities?

 

[PeterDonis]

Are they among those infinite number of possibilities?

S1 x R is (for either "small" or "large", since the S1 could be large), as is R x R2 = R3 (for "large"). (S1 x R4 with a "small" S1 was basically the idea behind Kaluza-Klein theory, with the S1 representing the U(1) symmetry of electromagnetism. Substitute a 6-dimensional Calabi-Yau space for S1 and you have the basic idea behind at least one version of how some string theorists think the standard model might emerge.)

I'm not sure about (0, 1) x R because topologically (0, 1) is R, so I'm not sure it's any different than R x R = R2.

 

[victorvmotti]

So, small S1 has nothing to with being compact?

Can you further explain how S1 can be either large or small?

Is it related to the metric?

How we define small or large here?

 

[PeterDonis]

So, small S1 has nothing to with being compact?

Can you further explain how S1 can be either large or small?

You're looking at it backwards. Why would it not be able to be any size? After all, we already know in GR that we can have solutions describing the whole universe that are spatially compact (a closed universe has topology S3 x R, with S3 being the spatial part).

The fact that S1 is compact actually allows it to be any size; the constraints come with non-compact manifolds like R, for which it doesn't really make sense to say that it is "small"--the topology R means, heuristically, that the manifold can extend in that "direction" indefinitely (or more precisely that if there is a constraint on extent in that direction, it will be dynamic, determined by the specific properties of a particular solution--such as the time extent of a closed universe that recollapses to a Big Crunch, which varies depending on the particular parameters of the solution--rather than a property that you can infer just from the topology).

 

[victorvmotti]

So are you saying that the S1 being small or large depends on how large it is measured relative to the other component in the product topology?

For example, we could have or imagine a manifold S3=S1 (large) X S1 (large) X S1 (small)?

 

[PeterDonis]

are you saying that the S1 being small or large depends on how large it is measured relative to the other component in the product topology?

Topology doesn't even have any notion of "size". You have to construct a more complete model. Again, there are an infinite number of possibilities.

 

[victorvmotti]

Is a manifold S3=S1 (large) X S1 (large) X S1 (small) possible?

 

[PeterDonis]

Is a manifold S3=S1 (large) X S1 (large) X S1 (small) possible?

There are an infinite number of possibilities. We could spend an infinite amount of time having you ask about them one by one. That's not a good use of this forum.

 

[PeterDonis]

The OP question has been answered as well as it can be given its open-ended nature.

Thread closed.