Metric of Manifold with Curled up Dimensions

In summary, a metric is a locally-defined structure which is reflected by the metric in the form of the "curling up" of dimensions.
  • #1
Markus Hanke
259
45
Would someone here be able to write down for me an example of a metric on a manifold with both macroscopic dimensions, and microscopic "curled up" dimensions with some radius R ? Number of dimensions and coordinates used don't matter.

Not going anywhere with this, I am just curious as to how such a metric could look like, mathematically.

Thanks in advance !
 
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  • #2
A metric is a locally-defined structure--that is, when defining one, we may consider regions as small as we like. So if one dimension is curled up into a circle of radius R, we could define our metric on balls of radius R/100000 for instance. At that scale, we can't even detect the curling.

The "curling-up" of dimensions is a global property of the space, and so (at least in the category of Riemannian manifolds) it is not reflected by the metric, which is local. There are geometric structures you can put on a manifold that, I think, do force a connection between the global and local structures. But I'm only starting to learn about that, so I'll let someone else jump in and say whether that's the right way to think about it.
 
  • #3
Tinyboss said:
The "curling-up" of dimensions is a global property of the space, and so (at least in the category of Riemannian manifolds) it is not reflected by the metric, which is local.

I am not an expert in this ( which is why I asked the question in the first place ), but somehow that statement does not appear to make sense. My thinking is that the extra dimension(s) would be a direct product between some "normal" manifold and the extra one, in which case they must appear in the metric tensor in some form.
 
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  • #4
The simplest example would be an infinitely long cylinder with a very small radius.
Take the axis of the cylinder to be the x-axis, the radius R, and [itex]\theta[/itex] the angle a line, pependicular to the axis, from the axis to a point on the cylinder makes with the z-axis. Then we can write [itex](x, y, z)= (x, R cos(\theta), R sin(\theta))[/itex].

Then [itex]dy= -Rsin(\theta)d\theta[/itex] and [itex]dz= Rcos(\theta)d\theta[/itex] and then [itex]ds^2= dx^2+ dy^2+ dz^2= dx^2+ R^2cos^2(\theta)d\theta+ R^2sin^2(\theta)d\theta= dx^2+ R^2d\theta^2[/itex].
The metric tensor is
[tex]\begin{bmatrix}1 & 0 \\ 0 & R^2\end{bmatrix}[/tex]
and the "curling up" will be for R very small.

More generally, given any metric tensor, you can "add" curled up dimensions by adding rows and columns with all 0s except that the values on the diagonal are very small numbers- almost 0.
 
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  • #5


Sure, I can provide an example of a metric on a manifold with both macroscopic dimensions and microscopic "curled up" dimensions. Let's consider a 4-dimensional spacetime manifold with 3 macroscopic dimensions (x,y,z) and 1 microscopic "curled up" dimension (θ) with radius R.

The metric for this manifold can be written as:

ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 + R^2(dθ^2)

Here, t represents the time coordinate and x, y, z represent the macroscopic spatial coordinates. The term R^2(dθ^2) represents the "curled up" dimension, where dθ is the infinitesimal change in the angle θ.

This metric can also be written in matrix form as:

gμν =
[-1, 0, 0, 0;
0, 1, 0, 0;
0, 0, 1, 0;
0, 0, 0, 1+R^2]

In this metric, the "curled up" dimension is compactified, meaning that it is "curled up" into a small circle with a radius of R. This allows for the existence of microscopic dimensions within the larger macroscopic dimensions.

One example of a physical system that can be described by this metric is a Kaluza-Klein theory, where the extra dimension θ is interpreted as a compactified dimension, and the metric describes the interaction between gravity and electromagnetism. Another example is in string theory, where the extra dimensions are also compactified and play a role in the fundamental interactions of particles.

I hope this helps to illustrate how a metric on a manifold with both macroscopic and microscopic dimensions with a radius R can look like.
 

1. What is a Metric of Manifold with Curled up Dimensions?

A Metric of Manifold with Curled up Dimensions is a mathematical concept used to describe the geometry of a space with multiple dimensions, some of which are "curled up" or compactified. This type of metric is often used in theories such as string theory and extra dimensions.

2. How is a Metric of Manifold with Curled up Dimensions represented mathematically?

A Metric of Manifold with Curled up Dimensions is typically represented using a mathematical object called a tensor. This tensor contains information about the distances and angles between points in the space, taking into account the curled up dimensions.

3. What is the significance of curled up dimensions in a Metric of Manifold with Curled up Dimensions?

The presence of curled up dimensions in a Metric of Manifold with Curled up Dimensions allows for the possibility of extra dimensions beyond the familiar three spatial dimensions and one time dimension. This can help in understanding the fundamental forces of nature and the underlying structure of the universe.

4. How are curled up dimensions different from the familiar three spatial dimensions?

Curled up dimensions are fundamentally different from the familiar three spatial dimensions in that they are compactified and do not have the same scale as our everyday dimensions. They are often described as being "rolled up" or "hidden" from our perception.

5. Can we observe curled up dimensions in our physical world?

Currently, there is no evidence that we can directly observe curled up dimensions in our physical world. However, theories such as string theory make predictions about the properties of these dimensions that may one day be testable through experiments or observations at a very small scale.

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