What does the metric of a 6D space with 3 compactified dimensions look like?

[JandeWandelaar]

TL;DR Summary

Imagine a 6D flat space. We compactify 3 of them to circles. How does the associated metric tensor look like?

I'm interested in describing a 6-dimensional space of which three are compactified to small circles. Globally this space looks 3-dimensional, like a 2-dimensional cylinder looks 1-dimensional globally.

Kaluza and Klein did a similar thing in the context of 4-dimensional spacetime. They extended the spacetime by attaching a tiny circle to every point. This was expressed in a 5-dimensional metric, containing eight extra off-diagonal metric components. They are the four components of the A four-vector for the electromagnetic field. An extra diagonal component is introduced but that appeared to be unphysical. The extra components imposed a vector bundle on the small circle.

How do we describe a 6D flat space of which we compactify three into small circles? It's easy to describe the metric of a flat 6D space, but how does the metric look like if three of them have been turned to circles? The 2D case would be a 2D flat space of which one dimension is compactified to a circle (a cylinder).

 

[martinbn]

It looks the same.

 

[JandeWandelaar]

It looks the same.

So only off-diagonal elements in the 6x6 metric tensor?

 

[martinbn]

So only off-diagonal elements in the 6x6 metric tensor?

What do you mean?

 

[JandeWandelaar]

What do you mean?

Kaluza and Klein introduced 8 off diagonal components in a 5D spacetime metric to describe the small extra circle. These were the four components of the A 4-vector. So the metric of the small circle was described by the four components. They formed a vector bundle over the circle.

How do 3 compactified dimensions (compactified to circles) show up in the 6x6 metric, which reduces to the 3x3 metric on large scales?

 

[martinbn]

Kaluza and Klein introduced 8 off diagonal components in a 5D spacetime metric to describe the small extra circle. These were the four components of the A 4-vector. So the metric of the small circle was described by the four components. They formed a vector bundle over the circle.

How do 3 compactified dimensions (compactified to circles) show up in the 6x6 metric, which reduces to the 3x3 metric on large scales?

The metric is something additional. It doesn't come from the manifold. I thought you were talking about the flat metric. If you mean another metric you need to say which. Just compactifying some dimensions does not determine a metric.

 

[JandeWandelaar]

Just compactifying some dimensions does not determine a metric.

But didn't Kaluza and Klein wrote a 5x5 spacetime metric for 5D spacetime for a 4D extended spacetime with one compactified to a circle?

 

[martinbn]

But didn't Kaluza and Klein wrote a 5x5 spacetime metric for 5D spacetime for a 4D extended spacetime with one compactified to a circle?

Yes.

 

[JandeWandelaar]

Yes.

So if we extend this to a 6x6 metric, for space only, we should add non-diagonal elements? So the 3x3 upper left matrix is the identity. But how would the rest of the matrix look? Say we start from 4D space and compactify one dimension to a circle. We then have the 3x3 identity matrix plus extra components "around" it. But how would these look? IF it can be done?

Or even more simple, a 2D flat space, and then we compactify one to a circle. (Excuse my English).

 

[martinbn]

So if we extend this to a 6x6 metric, for space only, we should add non-diagonal elements? So the 3x3 upper left matrix is the identity. But how would the rest of the matrix look? Say we start from 4D space and compactify one dimension to a circle. We then have the 3x3 identity matrix plus extra components "around" it. But how would these look? IF it can be done?

Or even more simple, a 2D flat space, and then we compactify one to a circle. (Excuse my English).

It depends on what you want to do. The point, which you are still missing, is that there isn't a unique way to define a metric.

 

[JandeWandelaar]

Well, I want to describe the motion of 3D "hyperspheres" in a 6D space (or 7D spacetime, but I first want to look at the space structure only, if possible). To visualize this it's easiest to look at 2D flat space, compactify one dimension to a circle, which obviously gives a cylinder. If we envision a particle as a circle on the cylinder, it looks like a point-like particle moving in 1D, if the radius of the circle is small (though this is relative). If we extend this to 6D flat space, compactify 3 of them to tiny circles, and imagine a particle to be a 3D "hypersphere" (to be compared with the circle on the cylinder), it looks globally as point-like particles traveling in 3D. But how to describe this mathematically? So at small scales the particle ain't point-like anymore?

 

[malawi_glenn]

it looks globally as point-like particles

What does "looks like" mean here?

 

[JandeWandelaar]

What does "looks like" mean here?

Like a string from afar can be seen as a point structure, won't a circle on a thin tube look like a point on a line? Or, if you have compactified two dimensions of a 4D space, a point in 2D? And then, if you compactify 3 dimensions of a 6D space to small circles, won't they look like points, if you place a 3D hyperstructure in it, like a circle on a tube. If you are part of the tube won't a particle look pointy in the sense it can move in one direction only (the length of the cylinder)? (No weed involved...)

 

[malawi_glenn]

Like a string from afar can be seen as a point structure, won't a circle on a thin tube look like a point on a line? Or, if you have compactified two dimensions of a 4D space, a point in 2D? And then, if you compactify 3 dimensions of a 6D space to small circles, won't they look like points, if you place a 3D hyperstructure in it, like a circle on a tube. If you are part of the tube won't a particle look pointy in the sense it can move in one direction only (the length of the cylinder)? (No weed involved...)

Look in terms of what? When we look at them with our eyes?
What will the classical equations of motion be, and how will you quantize?

 

[JandeWandelaar]

Look in terms of what? When we look at them with our eyes?

Look in terms of size. If the circles have a Planck diameter, the Planck hypervolumes will be so small that from afar, in the 3D large space, they look like points. But they can't form a truly point-like structure in a black hole. So a singularity won't be possible.

 

[malawi_glenn]

Look in terms of size. If the circles have a Planck diameter, the Planck hypervolumes will be so small that from afar, in the 3D large space, they look like points. But they can't form a truly point-like structure in a black hole. So a singularity won't be possible.

How do you measure size of something that is quantized?

 

[JandeWandelaar]

The Plancklength is Lorenz-invariant in this scenario. Space is non-discrete though. Distances closer than PL can't be measured, if the diameter of the circles is 2PL.

The question though was how to describe this 6D space, in which singularities can't form.

 

[malawi_glenn]

how to describe this 6D space, in which singularities can't form.

I would be very happy to see a source of this, with calculations.

 

[JandeWandelaar]

I would be very happy to see a source of this, with calculations.

So would I!

 

[malawi_glenn]

So would I!

How do you know singularities can't form unless it has been calculated?

 

[berkeman]

OP is on a 10-day holiday from PF. so this thread is closed.