Surface waves on a balloon and our possible extra dimensions.

[Spinnor]

Think about the dimensionality of a balloon with surface waves. Say these waves are of small amplitude compared with the radius of the balloon. Two coordinates label points on the balloon and a third labels radial position. Creatures who lived on the surface could make measurements that determined the curvature of their space. Let us say they would also be able to detect the waves that went about their space. They would say their space is two dimensional but they needed a real field to describe the waves on their surface. To we who live in 3 space dimensions we would say, what field, they live on a two dimensional surface that can vibrate. They need three coordinates to describe their world.

Giving our spacetime extra dimensions seems natural even if wrong.

Thanks for any thoughts, and apologies if this post is in the wrong place.

 

[Andy Resnick]

While what you say is a common conceptual device, it's important to remember that (in GR anyway), the 4 spacetime dimensions are not embedded in a Euclidean 5-dimensional manifold.

 

[Spinnor]

While what you say is a common conceptual device, it's important to remember that (in GR anyway), the 4 spacetime dimensions are not embedded in a Euclidean 5-dimensional manifold.

I'm not sure that changes anything I said.


Thanks.

 

[Spinnor]

While what you say is a common conceptual device, it's important to remember that (in GR anyway), the 4 spacetime dimensions are not embedded in a Euclidean 5-dimensional manifold.

In my simple example it seems there are two ways map this surface,

Radial distance as a function of theta and phi, or

A metric that describes curvature of the surface as a function of theta and phi.

Maybe 4D Theorists think in terms of intrinsic curvature and > than 4D Theorists think in terms of extrinsic curvature, two ways of thinking of different aspects of the same system?

Thanks for your help!

 

[Spinnor]

In my simple example it seems there are two ways map this surface,

Radial distance as a function of theta and phi, or

A metric that describes curvature of the surface as a function of theta and phi.

Maybe 4D Theorists think in terms of intrinsic curvature and > than 4D Theorists think in terms of extrinsic curvature, two ways of thinking of different aspects of the same system?

Thanks for your help!

In this simple example if it makes sense for a two dimensional creature to describe waves on the surface of his two dimensional world as a field (when in fact his two dimensional manifold is curved on both large and small scales) can the fields of the Standard Model correspond to various curvatures of our 4 dimensional spacetime manifold, thus eliminating the need for extra dimensions? This is what Einstein was after, all of physics in terms of geometry? Why does it not work if it doesn't work or is it more simple to think in terms of extra dimensions? Leave out quantum physics for the time being, that makes things messy and real, real messy?

Thanks for your help.

 

[Andy Resnick]

If I am understanding you, then yes- IIRC, Yang-Mills theory was an attempt to 'geometrize' electromagnetism by postulating a 5th dimension corresponding to charge. This led to the idea of 'wormholes' connecting positive and negative charges and a whole lot of extra complexity. Because of the huge increase in complexity and not much increase in understanding, that program was dropped.

That idea (geometrodynamics) is still being pursued under different guises: spin foams, 'pregeometry', loop quantum gravity.

 

[Galap]

As I understand it, our universe is thought to be finite in size but unbounded. Wouldn't this imply that it is folded over a 4th spatial dimension to allow it to 'wrap around'? Aren't there many different shapes we have proposed (i.e. saddle, horn, sphere, etc.)?