Confused about fourth spatial dimension

[sandesh trivedi]

thanks pervect

You might try the math forums, because your question apparently doesn't have anything to do with relativity at all (being instead about a fourth spatial dimension). Relativity only has three spatial dimensions, and a curved 4-dimensional space-time.

You seem to get grumpy when we keep introducing relativity into your non-relativistic question.

i was also thinking about posting this thread in maths section but i was confused where should i post it.can u please tell me where should i post it.

 

[yukcream]

Let me clarify something. There's nothing technically wrong with the problem solution you posted, it's just that the problem and its solution doesn't have much relevance to General Relativity and how it deals with curvature.

GR deals with curvature not from the viewpoint of someone looking at our universe from outside, but from the viewpoint of someone looking at it from inside. This is known as intrinsic curvature. That's why I stressed that viewpoint, because it is the one that GR uses.

Imagine a 2d surface of constant curvature - you probably imagine the surface of a sphere.

But a sphere requires periodic space and time coordiantes. To use a technical term, it's a compact manifold.

Try to imagine wrapping a piece of paper around a sphere, a piece of paper that's infinite in both directions (a plane). You can't do it in three dimensions.

You can wrap a narrow strip of paper (elastic paper) that's infinitely long but finitely wide around the sphere with no problem. But as you try to make the paper wider and wider, eventually you find that the paper has to pass through itself, something that you can't do. (Unless you add extra dimensions).

GR has to deal with non-compact manifolds all the time. You can't simply represent the curved 4-d space of GR as the surface of a 5 dimensional manifold. You need more than 5 dimensions to get the right sort of curvature, just as you need more than 3 dimensions to construct a plane that has a constant curvature everywhere.

This also makes the embedding non-unique. When you embed a n-dimensional maniofld in n+1 dimensions, you can solve the equaitons and find a unique solution, but when you embed a n-dimensional mainfold in n+2 or even higher dimensional space, you find the embedding is not unique.

I have few more question hope you ca answer me!
What is "manifold', what is the difference between manifold and dimensions?
What is non-compact manifolds?
Finally, be honest I am not quite understand the original question ~ once we use a paper to form a sphere, the paper is in 2-D while the shpere is 3-D~ is it means when "curvature" present the case no more within 2-D? If yes, how about the circle in on a plane (2-D) ?

yukyuk

 

[yukcream]

Our 3 spatial dimensions are Euclidean (not curved), except where there are strong gravitational fields.

No.

the second queston, why no?

 

[pervect]

I have few more question hope you ca answer me!
What is "manifold', what is the difference between manifold and dimensions?
What is non-compact manifolds?
Finally, be honest I am not quite understand the original question ~ once we use a paper to form a sphere, the paper is in 2-D while the shpere is 3-D~ is it means when "curvature" present the case no more within 2-D? If yes, how about the circle in on a plane (2-D) ?

yukyuk

You can think of a two dimensional manifold as composed of pieces of cloth (really finite two dimensional surfaces) that are "glued" or "sewn" together. The "sewing" process has some restrictions, the seams must be smooth.

This generalizes to three or more dimensions - an n dimensional manifold looks like small pieces of R^n "glued" together in a continuous manner.

http://mathworld.wolfram.com/Manifold.html

has a more formal definition of Manifolds.

"compact" is a bit harder to describe informally, but the surface of a sphere is compact (it has a finite area), while the surface of a plane is not compact (it has an infinite area). For a formal definition see

http://mathworld.wolfram.com/CompactManifold.html
http://mathworld.wolfram.com/CompactSpace.html

which includes some examples of 2-d compact manifolds (spheres, torii, klein bottles, etc.)

Any manifold has a dimension, which is the dimensionality of some small piece of the manifold - this is the same no matter which piece of the manifold one considers (this can be proven from the formal definition).

A circle on a plane would be a 1 dimensional manifold. It would not have a curvature, because you need to have at least a 2-dimensional manifold to define curvature.

 

[yukcream]

We get it. At least the part about the curvature, we get. it appears you may have some question? I don't understand what question you are asking, so far you have simply stated some facts, which are correct.

Fred has found how to measure "Gaussian" curvature, which is an intrinsic sort of curvature first defined/discoverd by (you guessed it) Carl Gauss.

See for instance the Wikipedia article at

http://en.wikipedia.org/wiki/Curvature

Where dose the equation comes from?
K = \lim_{r \rarrow 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}.

 

[yukcream]

To Pervect

Thanks so much~ your explanation is so clear~ you help me a lot thxs

yukyuk