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  • Setup of idea

I had somewhat of a thought question regarding general relativity. Consider a simple situation of a sphere and arrow. You hold the arrow and walk from the equator to the north pole. Turn right but hold the stick the same direction and walk back to the equator. Now turn right and return to your starting spot. The arrow now points in the direction of the equation, showing the sphere had curvature.

  • Modification of this experiment

Now consider your "surface" is not the surface of a sphere, but the whole of R3. Now instead of holding a 1 dimensional arrow, you hold a cylindrical rod and traverse the same path. However the rod has a red marking on top and a blue one on the side etc. Now, when you get back to the start, the arrow does not point in the same direction AND the red, which started on top, is now showing blue to be on top.

  • Theory of extra dimensions and its implications

Many theories predict extra dimensions such as string theory. So if there were some form of curvature in these extra dimensions, it seems like there would be direct experimental evidences of this in the form of red and blue. Some property of an object in our space would change even though we would not anticipate it. My question then is this:

  • Question

Is there any way to link unexpected properties of our Universe to an extra dimension? I know it a vague and abstract question, but I am still trying to figure out what it all could mean. Maybe another way of putting it: Are there transformations that we do to objects that cause it to change, but we don't know why that could be related to the concept of dimensions and curvature?

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  • $\begingroup$ I think the concept you're trying to get at in your modified experiment is the torsion tensor (en.wikipedia.org/wiki/Torsion_tensor). In general relativity we assume that the connection is such that the torsion tensor is zero. Einstein-Cartan theory is an extension of general relativity where we don't assume this. A torsion-free connection in 5 dimensions could, restricted to 4 dimensions, look like one with torsion, just like a a sphere sitting in flat space is still curved. $\endgroup$
    – Robin Ekman
    Commented Feb 12, 2016 at 17:46

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Your example of a sphere is an example of extrinsic curvature. The sphere is a two dimensional manifold and it's curved because it is embedded in a three dimensional manifold. The extra dimension is necessary for the curvature to exist.

However in general relativity the curvature is intrinsic curvature not extrinsic curvature. This does not require any extra dimension. Our four dimensional spacetime is not curved because it is embedded in some higher dimensional - the curvature is intrinsic to the four dimensions we observe.

If higher dimensions exist, and we occupy a submanifold with them (e.g. a braneworld), then there could be measurable effects due to the higher dimensions. However at the moment there is absolutely no experimental evidence to suggest that higher dimensions exist. They remain a purely theoretical idea.

If you're interested to know more it would be worth doing a search of this site as there are lots of questions and answers dealing with related issues.

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  • $\begingroup$ How does this intrinsic curvature work, exactly? Could this be demonstrated with a piece of paper (2-D) curving into three dimensions? $\endgroup$
    – Jiminion
    Commented Oct 14, 2016 at 15:17

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