Curved space-time and dimensions

In summary, the conversation discusses the concept of embedding in general relativity, specifically the Whitney embedding theorem which states that any Riemannian manifold can be isometrically embedded into a Euclidean space. The possibility of extra dimensions in the embedding space and their potential physical meaning is also brought up. However, it is stated that such dimensions do not have any meaning in general relativity. The conversation also briefly touches on Lorentzian embeddings, which differ from Riemannian embeddings and have less well-known bounds. Lastly, the idea of representing higher dimensional manifolds as lower dimensional ones with simpler criteria is mentioned.
  • #1
Unkraut
30
1
I don't know anything about GTR, nor do I know anything about differential geometry. But I have one maybe stupid question:
As far as I know space-time in general relativity is represented by a pseudo-riemannian manifold. And according to Whitney's (or Nash's? - don't know who is in charge here) embedding theorem every (Riemannian) manifold can be embedded (isometrically) into an euclidic R^n.
So, if we now have some curved space-time manifold and we find the smallest possible embedding R^n. Could the extra dimensions in the embedding space have any physical meaning? Or are they even known to?
In other words: Would it perhaps make any sense to suppose some "virtual" processes to happen within that embedding space, I mean, just like calculating with complex numbers but only taking real results "for real", that kind of thing?
I'm just a curious pseudo-mathematician trying to understand some physics and asking questions that come to my mind. Sorry if it's stupid.

Unkraut
 
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  • #2
They definitely don't have any meaning in general relativity. Einstein's equation defines a relationship between the metric and the stress-energy tensor (i.e. matter), but doesn't relate any of that to anything outside of space-time.

I suppose that what you have in mind is some kind of duality between mathematical objects that "live" in the manifold and mathematical objects that live in an embedding space. I have never heard of anything like that, but I don't see a way to rule out that something like that could be useful as a mathematical tool.
 
  • #3
Okay, thanks. That's what I expected.
 
  • #4
Someone who knows this stuff better than I do suggested that I should inform you that embedding theorems for Lorentzian manifolds are quite different from embedding theorems for Riemannian manifolds.

This is how he explained it to me:

Whitney embedding theorem: any smooth m-dimensional RIEMANNIAN manifold you are likely to meet can be embedded in 2m-dimensional euclidean space.

Wikipedia is actually pretty good for RIEMMANNIAN embeddings, but has nothing on LORENTZIAN embeddings, where the known bounds are MUCH worse. You can Google for an old sci.physics.research post which gives a citation for Lorentzian embeddings.

One other thing which might be worth mentioning is that one can sometimes "represent" higher dimensional manifolds with special properties (e.g. special kinds of solutions to the EFE) as lower dimensional manifolds with abstract coordinates, and satisfying a simpler criterion. For example you can look for a paper by H. J. Schmidt, formerly editor of Gen. Rel. Grav.
 

1. What is curved space-time?

Curved space-time refers to the concept in which the three dimensions of space (length, width, and height) and the dimension of time are interconnected and can be affected by the presence of matter and energy. This idea was first proposed by Albert Einstein in his theory of general relativity.

2. How does curved space-time affect the motion of objects?

Curved space-time can affect the motion of objects by causing them to follow curved paths rather than straight lines. This is because the presence of matter and energy can warp the fabric of space-time, altering the geometry of the space around it. The more massive an object is, the more it can curve space-time.

3. What are the implications of curved space-time for our understanding of gravity?

Curved space-time provides a new understanding of gravity, as it suggests that gravity is not a force between two objects, but rather the curvature of space-time caused by the presence of matter and energy. This explains why objects with mass are attracted to each other, as they are simply following the curved paths of space-time.

4. Are there more than three dimensions in curved space-time?

Yes, according to some theories, there may be more than three dimensions in curved space-time. These extra dimensions may be curled up or compactified, meaning they are so small that we cannot perceive them. However, they may play a role in explaining certain phenomena, such as the forces between particles.

5. How is curved space-time related to the concept of a black hole?

Curved space-time is essential in understanding the concept of a black hole. A black hole is a region of space where the curvature of space-time is so extreme that nothing, including light, can escape its gravitational pull. This is due to the massive amount of matter and energy concentrated at the center of a black hole, which causes an intense curvature of space-time.

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