B Can space be curved in relation to an absolute straight space?

[Nugatory]

I notice a curvature of (existing) space against what "I see"', which is probably a mathematical (non existing) straight XYZ "space".

What you "see" (which doesn't mean what you literally "see" - that would be a two-dimensional image on the retina of your eyes - it means what your brain figures out about the world around you based on that two-dimensional image on the retina of your eyes) is not a flat Euclidean space. It's a curved space in which the curvature is small enough that it generally goes unnoticed.

 

[Orodruin]

always occurs in a higher dimensional flat space

Is this really true? I would think you could define extrinsic curvature for any embedding regardless of the flatness of the space you are embedding in.

 

[Dale]

Is this really true? I would think you could define extrinsic curvature for any embedding regardless of the flatness of the space you are embedding in.

Well, I have to admit that I have not done an exhaustive study of the literature, but every embedding theorem that I have read about is embedding in a flat space. That may be selection bias since I usually search for "isometric" too.

 

[Ibix]

You must be able to embed manifolds in curved manifolds - e.g. the Earth's surface in spacetime. And it certainly seems to be extrinsically curved. Or am I being naive?

 

[Orodruin]

You must be able to embed manifolds in curved manifolds - e.g. the Earth's surface in spacetime. And it certainly seems to be extrinsically curved. Or am I being naive?

There certainly is nothing to stop you from doing such an embedding and your example for the Earth's surface (or rather its world tube) is a good one.

I would think that the reason you see embedding existence theorems with flat space is that the curved space is rather non-interesting (just embed $M$ in $M\times N$ where $N$ is any manifold).

 

[DParlevliet]

Most people look to the world with a straight mathematical XYZ-geometry, as Nugatory mentioned. That is the reference for what we "see". The problem is that they think this is equal to space, so get confused when space is curved.

Returning to my first statement: I notice that space is curved against a straight mathematical XYZ-geometry. Perhaps this is not the same value or structure as GT and not useful for GT calculations, but it is caused by space curvature. So it still is a right statement that space is curved with reference to a straight mathematical XYZ-geometry.

 

[Dale]

So it still is a right statement that space is curved with reference to a straight mathematical XYZ-geometry.

No, it is not. Please see above for details. The minimal straight space is more than 3 dimensional.

 

[trainman2001]

Interesting discussion as always. What I can't wrap my head around is that space curvature around a mass is in all directions 360 degrees in every conceivable orientation, but the rubber sheet analogy is still in 3D space and is only in one plane. It's as if the rubber sheet was everywhere around the mass and that I find very hard to visualize. I completely accept the outcome that mass curves space which is what we perceive as gravity, but I just can't get the imagery right in my mind. Does anyone have something to put me out of my angst?

 

[Nugatory]

Interesting discussion as always. What I can't wrap my head around is that space curvature around a mass is in all directions 360 degrees in every conceivable orientation, but the rubber sheet analogy is still in 3D space and is only in one plane. It's as if the rubber sheet was everywhere around the mass and that I find very hard to visualize. I completely accept the outcome that mass curves space which is what we perceive as gravity, but I just can't get the imagery right in my mind. Does anyone have something to put me out of my angst?

Try to forget that you ever saw the rubber sheet analogy. If it's not helping you, you don't need it.

Yes, the picture is drawn in three-dimensional space, but you can't draw a picture showing three-dimensional curvature in a three-dimensional space; this is why you're finding it impossible to visualize the curvature being the same in all directions and orientations. But even if you could make it past that hurdle you'd still be stuck because we aren't working with curved three-dimensional space. We're working with curved four-dimensional spacetime and there isn't any time axis in the rubber sheet analogy.

 

[DaveC426913]

but you can't draw a picture showing three-dimensional curvature in a three-dimensional space;

Why not?

 

[A.T.]

Why not?
View attachment 210173
View attachment 210175
View attachment 210174

Those pictures show no intrinsic 3D curvature. The only intrinsic curvature shown here is that of the 2D grid planes. The distorted 3D grid still encompasses the same total volume an undisturbed grid would (within the same boundary). But the key feature of intrinsically curved 3D space is that there is more volume inside a boundary that you would expect based on Euclidean geometry.

You cannot visualize intrinsic 3D curvature just by shifting grid nodes around, placed over flat 3D embedding space. You would need a higher dimensional embedding space, just like you need 3D embedding to show intrinsic 2D curvature.

We had some discussion on those pictures here:
https://www.physicsforums.com/threads/imagining-spacetime-curvature-more-accurately.753672/
https://www.physicsforums.com/threads/kind-of-newbie-question-about-gravity.782902/page-3

 

[P Hysicist]

A 2D surface can always be embedded in 3D Euclidean space. But in higher dimensions that's not always true. You are thinking of 2D curved spaces, and therefore think of their extrinsic curvature as necessary to describe their curvature. However, you can describe the curvature using intrinsic properties only. That's called intrinsic curvature. In higher dimensions that's the only way to go.

 

[Ibix]

A 2D surface can always be embedded in 3D Euclidean space.

I don't think this is true. A 2d surface of constant negative curvature cannot be embedded in 3d space. In general, I understand, you need 2N dimensions to embed an N-dimensional surface.

 

[abrogard]

you might not be able to draw a picture of 3D curvature but you can imagine it can't you? If you imagine a ball made of some stuff that increases in density as it gets closer to the centre.

no problem to imagine that. is there?

well that's the curvature.

 

[DrGreg]

you might not be able to draw a picture of 3D curvature but you can imagine it can't you? If you imagine a ball made of some stuff that increases in density as it gets closer to the centre.

no problem to imagine that. is there?

well that's the curvature.

Not really. "Density" and "curvature" are rather different concepts.

 

[l0st]

I don't think this is true. A 2d surface of constant negative curvature cannot be embedded in 3d space. In general, I understand, you need 2N dimensions to embed an N-dimensional surface.

If I was author of this thread, this is the question, I would ask after the initial explanation.
Can any intrinsic N-dimenional curvature be modeled as some f(N)-dimensional embedding into Euclidean space (apparently, f(N) = 2*N)? And could such an embedding have any useful physical meaning like sphere radius has for intrinsic curvature of sphere?
Especially, I'd be interested if that embedding could provide a more intuitive interpretation of curved time and gravity wells.
Would be super interesting to see 2D curved spacetime embedding, if its possible in 3D

 

[Ibix]

Can any intrinsic N-dimenional curvature be modeled as some f(N)-dimensional embedding into Euclidean space (apparently, f(N) = 2*N)?

Your terminology is a little off. With a few restrictions that are not relevant to physics, any smooth N-dimensional manifold can be embedded in a 2N-dimensional Euclidean space, according to Whitney's embedding theorem. This includes manifolds with curvature in the sense that general relativity uses it.

And could such an embedding have any useful physical meaning like sphere radius has for intrinsic curvature of sphere?

Depends what you mean by "physical meaning". Intrinsic curvature - which is the thing that has a meaning in general relativity - does not always correspond to the extrinsic curvature that appears in the embedding space. It does for a sphere. But take a piece of paper flat on a table and slide the edges slightly together so that it curves off the table. There's extrinsic curvature there (distance along the paper does not correspond to the straight line distance in the embedding space), but the intrinsic curvature is still zero. Triangles drawn on the paper still have angles summing to 180 etcetera etcetera.

So there's certainly physical meaning in the embedding space to the extrinsic curvature. But it doesn't necessarily correlate to the intrinsic curvature that matters to physics in the embedded space.

Especially, I'd be interested if that embedding could provide a more intuitive interpretation of curved time and gravity wells.

Not really, since you'd need an eight dimensional Euclidean space to embed a general 4d manifold.

Would be super interesting to see 2D curved spacetime embedding, if its possible in 3D

This kind of thing may be what you have in mind: https://www.physicsforums.com/threads/no-gravity.919084/page-2#post-5794900.

 

[A.T.]

Would be super interesting to see 2D curved spacetime embedding, if its possible in 3D

This kind of thing may be what you have in mind: https://www.physicsforums.com/threads/no-gravity.919084/page-2#post-5794900.

Note that the space-times in the link by Ibix do not have intrinsic curvature. That's why you can roll them out flat, as shown in the videos. This is valid for a small region, where tidal effects are negligible.

For embedding of space-time with intrinsic curvature over a larger region, see this:
http://www.adamtoons.de/physics/gravitation.swf
http://www.relativitet.se/Webtheses/tes.pdf

 

[PAllen]

Your terminology is a little off. With a few restrictions that are not relevant to physics, any smooth N-dimensional manifold can be embedded in a 2N-dimensional Euclidean space, according to Whitney's embedding theorem. This includes manifolds with curvature in the sense that general relativity uses it.

A little care is needed here. Whitney's embedding theorem says nothing about whether an embedding is isometric; in fact, it is a theorem about topological manifolds considered without metric. The answer to embeddings of Riemannian manifolds (i.e. with metric) that preserve the metric properties of the embedded manifold, are covered by the Nash embedding theorems, which are more complex, and the Euclidean space may require many more than 2N dimensions.