I Euclidean geometry and gravity

[pervect]

Is that "flat plane" two-dimensional or three-dimensional?

Two dimensional

 

[PeterDonis]

Then I read a reference to 'frame dragging' in rotational black holes. What effect, if any, will that have on the 'paper'?

If the paper is "hovering" statically, i.e., no radial or tangential motion, frame dragging has no effect.

 

[pervect]

In the side discussion of embedding a piece of a plane smoothly, isometrically, in a 3-sphere (following is part of a response to @pervect 's last post:

I thought I would make explicit the construction in that reference (which is presented it very compactly in any number of dimensions). Making it explicit for flat 2-surface section in a 3-sphere, the mechanism is simply to use a properly sized flat torus. As explained in:
https://en.wikipedia.org/wiki/Clifford_torus
a flat torus of just the right size will smoothly fit in a unit 3-sphere, both embedded in R4. But then, the embedding can be just ignored, and you are left with a smooth embedding of a piece of Euclidean plane in a 3-sphere.

The formalization and generalization of this is the general approach to embedding $E^n$ into higher dimesnsional spheres used in the reference I quoted earlier. The general achievement is a local embedding of $E^n$ into an m-sphere, with $m=2n-1$.

Thank you. Untangling this in terms I can more fully understand will be quite a problem. I'm suspecting the issue is fundamental in that I'm viewing curvature as being defined by a metric. Which isn't necessarily how mathematicains view it.

But I think the discussion has moved outside of what's helpful to the OP's question.

 

[PeterDonis]

I'm viewing curvature as being defined by a metric.

More precisely, by the Riemann curvature tensor that's derived from a metric via its Levi-Civita connection. There's no problem with that--that's how the term is being used in this discussion.

But the metric of what? Take the case of a 2-sphere embedded in Euclidean 3-space. The 2-sphere is curved. The 3-space is flat. But one is embedded in the other. How can that be? Taking what you appear to be trying to say at face value, that should be impossible.

The reason it's possible is that the 2-sphere embedded in Euclidean 3-space is a submanifold of that 3-space, and we can define a metric on the submanifold that is curved even though the metric of the 3-space overall is flat. They're simply two different metrics. Note that those metrics give different answers to questions like the distance between two points, even if we restrict the points under consideration to those on the 2-sphere.

What @PAllen has been describing is simply a case where that works in reverse: you have a curved 3-manifold, the 3-sphere, with a submanifold, the Clifford 2-torus, on which a flat metric can be defined. Again, you simply have two different metrics.

 

[PAllen]

The reason it's possible is that the 2-sphere embedded in Euclidean 3-space is a submanifold of that 3-space, and we can define a metric on the submanifold that is curved even though the metric of the 3-space overall is flat. They're simply two different metrics. Note that those metrics give different answers to questions like the distance between two points, even if we restrict the points under consideration to those on the 2-sphere.

When discussing isometric embeddings, there is a little more to it. The metric of the submanifold is taken to be induced from the manifold metric. This means that, e.g. the distance along a curve on the 2-sphere embedded in E3 is, in fact required to be that same whether you use the manifold metric or the induced metric on the 2-sphere. You get to choose manifold, its metric, and the definition of the submanifold embedded in it as a set of points. You do not then get to choose an arbitrary metric on the submanifold - it is taken to be that induced by the manifold metric.

 

[PeterDonis]

the distance along a curve on the 2-sphere embedded in E3 is, in fact required to be that same whether you use the manifold metric or the induced metric on the 2-sphere.

Yes. It's important to recognize also that the curve in question is a geodesic of the curved 2-sphere, but it is not a geodesic of the Euclidean 3-space. Considering the implications of that might help @pervect to see what's going on in such cases.

 

[Jaime Rudas]

a flat torus of just the right size will smoothly fit in a unit 3-sphere

What does "unit" mean in this context?

 

[PAllen]

What does "unit" mean in this context?

Radius of curvature = 1