Extrinsic properties of the curved space

[Dmitry67]

Example: take curved 2D space with positive constant curvature everywhere. You say, sphere with radius R? no, there are 2 different solutions in topology: sphere and half-sphere. Half sphere (1/2 of sphere where points across the 'equator' are connected to the opposite sides) can’t be 'embedded' in 3D 'continuously'. Both objects have different extrinsic properties and total volume (so the difference can be discovered by an observer ‘inside’) but the same curvature everywhere.

I’ve heard that for 3D, and especially hyperbolic 4D spaces (like ours) it is much worse – there are infinitely many different solutions with different extrinsic properties.

So, my question is – is anyone working on it? Any links, articles? I was always wondering… say, 2 sides of curved spacetime intersect in the embedded higher dimensional space. Does it mean that these 2 points meet in our physical spacetime? Or (if embedding is a pure abstraction) they can go thru each other without any interaction (like in the Klein bottle)

 

[bcrowell]

You're confusing "extrinsic" with "global." The two geometrical systems you're describing are spherical geometry and elliptic geometry. Spherical and elliptic geometric are different intrinsically, not just extrinsically. For example, an axiomatic treatment of elliptic geometry might have an axiom that says that two lines always intersect at exactly one point, whereas in spherical geometry two lines will intersect in two places. These two geometries have the same local intrinsic properties, but different global intrinsic properties.

 

[JesseM]

You're confusing "extrinsic" with "global." The two geometrical systems you're describing are spherical geometry and elliptic geometry. Spherical and elliptic geometric are different intrinsically, not just extrinsically.

Spherical geometry is a type of elliptic geometry, no? I think Dmitry is referring to topology, which also can be described in intrinsic terms without an embedding space.

 

[bcrowell]

Spherical geometry is a type of elliptic geometry, no? I think Dmitry is referring to topology, which also can be described in intrinsic terms without an embedding space.

No, note where the WP article says this: "[...]and points at each other's antipodes are considered to be the same point." Identifying antipodal points makes it a half-sphere rather than a sphere.

 

[JesseM]

No, note where the WP article says this: "[...]and points at each other's antipodes are considered to be the same point." Identifying antipodal points makes it a half-sphere rather than a sphere.

I understood that this statement was referring to the half-sphere, but I don't see how this quote is relevant to what I said. I was questioning your statement where you said "The two geometrical systems you're describing are spherical geometry and elliptic geometry", when Dmitry had been talking about the ordinary sphere vs. the half-sphere (with opposite points on the equator identified). I don't think it makes sense to say the difference between an ordinary sphere and a half-sphere is equivalent to the difference between elliptic geometry and spherical geometry, since it seems as though elliptic geometry doesn't refer to any specific way of identifying faraway points, and that spherical geometry is one type of elliptic geometry.

 

[bcrowell]

I don't think it makes sense to say the difference between an ordinary sphere and a half-sphere is equivalent to the difference between elliptic geometry and spherical geometry, since it seems as though elliptic geometry doesn't refer to any specific way of identifying faraway points, and that spherical geometry is one type of elliptic geometry.

You need to make the distinction between models of an axiomatic system and the axiomatic system itself. Elliptic geometry is a formal mathematical system with certain axioms. A half sphere (or a sphere with opposite points identified) is a model of that axiomatic system. The distinction between intrinsic and extrinsic properties is the distinction between properties that are specific to the model and properties that hold for all models of the system (because they can be proved from the axioms).

Spherical geometry is not a type of elliptic geometry. In spherical geometry, every pair of lines intersects in two points. In elliptic geometry, every pair of lines intersects in one point.

 

[JesseM]

You need to make the distinction between models of an axiomatic system and the axiomatic system itself. Elliptic geometry is a formal mathematical system with certain axioms. A half sphere (or a sphere with opposite points identified) is a model of that axiomatic system.

OK, thanks, I'd seen the notion of models of axiomatic systems in other contexts but didn't understand that the half-sphere was a model of elliptic geometry.

 

[Dmitry67]

So, if Universe is closed, does GR predicts space to bend as sphere or half-sphere (with the same curvature)?

 

[bcrowell]

So, if Universe is closed, does GR predicts space to bend as sphere or half-sphere (with the same curvature)?

Interesting question. I may be wrong, but I believe that answer is that GR doesn't make any prediction of this at all. GR simply predicts the local properties of space, via the Einstein field equations. Possibly relevant information here: http://en.wikipedia.org/wiki/Shape_of_the_Universe

 

[Dmitry67]

in half-spehere it is possible to cross the whole universe around before it collapses back. You have time to return to the same point (but you will be reflected in the mirror).

So statements about extrinsic properties are falsifiable and physical, even there are no local effects. So GR has an extra 'degree of freedom' - different solutions are possible.

 

[George Jones]

So, if Universe is closed, does GR predicts space to bend as sphere or half-sphere (with the same curvature)?

If topological identifications are allowed, the universe could be closed and have *negative* spatial curvature.

 

[JesseM]

in half-spehere it is possible to cross the whole universe around before it collapses back. You have time to return to the same point (but you will be reflected in the mirror).

So statements about extrinsic properties are falsifiable and physical, even there are no local effects. So GR has an extra 'degree of freedom' - different solutions are possible.

I still don't think it's right to say these are statements about "extrinsic" properties--you can differentiate the sphere from the half-sphere without reference to any extrinsic embedding space. I think the basic method of defining a half-sphere would be a topological one where you pick a region of space with an edge, then identify pairs of points on the edge, similar to how one can differentiate a flat finite space with the topology of a torus from an infinite flat space--see the tiling diagram in figure 8 of this article.

 

[Dmitry67]

Wait, half-sphere does not have edges.
All points on half-sphere, like on a full one, are not special.

 

[JesseM]

Wait, half-sphere does not have edges.
All points on half-sphere, like on a full one, are not special.

A torus whose curvature is flat everywhere doesn't have edges either, and no points are special. Where you draw the "edges" is arbitrary--look at fig. 8 in that article showing the repeating pattern of a flat space which has the topology of a torus, you could shift the position of the grid without shifting the underlying points in space (the bee and its tracks) and it wouldn't make a difference.

edit: similarly I think you could depict the half-sphere as a sort of tiling pattern on a regular sphere, with two copies of every point on opposite sides, and then the "equator" could be shifted to anywhere you like (any great circle would do), and this would be topologically equivalent to the space on one half of the sphere with the equator as an edge, and each point on this edge being identified with the opposite point on the edge.

 

[Dmitry67]

A torus whose curvature is flat everywhere doesn't have edges either, and no points are special. Where you draw the "edges" is arbitrary--look at fig. 8 in that article showing the repeating pattern of a flat space which has the topology of a torus, you could shift the position of the grid without shifting the underlying points in space (the bee and its tracks) and it wouldn't make a difference.

I don't deny it.
What I am saying is that you can experimentally verify if you are on infinite surface or torus.

 

[bcrowell]

So statements about extrinsic properties are falsifiable and physical, even there are no local effects. So GR has an extra 'degree of freedom' - different solutions are possible.

You're still using the term "extrinsic" incorrectly. See #2.

Re falsifiability, see the WP link at #9.

 

[JesseM]

I don't deny it.
What I am saying is that you can experimentally verify if you are on infinite surface or torus.

Yes, I agree with that. I was just saying you don't need to use an "extrinsic" description to distinguish different topologies with the same curvature, like infinite plane vs. flat torus, or sphere vs. half-sphere.