Intrinsic Curvature

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I have heard that intrinsic curvature does not require a manifold (is that the correct term?) to be embedded in another dimension for its curvature to be apparent ( and calculable ) to an observer inhabitant.

Parallel Transport on a globe is adduced as an example whereby a dweller on the surface can test for intrinsic curvature.

How can this be the case if as is described the vector which is being transported lies in the "tangent plane". ?

Does this tangent plane not lie in a higher dimension ?

How is it possible to have a tangent plane on the surface of a globe that does not presupppose the existence of an embedding 3rd dimension?

To carry on the idea (to an area I am extremely unfamiliar with) are there not supposed to be corresponding tangent planes in 3D+1 spacetime (tensors?) ?

Do not these tensors also presuppose a tangent plane which is situated in an embedding dimension (so a 5th dimension)?

To go back to my parallel transport on the globe ,is it really possible for a 2D inhabitant to calculate its curvature without recourse to a "bird's eye" view?

I mean ,how can he measure the angles of triangles for instance without looking down at the triangle he has drawn from above (which presupposes a 3rd dimension) ?

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I have heard that intrinsic curvature does not require a manifold (is that the correct term?) to be embedded in another dimension for its curvature to be apparent ( and calculable ) to an observer inhabitant.

Parallel Transport on a globe is adduced as an example whereby a dweller on the surface can test for intrinsic curvature.

How can this be the case if as is described the vector which is being transported lies in the "tangent plane". ?

Does this tangent plane not lie in a higher dimension ?

How is it possible to have a tangent plane on the surface of a globe that does not presupppose the existence of an embedding 3rd dimension?

To carry on the idea (to an area I am extremely unfamiliar with) are there not supposed to be corresponding tangent planes in 3D+1 spacetime (tensors?) ?

Do not these tensors also presuppose a tangent plane which is situated in an embedding dimension (so a 5th dimension)?

To go back to my parallel transport on the globe ,is it really possible for a 2D inhabitant to calculate its curvature without recourse to a "bird's eye" view?

I mean ,how can he measure the angles of triangles for instance without looking down at the triangle he has drawn from above (which presupposes a 3rd dimension) ?

Have you looked at posts 24 through 27 here?

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Thanks. I will.

Have you looked at posts 24 through 27 here?

That was not new material for me (the Elementary Geometry extract) but I think I have now overcome a road block in my mind since I have realized that it is possible to calculate angles without leaving the surface of the globe .
If seems possible to measure angles simply by measuring and comparing lengths of lines . I had only(well practically exclusively) looked at angles in a visual way up till now.
I am now comfortable with the notion that it is possible to gauge curvature of a surface in this intrinsic way.
I am not sure how this is accomplished practically in 3d+1 geometry but I guess the theory is no longer so mysterious and probably the corresponding distances are spacetime distances?
I am still a bit unsure ,mind you about this idea of tangents to the curvature. Does that not require an extra (embedding) dimension? Or is it simply a mathematical device of some kind?

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You can determine intrinsic geometry with the Einstein elevator. If you drop two objects they will only drop on parallel paths in flat geometry. If your geometry is curved ie over a larger seperation distance over say the curvature of the Earth. The freefall path of the two objects will converge. (google GR tidal effect)

https://en.m.wikipedia.org/wiki/Introduction_to_general_relativity

This principle of freefall path and parallel transport can be used to detect any type of spacetime curvature.

Edited by Mordred

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Thanks. I will.

That was not new material for me (the Elementary Geometry extract) but I think I have now overcome a road block in my mind since I have realized that it is possible to calculate angles without leaving the surface of the globe .
If seems possible to measure angles simply by measuring and comparing lengths of lines . I had only(well practically exclusively) looked at angles in a visual way up till now.
I am now comfortable with the notion that it is possible to gauge curvature of a surface in this intrinsic way.
I am not sure how this is accomplished practically in 3d+1 geometry but I guess the theory is no longer so mysterious and probably the corresponding distances are spacetime distances?
I am still a bit unsure ,mind you about this idea of tangents to the curvature. Does that not require an extra (embedding) dimension? Or is it simply a mathematical device of some kind?

OK so you say you are happy with paragraph in the other thread.

The vital thing to know and remember is that

The tangents do not exist in the same universe as we do.

They have their own complete coordinate system and touch our universe at one point only,

Further the tangent system at each point has its own universe and coordinate system, different from the universe and tangent system touching at any other point of our universe.

This is what is meant by a local coordinate system.

All the points in our universe make up the global coordinate system that we use in relativity.

I have said this before but the relationship between the local coordinate system at any one point containing a tangent (or other) system and the global one is mathematically called a chart.

Each point therefore has its own chart.

The relationship between two charts is called a connection. This is usually established by the 'parallel transport rule'

Without such a rule we could not relate or connect measurements made at one point with those made at another.

The 1 :1 mapping of the tangent plane (or other surface) onto the base universe is called a patch.

In general you need more than one patch to 'cover' the entire universe.

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Under GR the connection Studiot is referring to is called the Levi-Civitta connection. The patches Studiot mentioned is tangent bundles. Which is rather tricky to define in curved space. Tangent bundles is a type of fibre bundle but once again these terms are difficult to describe on a forum. Other key terms involved is tangent space, and tangent map.

Edited by Mordred

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OK so you say you are happy with paragraph in the other thread.

The vital thing to know and remember is that

The tangents do not exist in the same universe as we do.

They have their own complete coordinate system and touch our universe at one point only,

.........................

Thanks(I suppose ). That seems so far beyond what I have begun to come to terms with that I will treat it as a long term project.

I will keep an ear to the ground and perhaps I may worm a way into the subject eventually.A direct approach would be sure to fail in my case.

Still ,thanks for the "The tangents do not exist in the same universe as we do." warning. That is probably one less stumbling block and source of grievance for me now

Edited by geordief