"Our space is curved"

[studiot]

1 hour ago, Genady said:

Just a little anecdote here. I was once "attacked" by somebody for saying that CMB data indicate that our universe is perhaps flat. That person decided that I'm a flat earther.

With regard to 'flat' this post was wasted in Marius' threads but is highly pertinent here,

So I will reproduce it in full.

It is important to distinguish the measure and the measurand in all circumstances.
That is the ruler is not the same as that which is being measured.
Failure to do this leads to much misunderstanding.

 

Here are two 'straight line' graphs.

Or are they ?

Straight is the one dimensional version of not curved or zero curvature ie flat.

 

I would be interested in your response to the question.

Edited January 3, 2022 by studiot

[Genady]

31 minutes ago, studiot said:

Straight is the one dimensional version of not curved or zero curvature ie flat.

We need to distinguish between intrinsic and extrinsic geometry. GR and differential geometry refer to the intrinsic one.

Intrinsically, all 1D manifolds are flat = have zero curvature = locally indistinguishable, regardless if they look straight or curved extrinsically, or even close on themselves like a loop.

[studiot]

4 hours ago, Genady said:

We need to distinguish between intrinsic and extrinsic geometry. GR and differential geometry refer to the intrinsic one.

Intrinsically, all 1D manifolds are flat = have zero curvature = locally indistinguishable, regardless if they look straight or curved extrinsically, or even close on themselves like a loop.

Indeed so, or at least nearly so as I have said to several members, several times in the past.

This fact also has implications for the thread about 'time travel', which I have been trying to bring out, but others seem to have abandoned that thread.

Note that whilst a line may have zero Gaussian/Riemannian curvature, it may still have torsion, which is a similar but different thing.

 

A to the question of are all 1D manifolds flat, that was the point of my figures A and B.

A is a plot of a second order differential equation   -  that of the rate of a chemical reaction  -  the reciprocal of concentration is plotted against time  - one axis is non linear
B is a plot of a first order differential equation   -  that of radioactive decay  -   log(activity) is plotted against time  -  again one axis is non linear

 

 

[Genady]

7 minutes ago, studiot said:

Note that whilst a line may have zero Gaussian/Riemannian curvature, it may still have torsion, which is a similar but different thing.

I don't think one-dimensional line may have a torsion. Seems to me you need some points off axis to twist them relative to each other.

(I don't see a connection between the graphs and the curvature question, sorry.)

[studiot]

41 minutes ago, Genady said:

I don't think one-dimensional line may have a torsion. Seems to me you need some points off axis to twist them relative to each other.

(I don't see a connection between the graphs and the curvature question, sorry.)

Well think again

Quote

Wiki

In the elementary differential geometry of curves in three dimensions, the torsion of a curve measures how sharply it is twisting out of the plane of curvature. Taken together, the curvature and the torsion of a space curve are analogous to the curvature of a plane curve. For example, they are coefficients in the system of differential equations for the Frenet frame given by the Frenet–Serret formulas.

Sorry the graphs are to do with manifolds more generally.

[Genady]

3 minutes ago, studiot said:

Well think again

I did. There is no contradiction. In "differential geometry of curves in three dimensions" their curvature is an extrinsic curvature. What I refer to is an intrinsic curvature, which is a curvature of a one dimensional curve itself, not as being embedded in a higher dimensional space. Generally, extrinsic curvature is intrinsic curvature plus something (could look up the formula.)

Example: intrinsically, a plane and a cylinder are both flat, with zero curvature. 

[studiot]

47 minutes ago, Genady said:

I did. There is no contradiction. In "differential geometry of curves in three dimensions" their curvature is an extrinsic curvature. What I refer to is an intrinsic curvature, which is a curvature of a one dimensional curve itself, not as being embedded in a higher dimensional space. Generally, extrinsic curvature is intrinsic curvature plus something (could look up the formula.)

Example: intrinsically, a plane and a cylinder are both flat, with zero curvature. 

Torsion is not a form of curvature.

Further the direction vectors for both torsion and curvature do not live in the same 'space' as the line itself.

[Markus Hanke]

12 hours ago, Genady said:

Here it is:

I see. My immediate reaction to this would be that the two cases are interchangeable only in the classical domain; once you take into account the quantum fields that make up matter and vacuum (which GR of course does not do), then you will find that neither the strong nor the weak interaction are invariant under scaling of this type, so it is difficult to make sense of ‘rubbery’ measuring intervals. As such, the curved spacetime view is the more general one, as it applies to a wider domain within the real world.

[Genady]

26 minutes ago, Markus Hanke said:

I see. My immediate reaction to this would be that the two cases are interchangeable only in the classical domain; once you take into account the quantum fields that make up matter and vacuum (which GR of course does not do), then you will find that neither the strong nor the weak interaction are invariant under scaling of this type, so it is difficult to make sense of ‘rubbery’ measuring intervals. As such, the curved spacetime view is the more general one, as it applies to a wider domain within the real world.

Thank you again. I also tend to think so, but I don't have a proof, so good to have a supporting opinion :). 

Edited January 3, 2022 by Genady