Curved or non-curved space, what is more fundamental?

[exponent137]

curved or non-curved space, what is more fundamental?

By intuition it seems that non-curved space is more fundamental.

By general relativity and difeomorphism it seems that no one is more fundamental.

But how it is with this?

 

[Dale]

What do you mean by "more fundamental"?

 

[exponent137]

What do you mean by "more fundamental"?

1. According to general relativity there is no difference in priority between curved and uncurved space? Is this true?
2. Area of rectangle in non-curved space is S=a x b. In curved space this is not true.
3. According to 2., Pythagoras theorem is fundamental property of geometry, if noncurved space is more fundamental. If this is not true, Pythagoras theorem is consequence of physics.

 

[Dale]

1. According to general relativity there is no difference in priority between curved and uncurved space? Is this true?

"Difference in priority"? You mean that when curved and uncurved space arrive at a formal ball which one is announced first? You keep using very strange words here. Perhaps you are a non-native english speaker?

If so, let me suggest the word "general". Curved spaces are more general than flat spaces because a flat space is a special case of a curved space where the curvature goes to 0.

 

[exponent137]

Yes, I am not native speaker, but it is not easy find words here in any language.

1. We know, that there is no difference between accelerated frame and gravity field. Is similarly a difference between curved and non-curved space.
2. Or maybe differently: I suppose that geometry is a consequence of physics. So I suppose that Pythagorean theorem is a consequence of physics.
If it is a consequence of geometry, not physics, then non-curved space is more fundamental?

Kinetic energy as sum of two orthogonal directions is:
mv_x^2+mv_y^2=mv^2
and it gives pythagoras theorem without geometry.

 

[atyy]

There is an explicit statement in Kip Thorne's popular book that GR can be formulated equivalently as a flat spacetime or a curved spacetime theory. I am not sure this is true without qualification. The closest I have been able to find is Eq 62 of http://www.emis.de/journals/LRG/Articles/lrr-2006-3/ which comes with the proviso that the spacetime can be covered by harmonic coordinates. (MTW also says the same thing as Thorne's popular book and gives a reference to ... )

Work like http://arxiv.org/abs/1008.3177 seems to indicate the the restriction to harmonic coordinates does not give the full solution space of GR, and that generalized harmonic coordinates are needed for that.

 

[Dale]

Yes, I am not native speaker, but it is not easy find words here in any language.
...
then non-curved space is more fundamental?

The word "fundamental" simply doesn't apply to spaces. AFAIK, you can use the word "fundamental" to distinguish between two theories or two particles. In the case of theories, if theory A reduces to theory B in some specific limit then theory A could be called more "fundamental" than theory B. It could also be called more "general" than B, so in this context "fundamental" and "general" are synonymous which is why I suggested the word "general" instead of "fundamental". Clearly theories which permit curved spacetimes are more general than theories which permit only flat spacetimes since they will reduce to the flat-spacetime theories in the appropriate limits.

If this concept is not what you are getting at then you will need to carefully explain what you mean by the word "fundamental".

 

[Vanadium 50]

We know, that there is no difference between accelerated frame and gravity field.

We certainly don't know that, because it is not true. There is a difference. Gravitational fields have tides.