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"Large" diffeomorphisms in general relativity
We had a discussion regarding "large diffeomorphisms" in a different thread but it think we should ask this question here.
For a 2-torus there are the so-called "Dehn twists"; a Dehn twist is generated via cutting the 2-torus, rotating one of the two generated circles by some angle theta and gluing the two circles together again.
Two things are interesting:
A) using an arbitrary angle theta this is not a homeomorphism (and therefore not a diffeomorphisms either) as neigboured points are not mapped to neighboured points. Nevertheless the torus is mapped to a torus.
B) using an angle theta which is amultiple of 360° this is a diffeomorphism, but it seems that it should be called a "large" diffeomorphism as the two ccordinate systems are not transformed into each other via l"local" deformations.
Now in GR we expect everything to be invariant regarding diffeomorphisms. The Dehn twist is a rather simple example but one can easily construct similar transformations in higher dimensional spaces.
Questions:
In the case A) the twist is not a diffeomorphism, therefore we need not expect invariance; but can this case A) be generated via dynamics in GR? Or are there "diffeomorphic superselection sectors"?
In the case B) we have a diffeomorphism, but nevertheless it seems that there is a discrete structure regarding the different N*360° rotations labelling "different" (but diffeomorphic) tori. Again: are such "different" but diffeomorphic manifolds of any relevance.
General question: is there some topology of the diffeomorphism group in n dimensions which is related to these Dehn twists and other "lagre diffeomorphisms" in higher dimensions?
Thanks
Tom
We had a discussion regarding "large diffeomorphisms" in a different thread but it think we should ask this question here.
For a 2-torus there are the so-called "Dehn twists"; a Dehn twist is generated via cutting the 2-torus, rotating one of the two generated circles by some angle theta and gluing the two circles together again.
Two things are interesting:
A) using an arbitrary angle theta this is not a homeomorphism (and therefore not a diffeomorphisms either) as neigboured points are not mapped to neighboured points. Nevertheless the torus is mapped to a torus.
B) using an angle theta which is amultiple of 360° this is a diffeomorphism, but it seems that it should be called a "large" diffeomorphism as the two ccordinate systems are not transformed into each other via l"local" deformations.
Now in GR we expect everything to be invariant regarding diffeomorphisms. The Dehn twist is a rather simple example but one can easily construct similar transformations in higher dimensional spaces.
Questions:
In the case A) the twist is not a diffeomorphism, therefore we need not expect invariance; but can this case A) be generated via dynamics in GR? Or are there "diffeomorphic superselection sectors"?
In the case B) we have a diffeomorphism, but nevertheless it seems that there is a discrete structure regarding the different N*360° rotations labelling "different" (but diffeomorphic) tori. Again: are such "different" but diffeomorphic manifolds of any relevance.
General question: is there some topology of the diffeomorphism group in n dimensions which is related to these Dehn twists and other "lagre diffeomorphisms" in higher dimensions?
Thanks
Tom