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Large diffeomorphisms in general relativity 
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#19
Mar211, 04:01 PM

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Normally you would have b) if you have a) 


#20
Mar211, 04:15 PM

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I tend to agree with a) But that means that GR is NOT invariant w.r.t. all diffeomorphisms but only w.r.t. "restricted" diffeomorphisms (like small and large gauge transformations, where large gauge trf's DO generate physical effects)



#21
Mar211, 04:24 PM

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PF Gold
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#22
Mar211, 04:35 PM

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#23
Mar211, 04:54 PM

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PF Gold
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My understanding is that topology of a differentiable manifold is encoded in how coordinate patches overlap. So, if we don't change this (and we don't need to for the Dehn twist), and we don't change anything computable from the metric, what can change? In my (1) and (2) I was trying to get at the idea of making the operation 'real' so it does change geometry, versus treating as a pure coordinate transform, such that the corresponding metric transform preserves all geometric facts. I've heard the terms active versus passive difffeomorphism. I don't fully understand this, but I wonder if it is relevant to this distinction. 


#24
Mar211, 04:58 PM

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#25
Mar211, 05:11 PM

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PF Gold
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Then, the physics question becomes that conventionally formulated GR would attach no meaning to this additional structure, it would become physically meaningful only in the context of an extension to GR that gave it meaning. This is what some of the classical unified field theory approaches did. 


#26
Mar211, 05:31 PM

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I found an explanation on Baez "this week's finds", week 28: http://math.ucr.edu/home/baez/week28.html 


#27
Mar211, 06:04 PM

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PF Gold
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If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant. Is it some form of coordinate dependent torsion? 


#28
Mar211, 06:14 PM

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#29
Mar211, 06:27 PM

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#30
Mar211, 07:19 PM

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PF Gold
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I found two possibly relevant papers, both focusing on 2+1 dimensions:
http://relativity.livingreviews.org/...es/lrr20051/ http://matwbn.icm.edu.pl/ksiazki/bcp/bcp39/bcp3928.pdf If I am reading section 2.6 of the Carlip paper (first above) correctly, it suggests that GR is invariant under large diffeomorphisms, as I guessed above. 


#31
Mar311, 12:15 AM

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Thanks a lot for checking that and providing the two links. Looks very interesting.



#32
Mar311, 01:11 AM

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This material has always confused me, and its hard to find good references. Over the years i've asked a few specialists but it hasn't helped me much.
In 2+1 dimensions, the whole mapping class group sort of makes good intuitive sense, but then I rarely see it generalized in 4d. Further, the real subleties, at least to me, arise when the diffeomorphisms change the asymptotic structure of spacetime. Its not clear whether bonafide 'observables', are invariant under these 'gauge' transformations (incidentally, to avoid confusion, the notion of a large gauge transformation is afaik typically done where you fix a spacetime point, fix a vielbein and treat the diffeomorphism group acting on these elements in an analogous way to intuition from gauge theory) 


#33
Mar311, 01:17 AM

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#34
Mar311, 03:27 AM

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And yet I'd say this is a vital point, the theoretical base of many yet unobserved physics, such as that of black holes(see the Carlip cite in section 2.6) depends on whether spacetimes are considered as invariant or not for these large diffeomorphisms. Of course some people that are not very fond of thinking for themselves would rather just obey the conventional opinion on this and let it be like that, so your question Tom, touches a very sensitive spot. 


#35
Mar311, 04:01 AM

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This is similar to "QCD is invariant w.r.t. SU(3) gauge transformations". As long as one studies infinitesimal ones everything is fine, but as soon as you try to study large gauge transformations on different topologies it becomes interesting (in the axial gauge "Ał(x)=0" you cannot gauge away a dynamical zero mode ał=const., Gribov ambiguities, winding numbers, instantons and merons, center symmetry, ...).
There is an important question: gauge transformations arise due to unphysical degrees of freedom (in contradistinction to other global symmetries like flavor) which have to be gaugefixed (e.g. via Dirac's constraint quantization in the canonical formulation). But it seems that in gauge theories unphysical local and physical global aspects are entangled). I guess that soemthing similar will happen in GR as well. Many aspects in gauge theory become visible during quantization. So as long as we do not fully understand QG, some aspects may be hidden or irrelevant; example: what is the physical meaning of Kruskal coordinates? we don't care classically  but we would have to as soon as during BH evaporation the whole Kruskal spacetime has to be taken into account in a PI or whatever. If spacetime will be replaced by some discrete structure many problems may vanish, but if spacetime as a smooth manifold will survive quantization than these issues become pressing (diffeomorphisms in 4dim. are rather complicated  see Donaldson's results etc. ) 


#36
Mar311, 04:11 AM

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http://www.samjordan.ch/download/ph...esentation.pdf 


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