# "Large" diffeomorphisms in general relativity

by tom.stoer
Tags: diffeomorphisms, relativity
P: 2,863
 Quote by tom.stoer Let's make an example. Assume R*Tł solves Einstein equations in vacuum (it does not, but that doesn't matter here). Assume we have a closed geodesic curve of a test object with winding numbers (1,0); this should be OK as Tł is flat and therefore a straight line with (0,1) should work. No let's do the cut-twist-glue procedure. What we get back is a different closed curve with winding number (1,1). Questions: a) does this generate a new, physically different spacetime? b) does this generate a different path of a test object on the same spacetime? c) did I miss something, e.g. did I miss to check whether this new curve can still be a geodesic?
Mainstream says none of the above, I'd say a)
Normally you would have b) if you have a)
 Sci Advisor P: 5,295 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)
PF Gold
P: 4,785
 Quote by tom.stoer 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)
I think this would have to show up somehow in the process of transforming the metric, e.g. unavoidable singularities. Otherwise it is just arithmetic that two curves of some length, and orthogonal to each other, and with one intersection, preserve all those feature in new coordinates with properly transformed metric.
P: 5,295
 Quote by PAllen I think this would have to show up somehow in the process of transforming the metric, e.g. unavoidable singularities.
Why? It is a diffeomorphism and does not create a singularity
PF Gold
P: 4,785
 Quote by tom.stoer Why? It is a diffeomorphism and does not create a singularity
Well, then it can't change the geometry, at least as defined by anything you can compute using the metric. This really has nothing to do with GR, it is differential geometry.

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.
P: 5,295
 Quote by PAllen Well, then it can't change the geometry, at least as defined by anything you can compute using the metric. This really has nothing to do with GR, it is differential geometry.
Du you agree that it changes the winding number of a closed curve?
PF Gold
P: 4,785
 Quote by tom.stoer Du you agree that it changes the winding number of a closed curve?
Certainly, it is changed in my case (1) of my post #17. I'm not sure about as described in case(2) of that post. If you can compute winding number from the metric and topology as encoded in coordinate patch relationships, then it would seem mathematically impossible. If this is an example of geometrical fact independent of the metric and patch relationships, then we would need some definition how to compute it intrinsically, and it would seem to necessitate adding some additional structure to the manifold. In this case, it may well be possible, having specifically introduced non-metrical geometric properties not preserved by coordinate transforms.

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.
P: 5,295
 Quote by PAllen Certainly, it is changed in my case (1) of my post #17. I'm not sure about as described in case(2) of that post. If you can compute winding number from the metric and topology as encoded in coordinate patch relationships ...
It does even in case (2)

I found an explanation on Baez "this week's finds", week 28:

http://math.ucr.edu/home/baez/week28.html

 Quote by Baez,#28 Now, some diffeomorphisms are "connected to the identity" and some aren't. We say a diffeomorphism f is connected to the identity if there is a smooth 1-parameter family of diffeomorphisms starting at f and ending at the identity diffeomorphism. In other words, a diffeomorphism is connected to the identity if you can do it "gradually" without ever having to cut the surface. To really understand this you need to know some diffeomorphisms that aren't connected to the identity. Here's how to get one: start with your surface of genus g > 0, cut apart one of the handles along a circle, give one handle a 360-degree twist, and glue the handles back together! This is called a Dehn twist. ... In other words, given any diffeomorphism of a surface, you can get it by first doing a bunch of Dehn twists and then doing a diffeomorphism connected to the identity.
So we can now concentrate on the physical role of these "large" diffeomorphisms.
PF Gold
P: 4,785
 Quote by tom.stoer It does even in case (2) I found an explanation on Baez "this week's finds", week 28: http://math.ucr.edu/home/baez/week28.html So we can now concentrate on the physical role of these "large" diffeomorphisms.
This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?

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?
P: 5,295
 Quote by PAllen This was very interesting, but I didn't find any answer to my question my key question: how is it winding number of closed curve computed / defined against the definition of a differentiable manifold?
Yes, not a single word.

 Quote by PAllen If it can change while the manifold is considered identical, then it must be computed in a way that is not invariant.
I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa - which does not hold in higher dimensions) then the two manifolds before and after the twist are identical - there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.
PF Gold
P: 4,785
 Quote by tom.stoer Yes, not a single word. I don't agree. If the Dehn twist is a global diffeomorphism (and if we agree that in 2 dimensions homeomorphic manifolds are diffeomorphic and vice versa - which does not hold in higher dimensions) then the two manifolds before and after the twist are identical - there is no way to distinguish them. Now suppose we cannot compute the winding numbers (m,n) but only their change under twists. Then this change is not a property of the manifold but of the diffeomorphism. So we don't need a way to compute the winding numbers from the manifold but a way to compute their change from the diffeomorphism (this is similar to large gauge transformations where the structure is encoded in the gauge group, not in the base manifold). I think for large diffeomorphisms there is some similar concept.
Thankyou! Very interesting. Then I spout my opinion of the physics issue (assuming something like this is what is going on). Conventional GR only gives meaning to metrical quantities, so this aspect of the diffeomorphism would have no physical significance, and anything metrically defined would be preserved. And I come back to the idea that this sort of thing provides an opportunity to extend conventional GR- without changing any of its predictions, you can add new content.
 Sci Advisor PF Gold P: 4,785 I found two possibly relevant papers, both focusing on 2+1 dimensions: http://relativity.livingreviews.org/...es/lrr-2005-1/ 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.
 Sci Advisor P: 5,295 Thanks a lot for checking that and providing the two links. Looks very interesting.
 Sci Advisor P: 1,652 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)
P: 5,295
 Quote by Haelfix This material has always confused me, ... 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
I agree with you. I can't remember why we started to discuss these issues in GR, but thinking about it, it became clear to me that these large diffeomorphisms are not well-understood (at least not by me).
P: 2,863
 Quote by tom.stoer I agree with you. I can't remember why we started to discuss these issues in GR, but thinking about it, it became clear to me that these large diffeomorphisms are not well-understood (at least not by me).
It seems clear to me that large difeomorphisms in the GR context are not well understood, and yet all the mainstream experts have decided that spacetimes are invariant under these large diffeomorphisms without offering any real reason.
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.
 Sci Advisor P: 5,295 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 gauge-fixed (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 4-dim. are rather complicated - see Donaldson's results etc. )
P: 2,863
 Quote by tom.stoer 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 4-dim. are rather complicated - see Donaldson's results etc. )
Here is a slide presentation that treats the Kruskal coordinates in terms of gauge and local versus global gauge freedom that can be related to the small vs. large diffeomorphisms, see for instance slides 27-28.