"Large" diffeomorphisms in general relativity

tom.stoer

I think 1) is irrelevant as we know that the embedding of T² in R³ does itself change the metric; T² admits a flat metric, but the embedding in R³ does not. So 1) is an artefact of the embedding.

TrickyDicky

Mainstream says none of the above, I'd say a)
Normally you would have b) if you have a)

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)

PAllen

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.

tom.stoer

Why? It is a diffeomorphism and does not create a singularity

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.

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.

tom.stoer

Du you agree that it changes the winding number of a closed curve?

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, 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.

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.

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?

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?

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.

PAllen

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.

PAllen

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.

tom.stoer

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

Haelfix

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)

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).

TrickyDicky

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.